A study on quantum gases: bosons in optical lattices and the one-dimensional interacting Bose gas
UUNIVERSIDADE DE SÃO PAULOINSTITUTO DE FÍSICA DE SÃO CARLOSFelipe Taha Sant’Ana
A study on quantum gases: bosons in optical lattices andthe one-dimensional interacting Bose gas
São Carlos2020 a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l elipe Taha Sant’Ana A study on quantum gases: bosons in optical lattices andthe one-dimensional interacting Bose gas
Thesis presented to the Graduate Programin Physics at the Instituto de Física de SãoCarlos, Universidade de São Paulo, to obtainthe degree of Doctor in Science.Concentration area: Theoretical and Experi-mental PhysicsAdvisor: Prof. Dr. Francisco Ednilson Alvesdos Santos
Corrected version(Original version available on the Program Unit)São Carlos2020
AUTHORIZE THE REPRODUCTION AND DISSEMINATION OF TOTAL ORPARTIAL COPIES OF THIS DOCUMENT, BY CONVENTIONAL OR ELECTRONICMEDIA FOR STUDY OR RESEARCH PURPOSE, SINCE IT IS REFERENCED.
Sant'Ana, Felipe Taha A study on quantum gases: bosons in optical latticesand the one-dimensional interacting Bose gas / FelipeTaha Sant'Ana; advisor Francisco Ednilson Alves dosSantos - corrected version -- São Carlos 2020. 151 p. Thesis (Doctorate - Graduate Program in Theoreticaland Experimental Physics) -- Instituto de Física de SãoCarlos, Universidade de São Paulo - Brasil , 2020. 1. Optical lattices. 2. Bose-Hubbard model. 3. Quantumphase transition. 4. Interacting Bose gas. 5. Tonks-Girardeau gas. I. dos Santos, Francisco Ednilson Alves,advisor. II. Title. o my parents, Leila and Mário, for believing in me.
CKNOWLEDGEMENTS
There are many people whom I should thank for different reasons. Firstly, I mustthank my family for the basis and for being present when I most needed: I am very gratefulto my mother, Leila, for being so caring, to my father, Mário, for encouraging me in mystudies during my whole life, and to my brother, Vitor, for sharing his enthusiasm and therock ’n roll.A very special thanks goes to my wife, Fernanda, for being patient, comprehensive,and, above all, an awesome companion.I am very grateful to my friend Michael Melo for sharing his passion for physicsand, also, for great times with beer, chess, philosophy, and countless epic talks.Academically speaking, I am very grateful to my supervisor, Francisco Ednilson,for being present when I needed advices, for teaching and explaining physics when I didnot understand, and for guiding me during my doctorate. Thanks a lot, Ednilson, for that!I am also very thankful to Axel Pelster, for sharing discussions, for supporting me whenI wanted to go to Germany for a full doctorate, and, especially, for introducing me doEdnilson. In the France side of the story, I am very grateful to Mathias Albert and PatriziaVignolo for receiving me, for helping me with all I needed to adapt myself in the newenvironment, and for supervising my studies during my one-year work at InPhyNi. Thanksa lot for that! Also, I must thank Frédéric Hébert for useful discussions and, especially,for providing my first lunch at InPhyNi. Thanks, Fred! Finally, I must thank RomainBachelard for bringing me the opportunity to go to France and make it possible.During my doctoral studies I have made many friends. In the early days at IFSC, Icannot thank enough my dearest friends Marios and Sasha for sharing very good times andbeers. Also, I must thank Julián and Tiago for the "bandejão" sessions and the barbecues.Then, in France, I really had a great time at InPhyNi. I am very grateful to every singleperson I met there and who made me feel comfortable in the new environment. Firstly,thanks a lot to Ana and Julián for the hikings, they were incredible! I must thank thefootball team members Antonin, Julián, Mathias, and Vittorio, for making the footballsessions absolutely amazing. Thanks, guys! Also, I can not forget to thank Marius for theunstoppable humorous talks, those were really fun.This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoalde Nível Superior – Brasil (CAPES) – Finance Code 001.
OREWORD
I cannot remember a period of time in my life where I have learned about manydifferent topics as much as I learned writing this thesis. Every single page came to life aftermany hours of research, reading, and learning, so that I could disseminate the amount ofknowledge I acquired during this process. I wish I had more time to keep writing because Iwould appreciate very much in going deeper in some subjects that I found very interestingand important as a link to understand another one. In some parts of this work I spentmany lines in details and algebraic manipulations that I considered to be relevant for theunderstanding of the reader, while in other parts I simply omitted the details because Iconsidered them to be straightforward and not crucial to the understanding of the specificidea. I apologize if I failed in doing so.This thesis is composed of two parts. The first part is concerned to the study ofthe quantum phase transition between the Mott insulator and the superfluid regardingbosonic atoms loaded in optical lattices. This part of my doctoral studies was conducted atthe São Carlos Institute of Physics in Brazil under the supervision of Francisco Ednilsonand with collaboration of Axel Pelster from the Technical University of Kaiserslauternin Germany. Then, I traveled to France, where I spent one year performing research onthe one-dimensional interacting Bose gas at the Institut de Physique de Nice under thesupervision of Mathias Albert and Patrizia Vignolo and with the collaboration of FrédéricHébert. The research conducted in France corresponds to the second part of this thesisand is described in Chap. 6.
Education is the kindling of a flame,not the filling of a vessel.”Socrates
BSTRACT
SANT’ANA, F. T.
A study on quantum gases: bosons in optical lattices andthe one-dimensional interacting Bose gas . 2020. 151p. Thesis (Doctor inScience) - Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 2020.Bosonic atoms confined in optical lattices are described by the Bose-Hubbard modeland can exist in two different phases, Mott insulator or superfluid, depending on thestrength of the system parameters, such as the on-site interaction between particles andthe hopping parameter. Differently from classical phase transitions, the Mott-insulator-superfluid transition can happen even at zero temperature, driven by quantum fluctuations,thus characterizing a quantum phase transition. For the homogeneous system, we canapproximate the particle excitations as a mean-field over time, thus providing a localHamiltonian, which makes possible the evaluation of physical properties from a singlelattice site. From the Landau theory of second-order phase transitions, it is possible toexpand the thermodynamic potential in a power series in terms of the order parameter,giving rise to the Mott-insulator-superfluid phase diagram. As the condensate density goesfrom a finite value to a vanishing one when the system transits from superfluid to a Mottinsulator, it can be considered as the order parameter of the system. In the vicinity ofthe phase boundary, it is possible to consider the hopping term as a perturbation, sinceit contains the order parameter. Thence, one can apply perturbation theory in order tocalculate important physical quantities, such as the condensate density. However, due todegeneracies that happen to exist between every two adjacent Mott lobes, nondegenerateperturbation theory fails to give meaningful results for the condensate density: it predictsa phase transition due to the vanishing of the order parameter in a point of the phasediagram where no transition occurs. Motivated by such a misleading calculation, wedevelop two different degenerate perturbative methods to solve the degeneracy-relatedproblems. Firstly, we develop a degenerate perturbative method based on Brillouin-Wignerperturbation theory to tackle the zero-temperature case. Afterwards, we develop anotherdegenerate perturbative method based on a projection operator formalism to deal withthe finite-temperature regime. Both methods have the common feature of separating theHilbert subspace where the degeneracies are contained in from the complementary one.Therefore, such a separation of the Hilbert subspaces fixes the degeneracy-related problemsand provides us a framework to obtain physically consistent results for the condensatedensity near the phase boundary. Moreover, we study the one-dimensional repulsivelyinteracting Bose gas under harmonic confinement, with special attention to the asymptoticbehavior of the momentum distribution, which is a universal k − decay characterized by theTan’s contact. The latter constitutes a direct signature of the short-range correlations insuch an interacting system and provides valuable insights about the role of the interparticlenteractions. From the known solutions of the system composed of two particles, we areable to acquire important knowledge about the manifestation of the interaction, e.g. , thecusp condition that implies the vanishing of the many-body wave function whenever twoparticles meet. Then, we investigate the system constituted of N interacting particles inthe strongly interacting limit, also known as Tonks-Girardeau gas . In such a regime, thestrong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas,an effect known as fermionization . Because of the difficulty in analytically solving thesystem for N particles at finite interaction, the Tonks-Girardeau regime provides, throughthe fermionization of the bosons, a favorable scenario to probe the Tan’s contact. Therefore,within such a regime, we are able to provide an analytical formula for the Tan’s contact interms of the single-particle orbitals of the harmonic oscillator. Furthermore, we analyzethe scaling properties of the Tan’s contact in terms of the number of particles N in thehigh-temperature regime as well as in the strongly interacting regime. Finally, we compareour analytical calculations of the Tan’s contact to quantum Monte Carlo simulationsand discuss some fundamental differences between the canonical and the grand-canonicalensembles. Keywords : Optical lattices. Bose-Hubbard model. Quantum phase transition. InteractingBose gas. Tonks-Girardeau gas.
ESUMO
SANT’ANA, F. T.
Um estudo sobre gases quânticos: bósons em redes ópticase o gás interagente e unidimensional de Bose . 2020. 151p. Tese (Doutorado emCiências) - Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 2020.Átomos bosônicos confinados em redes ópticas são descritos pelo modelo de Bose-Hubbarde podem existir em duas diferentes fases, isolante de Mott ou superfluido, dependendo daforça dos parâmetros do sistema, tais como a interação local entre partículas e o parâmetrode salto. Diferentemente das transições de fase clássicas, a transição entre isolante deMott e superfluido pode ocorrer mesmo a temperatura zero, impulsionada por flutuaçõesquânticas, caracterizando uma transição de fase quântica. Para o sistema homogêneo,podemos aproximar as excitações de partículas a um campo médio ao longo do tempo,fornecendo um Hamiltoniano local, o que torna possível a avaliação de propriedades físicasa partir de um único sítio da rede. A partir da teoria de Landau de transições de fase desegunda ordem, é possível expandir o potencial termodinâmico em uma série de potênciasem termos do parâmetro de order, dando origem ao diagrama de fase. Como a densidadede condensado passa de um valor finito para um valor nulo quando o sistema transita desuperfluido para isolante de Mott, este pode ser considerado como sendo o parâmetro deordem do sistema. Nas proximidades da fronteira de fase, é possível considerar o termode salto como uma perturbação, uma vez que este contém o parâmetro de ordem. Daí,pode-se aplicar teoria de perturbação para calcular quantidades físicas importantes, comoa densidade de condensado. No entanto, devido a degenerescências que existem entredois lóbulos de Mott adjacentes, teoria de perturbação não degenerada falha em fornecerresultados significativos para a densidade de condensado: esta prevê uma transição defase devido ao desaparecimento do parâmetro de order em um ponto do diagrama de faseonde nenhuma transição ocorre. Motivados por esse cálculo enganoso, desenvolvemos doismétodos perturbativos degenerados diferentes para resolver os problemas relacionados àsdegenerescências. Em primeiro lugar, desenvolvemos um método perturbativo degeneradobaseado em teoria de perturbação de Brillouin-Wigner para solucionar o sistema a temper-atura zero. Posteriormente, desenvolvemos outro método perturbativo degenerado baseadoem um formalismo de operadores de projeção para lidar com o regime a temperatura finita.Ambos os métodos têm a característica comum de separar o subespaço de Hilbert onde asdegenerescências estão contidas de seu complementar. Portanto, essa separação dos sube-spaços de Hilbert corrige os problemas relacionados às degenerescências e nos fornece umaestrutura para obter resultados fisicamente consistentes para a densidade de condensadopróximo à fronteira da fase. Além disso, estudamos o gás de Bose unidimensional cominteração repulsiva entre partículas sob confinamento harmônico, com especial atenção aocomportamento assintótico da distribuição de momento, que é um decaimento universale k − caracterizado pelo contato de Tan. Este último constitui uma assinatura diretadas correlações de curto alcance em tal sistema interagente e fornece informações valiosassobre o papel das interações entre partículas. A partir das conhecidas soluções do sistemacomposto de duas partículas, somos capazes de adquirir conhecimentos importantes sobrea manifestação da interação, e.g. , a condição de cúspide que implica no desaparecimentoda função de onda de muitos corpos sempre que duas partículas se encontram. Em seguida,investigamos o sistema constituído de N partículas fortemente interagentes, tambémconhecido como gás de Tonks-Girardeau . Nesse regime, a forte interação entre partículasfaz com que os bósons se comportem de maneira semelhante ao gás ideal de Fermi, umefeito conhecido como fermionização . Devido à dificuldade em resolver analiticamenteo sistema com N partículas com interação finita, o regime de Tonks-Girardeau fornece,através da fermionização dos bósons, um cenário favorável para o estudo do contato deTan. Portanto, dentro de tal regime, somos capazes de fornecer uma fórmula analítica parao contato do Tan em termos dos orbitais de uma única partícula do oscilador harmônico.Além disso, analisamos as propriedades de escalonamento do contato do Tan em termos donúmero de partículas N nos regimes de altas temperaturas e fortes interações. Finalmente,comparamos nossos cálculos analíticos do contato de Tan a simulações de Monte Carloquântico e discutimos algumas diferenças fundamentais entre os conjuntos canônico emacrocanônico. Palavras-chave : Redes ópticas. Modelo de Bose-Hubbard. Transição quântica de fase.Gás de Bose interagente. Gás de Tonks-Girardeau.
IST OF FIGURES
Figure 1 – Observation of the two first Bose-Einstein condensates by absorptionimaging from the rubidium Rb experiment
54, 58 in (a) and from thesodium Na experiment
19, 55 in (b). . . . . . . . . . . . . . . . . . . . . 31Figure 2 – Schematic drawing of optical lattices in the two different phases: thesuperfluid in (a) and the Mott insulator in (b). The superfluid phase ischaracterized by a high delocalization of the atoms, implying well knownvalues of the momentum, generating well defined peaks in the momentumspace. On the other hand, the Mott insulator phase is characterized bya high localization of the atoms, thence their momentum space imageconsists in a blur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 3 – Time-of-flight absorption images for the potential depths V of: (a) 0,(b) 3 E R , (c) 7 E R , (d) 10 E R , (e) 13 E R , (f) 14 E R , (g) 16 E R , and (h) 20 E R . 33Figure 4 – Hopping energy from the Mathieu solution (2.39) (blue line) and fromthe harmonic approximation (2.44) (yellow line). . . . . . . . . . . . . . 46Figure 5 – Landau expansion of the thermodynamic potential from (3.2). The blue,yellow, and green curves correspond to, respectively, a > a = 0,and a <
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 6 – Phase diagrams for the inverse temperatures β = 5 /U (black), β = 10 /U (red), β = 30 /U (green), and β → ∞ (blue). . . . . . . . . . . . . . . . 55Figure 7 – Condensate density (a) from (3.37) and particle density (b) from (3.38)via NDPT as functions of µ/U for J z/U = 0 . β = 5 /U (dotted-dashed black), β = 10 /U (dashed red), β = 30 /U (dotted green),and β → ∞ (continuous blue). . . . . . . . . . . . . . . . . . . . . . . . 56Figure 8 – Unperturbed ground state energies E (0) n = E n − J z | Ψ | . Different linescorrespond to different values for n from smaller to larger slope: n = 1(red), n = 2 (blue), n = 3 (green), and n = 4 (purple). Vertical dashedblack lines correspond to the points of degeneracy. Solid colored linesrepresent realized lowest energies, while dashed colored lines indicatethe continuation of the energy lines. . . . . . . . . . . . . . . . . . . . . 57igure 9 – Condensate densities from nondegenerate perturbation theory (dashedlines) in comparison to the condensate densities from degenerate pertur-bation theory according to (3.40) (dotted lines) with µ = U n + ε and n = 1 for the left part (negative ε/U ) and n = 2 for the right part (posi-tive ε/U ). The hopping strengths are, from the spacing inside to outside, J z/U = 0 .
02 (red),
J z/U = 0 .
08 (blue), and
J z/U = 0 .
101 (green).The dashed plots vanish at the mean-field phase boundary, yieldinga nonphysical behavior at the degeneracy. Also, they have increasingmaxima for increasing
J z/U , and for
J z/U = 0 .
101 and ε/U = 0 . ∗ Ψ = 0 . J z/U and close to the phase boundary, the plots coincide. . . . . . . . 58Figure 10 – Zero-temperature phase boundaries for bosons in optical lattices fromdifferent treatments. The nondegenerate theory yields the dashedorange plot, while the degenerate one results in the dotted magentaplot. Inside the lobes the system is in the Mott insulator phase, whileoutside the lobes the superfluid phase takes place. The number ofparticles per site, n , increases from left to right by one per lobe. Thethree horizontal continuous lines correspond to, from bottom to top, J z/U = 0 .
02 (red),
J z/U = 0 .
08 (blue), and
J z/U = 0 .
101 (green).They all start at the line
J z/U = − µ/U , which indicates n = 0, andend at µ/U = 2 .
15. The inset shows the zoomed region between thefirst two Mott lobes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 11 – Condensate density from the one-state approach for n = 1 (negative ε/U , purple squares) and n = 2 (positive ε/U , red circles), with thehopping strength of J z/U = 0 .
08. . . . . . . . . . . . . . . . . . . . . . 68Figure 12 – Perturbed ground state energies E n /U up to O ( λ ) between the Mottlobes, inside the superfluid regions, for three different hopping values: J z/U = 0 .
02 (red circles),
J z/U = 0 .
08 (blue crosses), and
J z/U =5 − √ ≈ .
101 (green rings). At
J z/U = 5 − √
6, the second lobeachieves its tip. (a) Superfluid energies between the first two Mottlobes. For a better visualization, the linear equation 0 .
15 + 1 . µ/U ,which scales the outmost points of the green plot to zero, is addedto the energy. (b) Zoomed region centered around the degeneracy byintroducing µ = U + ε . For a better visualization, the linear equation1 .
15 + 1 . ε/U , which scales the outmost points of the green plot tozero, is added to the energy. . . . . . . . . . . . . . . . . . . . . . . . . 70igure 13 – Particle densities − ∂E n /∂µ as functions of the chemical potential µ/U according to the corresponding hopping values. Horizontal lines corre-spond to Mott-insulating regions, while ascending curves correspondto superfluid regions. The higher the hopping, the rounder the curvesbecome. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 14 – Condensate densities as functions of ε/U = µ/U − O ( λ ) (red circles), O ( λ )(blue squares), O ( λ ) (green rings), and O ( λ ) (purple triangles). Forsmall values of J z/U , and thus close to the degeneracy, the third- (greenrings) and fourth-order (purple triangles) data coincide. . . . . . . . . . 72Figure 15 – Condensate densities Ψ ∗ Ψ as functions of ε/U = µ/U − O ( λ )between the first and the second Mott lobes for different hopping values:from J z/U = 0 .
01 (innermost points) until
J z/U = 0 .
20 (outermostpoints) with a step size of 0.01. . . . . . . . . . . . . . . . . . . . . . . 72Figure 16 – Ground-state energy E out of the one-state approach for µ = 0 . U .From top to bottom, the curves represent the numerical diagonalizationcalculation (blue line) as well as the perturbative analytical calculationsup to O ( λ ) (yellow line), O ( λ ) (red line), and O ( λ ) (green line).Here, N s represents the number of lattice sites and F stands for thezero-temperature free energy. . . . . . . . . . . . . . . . . . . . . . . . 73Figure 17 – Graphical approach for the matrix elements (4.31) from the effectiveHamiltonian (4.21) for the Bose-Hubbard mean-field Hamiltonian (3.7)up to fifth order in the hopping term for the two-state approach. . . . . 74Figure 18 – Equations of state N = N (˜ µ ) for the following parameters: m = 87 u , a = 400nm, and ω = 48 π Hz. From left to right, the hopping values are:
J z/U = 0 .
02 (dashed red line),
J z/U = 0 .
101 (dotted green line), and
J z/U = 0 .
08 (continuous blue line). . . . . . . . . . . . . . . . . . . . . 78Figure 19 – Condensate densities as functions of µ/U evaluated from FTDPT via(5.28) for four different temperatures: (a) β = 5 /U , (b) β = 10 /U ,(c) β = 30 /U , and (d) T = 0. Different data styles correspond todifferent hoppings: J z/U = 0 . J z/U = 0 .
15 (orangesquares),
J z/U = 0 . J z/U = 0 .
05 (red triangles),and
J z/U = 0 .
01 (purple inverted triangles). . . . . . . . . . . . . . . . 86igure 20 – Comparison between the condensate densities calculated via FTDPT(dots) and NDPT (lines) for the temperatures β = 30 /U (left panel)and T = 0 (right panel), and for the hoppings J z/U = 0 . J z/U = 0 .
15 (orange squares and dashedorange lines), and
J z/U = 0 . n = 0 and 1, while (c) and(d) correspond to the region between the first and second lobes. . . . . 87Figure 21 – Equation of state for the hopping strengths (a) J z/U = 0 .
05 and (b)
J z/U = 0 . T = 0 (continuous blue), β = 30 /U (dotted green), β = 10 /U (dashed red), and β = 5 /U (dotted-dashedblack). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 22 – The fundamental and the first three excited states of the harmonicoscillator wave function (6.11). . . . . . . . . . . . . . . . . . . . . . . 93Figure 23 – Relative motion wave functions for different interaction strength ˜ g .Different colors correspond to different states, from the fundamentalstate to the third excited one. In growing order of excitation theycorrespond to the blue, yellow, green, and red curves. . . . . . . . . . . 96Figure 24 – f ( ν ) + 1 / ˜ g as a function of ν for different adimensional interactionstrengths ˜ g . The zoomed inset helps us recognize the vanishing of thefunction between ν = 0 (weakly interacting limit) and ν = 1 (stronglyinteracting limit). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 25 – Tan’s contact from (6.52) as a function of the adimensional temperature k B T / (cid:126) ω for different values of the adimensional interaction parameter ˜ g .100Figure 26 – Zero-temperature Tan’s contact from (6.62) as a function of the adi-mensional interaction parameter ˜ g . . . . . . . . . . . . . . . . . . . . . 102Figure 27 – Momentum distributions for different number of particles as well asdifferent temperatures. The insets show the tails of the curves. . . . . . 108Figure 28 – Tonks-Girardeau contact in the canonical ensemble (empty symbols)from Eq. (6.95) and in the grand-canonical ensemble (filled symbols)from Ref. 209 for N = 2 (violet squares), N = 3 (green circles), N = 4(light-blue triangles), and N = 5 (orange inverted triangles). Here, τ ≡ T /T F is the adimensional temperature and a ho ≡ q (cid:126) /mω is theharmonic oscillator length. . . . . . . . . . . . . . . . . . . . . . . . . . 109igure 29 – Tan’s contacts in the Tonks-Girardeau limit from Eq. (6.95) in adi-mensional units as functions of the reduced temperature τ scaled bythe generalized conjecture s ( N ) ≡ N / − N / − /τ )] for the re-spective number of particles: N = 2 (violet squares), N = 3 (greencircles), N = 4 (blue triangles), and N = 5 (orange inverted trian-gles). The black cross corresponds to the zero-temperature Tonks-Girardeau two-boson contact from Eq. (6.63) rescaled by s ( N ) for τ = 0:(2 / − / ) − C (˜ g → ∞ , T →
0) = (2 / − / ) − (2 /π ) / = 0 . √ τ /π / , while the black continuousline is simply the contact rescaled by the generalized scaling factor, i.e. ,the implicit proportionality factor in Eq. (6.115). . . . . . . . . . . . . 112Figure 30 – Tan’s contacts from QMC simulations as functions of the adimensionaltemperature τ for z = ˜ g = 1. The panels (a), (b), and (c) correspondto the rescaling of the contact regarding the low-temperature factor N / − N / , the high-temperature factor N / − N / , and the all-range-temperature factor s ( N ) ≡ N / − N / − /τ )] , respectively.In panel (d), the QMC data is rescaled by the TG-limit contact from Eq.(6.95). The symbol styles correspond to: N = 2 (violet squares), N = 3(green circles), N = 4 (blue triangles), and N = 5 (orange invertedtriangles). The continuous yellow line corresponds to the two-bosoncontact obtained by Eq. (6.52). The QMC error bars are smaller thanthe symbol sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 31 – Tan’s contacts from QMC simulations as functions of the adimensionaltemperature τ for z = ˜ g = 2 .
5. The panels (a), (b), and (c) correspondto the rescaling of the contact regarding the low-temperature factor N / − N / , the high-temperature factor N / − N / , and the all-range-temperature factor s ( N ) ≡ N / − N / − /τ )] , respectively.In panel (d), the QMC data is rescaled by the TG-limit contact from Eq.(6.95). The symbol styles correspond to: N = 2 (violet squares), N = 3(green circles), N = 4 (blue triangles), and N = 5 (orange invertedtriangles). The continuous yellow line corresponds to the two-bosoncontact obtained by Eq. (6.52). The QMC error bars are smaller thanthe symbol sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 32 – Canonical (empty symbols) contact in the TG limit from (6.95) rescaledby N / − N / and grand-canonical (filled symbols) contact from Ref.209 rescaled by N / as functions of τ for the respective number ofparticles: N = 2 (violet squares), N = 3 (green circles), N = 4 (bluetriangles), and N = 5 (orange inverted triangles). The black continuouscurve corresponds to √ τ /π / . . . . . . . . . . . . . . . . . . . . . . . . 116igure 33 – Tan’s contacts evaluated from canonical QMC simulations (empty sym-bols) and from grand-canonical QMC simulations (filled symbols) asfunctions of τ in the weak-intermediate regime z = ˜ g = 0 . N = 2 (violet squares), N = 3 (green cir-cles), N = 4 (blue triangles). QMC error bars in the canonical-ensemblecalculation are smaller than the symbols size. . . . . . . . . . . . . . . 117Figure 34 – QMC grand-canonical contact rescaled by the TG grand-canonicalcontact from Ref. 209 as a function of τ for the respective number ofparticles: N = 2 (violet squares), N = 3 (green circles), and N = 4(blue triangles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 IST OF ABBREVIATIONS AND ACRONYMS
BEC Bose-Einstein Condensate/CondensationBH Bose-HubbardBWPT Brillouin-Wigner Perturbation TheoryFTDPT Finite-Temperature Degenerate Perturbation TheoryMI Mott InsulatorNDPT Non-Degenerate Perturbation TheoryOP Order ParameterQMC Quantum Monte CarloRSPT Rayleigh-Schrödinger Perturbation TheorySF SuperfluidTG Tonks-Girardeau
IST OF PHYSICAL AND MATHEMATICAL CONSTANTS (cid:126) = 1 . × − J . s Reduced Planck’s constant k B = 1 . × − J . K − Boltzmann’s constant e = 1 . × − C Electronic charge ε = 8 . × − F . m − Vacuum permittivity c = 299792458 m . s − Speed of light π = 3 . . . . Archimedes’ constante = 2 . . . . Euler’s number
ONTENTS1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.1 Bosonic atoms loaded in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
362 BOSONS IN OPTICAL LATTICES . . . . . . . . . . . . . . . . . . 392.1 Atom-laser interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
765 THE SYSTEM AT FINITE TEMPERATURE . . . . . . . . . . . . . 795.1 The projection operators method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
886 ONE-DIMENSIONAL INTERACTING BOSE GAS . . . . . . . . . . 916.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX A – ATOMIC COLLISIONS IN COLD GASES . . . . 139A.1 The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ANNEX A – PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . 151 Before the quantum revolution in the early days of the twentieth century, whichwas motivated during the 19th century by the studies of Thomas Young in the famousdouble-slit experiment, the black-body radiation problem stated by Gustav Kirchhoff, the Ludwig Boltzmann’s work on the statistics of possible energies of atoms and moleculesin a gas, and the works of Max Planck on the quantum hypothesis of energy, theworld was described by the fundamental laws of Isaac Newton on gravitation and classicalmechanics, the unified electromagnetism theory of James Clerk Maxwell, and theclassical thermodynamics developed in the 17th century. Due to the absence of a completequantum theory in those days, there was a limited access to some physical properties ofmatter, such as the states of matter known at that time: liquid, solid and gaseous. However,such a knowledge about the forms that matter can acquire was about to drastically changeas a result of the advent of quantum mechanics.The concept of Bose-Einstein condensation (BEC) originated in 1925 when A.Einstein, on the basis of the work of S. N. Bose, which described the quantum statisticaltheory of light, wrote a paper about the quantum theory of the ideal monoatomic gas, predicting the occurrence of a phase transition at low enough temperatures. In 1938, F.London argued that the phenomenon of Bose-Einstein condensation was intimately relatedto the occurrence of superfluidity in He. Later on, he also suggested that BEC andsuperconductivity phenomena were closely related. An important aspect towards the understanding of the Bose-Einstein condensationis the thermal wavelength,
17, 18 that relates, by means of the de Broglie relation, the wavecharacter of a particle with mass m to its temperature T through the formula λ = s π (cid:126) mk B T . (1.1)As the temperature of the particles decreases, its associated de Broglie wavelength increases.As a result, the many wave packages related to different particles in the system begin tooverlap with one another, until a critical point is reached and all the atoms behave as asingle macroscopic wave, which is the characterization of a Bose-Einstein condensate. Thetransition to the BEC phase occurs when the thermal wavelength of the particles in theatomic gas becomes comparable to the interatomic spacing between them λ ∼ n − / , where n is the density of atoms. Consequently, the critical temperature in order to achievea BEC is of the order of T c ∼ n / π (cid:126) mk B . (1.2)To create a BEC one must not just reach the critical temperature, but also achieve Chapter 1 Introduction low densities, otherwise the atomic gas would simply condensate into a more conventionalliquid or solid. The density of a dilute gas that provides an adequate environment forthe emergence of a BEC must be of the order of a hundred-thousandth of the density ofnormal air, which is around 10 cm − . Typically, the particle density at the center of aBEC is about 10 − cm − . Thence, the temperature one must accomplish in orderto realize a BEC is around 10 − K.Obviously, it took great amounts of scientific work throughout the twentieth centuryfor the purpose of achieving such low temperatures. A landmark in the history of atomiccooling techniques are the works performed during the 1970s and 1980s on laser cooling.A seminal work on laser cooling techniques was performed by T. W. Hänsch and A. L.Schawlow in 1975, where they were able to achieve temperatures around 0 .
24 K. Lateron, in 1978, D. J. Wineland et al. and A. Ashkin succeeded in obtaining temperaturesin the milikelvin, 10 − K, and microkelvin scales, 10 − K, respectively. After that, manyworks followed towards the advance of laser cooling techniques, e.g. , S. Chu et al. reported the cooling of neutral sodium atoms in three dimensions via radiation pressure ofconterpropagating laser beams, attaining temperatures around 240 µ K, while A. Aspect et al. described a scheme that allowed the cooling of He to a temperature of 2 µ K. Allthese developments in laser cooling culminated in the 1997 Nobel prize awarded to S.Chu, C. N. Cohen-Tannoudji, and W. D. Phillips. The field of laser cooling includes different techniques:
Sisyphus cooling ; re-solved sideband cooling ;
22, 31–36
Raman sideband cooling ;
37, 38 velocity-selective coherentpopulation trapping ; gray molasses ; cavity mediated cooling ; Zeeman slower ;
46, 47 electromagnetically induced transparency ; Doppler cooling .
21, 22, 28, 53
The traditionaland still most used is the latter one. The principle behind the
Doppler cooling technique isthe following: when an atom interacts with a photon —the interaction process is consistedof the absorption and thereafter the emission of a photon —the velocity of the atom ischanged due to momentum conservation. Depending on the direction of the photon relativeto the direction of the atom velocity, it can either increase or decrease its momentum. So,in order to eliminate the undesirable result of increasing the atomic temperature, one hasto tune the laser frequency just below the electronic transition frequency of the atom. Thisprocess results in an overall reduction of the gas temperature, since the atoms will onlyinteract with counterpropagating photons as a result of the
Doppler effect .Such advances in laser cooling made possible the accomplishment of the
Bose-Einstein condensate in 1995 by the group of Eric Cornell and Carl Wieman at the Universityof Colorado Boulder in a gas of rubidium Rb atoms and by the group of WolfgangKetterle at MIT in a gas of sodium Na atoms, Fig. 1. In order to achieve cold enoughtemperatures to realize the BEC, it was necessary to combine different stages: in theprecooling stage, laser cooling techniques were used to make the atoms cold enough to be .1 Bosonic atoms loaded in optical lattices confined in a wall-free magnetic trap; then forced evaporative cooling was applied as thesecond stage,
56, 57 which consists in reducing the trap depth so that the most energeticatoms can escape, thus decreasing the temperature of the whole gas. For their achievementson BEC, Eric Cornell, Carl Wieman and Wolfgang Ketterle were awarded the 2001 Nobelprize in physics.
19, 58
Differently from bosons, fermionic atoms are subject to the
Pauli exclusion principle , which forbids two or more fermions of occupying the same quantum-mechanical state.However, in view of the formation of the so-called Cooper pairs , the realization of BEC ininteracting Fermi gases was achieved in 2003 by Greiner et al. with potassium K atoms,while it was also achieved thereafter with lithium Li atoms.
Such a realization infermionic gases represented the accomplishment of a long-standing goal of ultracold atomsresearch, the celebrated
BCS-BEC crossover , referring to the Bardeen–Cooper–Schrieffer(BCS) theory of superconductivity.
66, 67 (a) The respective temperatures are, fromleft to right, 0 . µ K, 0 . µ K, and0 . µ K. (b) From left to right, the temperatures are
T > µ K, T ∼ µ K, and
T < µ K. Figure 1 – Observation of the two first Bose-Einstein condensates by absorption imagingfrom the rubidium Rb experiment
54, 58 in (a) and from the sodium Naexperiment
19, 55 in (b).Source: (a) CORNELL et al. ; (b) KETTERLE. Optical lattices are laser arrangements which enable a spatially periodic trappingof atoms due to the interaction between the external electric field and the induced dipolemoment of the atoms.
20, 68–71
Such artificial laser-generated periodic potentials create apropitious environment to probe ultracold atoms and provide clean access to physicalquantities, in contrast to natural crystal lattices, where disorder of many kinds, e.g. , latticevibrations, or the so-called phonons , contribute to undesirable features in the effectiveHamiltonian that one needs to take into account to describe the dynamics of the systemas well as to calculate its physical properties. In principle, it is always possible to takeonly one kind of disorder into account and apply perturbation theory to evaluate physical Chapter 1 Introduction quantities, but this is simply the reduced version of the story: there happens to existinnumerable kinds of noise in a crystal lattice that it is theoretically impossible to accountfor all their contributions simultaneously. Even if one uses numerical methods, the amountof computational capacity is usually beyond what the computers nowadays can achieve.So, optical lattices provide a suitable setting for the realization of simplified models ofcondensed-matter systems and the study of many-body systems. Also, optical lattices allowthe implementation of Richard Feynman’s pioneering idea of “quantum simulation”:
72, 73 using one quantum system to investigate another one. In addition, the controllability ofoptical lattices is much higher than most condensed-matter systems, thus providing aneasy control of important parameters, such as the strength of the interatomic interactions.Figure 2 – Schematic drawing of optical lattices in the two different phases: the superfluidin (a) and the Mott insulator in (b). The superfluid phase is characterized by ahigh delocalization of the atoms, implying well known values of the momentum,generating well defined peaks in the momentum space. On the other hand,the Mott insulator phase is characterized by a high localization of the atoms,thence their momentum space image consists in a blur.Source: BLOCH. A gas composed of bosonic atoms in an optical lattice can be described by the Bose-Hubbard model,
70, 75 which has three main parameters: the on-site interaction parameter,the hopping parameter, and the chemical potential. Depending on the magnitude ofthe parameters, the system can realize two different phases, the Mott insulator or thesuperfluid phase, as illustrated in Fig. 2. If the on-site interaction parameter is muchlarger than the hopping parameter, the system is in the Mott insulator (MI) phase. Thisphase is characterized by a high localization of the atoms, implying an integer number of .1 Bosonic atoms loaded in optical lattices particles n per lattice site and zero compressibility, ∂n/∂µ = 0. Also, the Mott insulatorpresents an energy gap for both particle and hole excitations, due to the restricted mobilitybetween neighboring sites. By decreasing the amplitude of the periodic potential, so thatthe hopping parameter becomes much larger than the atom-atom interaction parameter,the system undergoes a phase transition to a superfluid (SF) phase, where a fractionof the atoms become delocalized. Such a phase is characterized by zero viscosity, i.e. ,superfluidity signifies the ability of carrying currents without dissipation, analogously tosuperconductivity. Such differences in the localization of the bosons make it possible tomeasure the phase the system is currently in through time-of-flight experiments,
78, 87 whichare depicted in Fig. 3. The MI-SF transition can happen even at zero-temperature, drivenby quantum mechanical fluctuations, thus characterizing a quantum phase transition. Figure 3 – Time-of-flight absorption images for the potential depths V of: (a) 0, (b) 3 E R ,(c) 7 E R , (d) 10 E R , (e) 13 E R , (f) 14 E R , (g) 16 E R , and (h) 20 E R .Source: GREINER et al. Following this simplification, Rayleigh-Schrödingerperturbation theory (RSPT) is typically used for obtaining the mean-field phase diagramat zero temperature. However, there are problems that arise from RSPT, since it doesnot properly deal with degeneracies that occur between two consecutive Mott lobes. Oneof such RSPT problems concerns the calculation of the condensate density, which falselyvanishes between two consecutive Mott lobes. Also, other methods have been suggested in order to improve the mean-fieldquantum phase diagram for bosons in optical lattices, e.g. , a variational method that uses a Chapter 1 Introduction field-theoretic concept of the effective potential. In addition, the MI-SF phase transition atarbitrary temperatures was investigated using an effective-action approach. Furthermore,a similiar method was derived for the Bose-Hubbard model within the Schwinger-Keldyshformalism in order to handle time-dependent problems at finite temperature.
92, 93
Likewise,Ref. 89 implemented a nearly degenerate perturbation theory for the zero-temperature case,which led to better results for the order parameter (OP) when compared to those from theRSPT calculations. The authors of Ref. 94 applied the Floquet theory in order to analyzethe effects of a periodic modulation of the s -wave scattering length upon the quantumphase diagram of bosons in 2D and 3D optical lattices. It turns out that nondegeneratefinite-temperature perturbation theory
91, 95 also presents degeneracy problems similar toRSPT. Indeed, RSPT is equivalent to the usual finite-temperature perturbation theory inthe zero-temperature limit. Therefore, degeneracy-related problems are also expected toappear at low enough temperatures. Moreover, beyond our considered bosonic gas in anoptical lattice, the degeneracy-related problems also emerge in other systems.
The study of low dimensional systems is motivated by the fact that many three-dimensional theories completely fail in lower dimensions, e.g. , the Landau-Fermi liquidtheory describing interacting fermions in low temperature systems (such as metals andthe liquid helium-3, He) simply breaks down in one dimension. The explanation for sucha failure and an adequate treatment of interacting fermions in 1D was firstly given by S.Tomonaga in 1950, and thereafter reformulated in 1963 by J. M. Luttinger, with thetheory receiving their names as
Tomonaga-Luttinger liquid theory . The works of Tomonagaand Luttinger were complemented by D. C. Mattis and E. H. Lieb in 1964, when theynoted a paradox regarding the density operator commutators and used their observationsto solve the model and obtain the exact solution of the one-dimensional many-fermionsystem.Contrarily to higher dimensions, the role of interactions are particularly importantin 1D systems. This aspect can be easily elucidated by visualizing that, in higher dimensions,the particles can ipso facto avoid each other, which is completely different in 1D, where thedimensional constraint assembles the inevitability of interparticle rendezvous. This impliesthat whenever there is a single-particle excitation, a collective one will automaticallyemerge in the one-dimensional system.Another important consequence of the reduced dimensionality is the absence ofBEC in a uniform infinite system at nonzero temperature,
T >
0, in low dimensions d ≤ Even at zero temperature, the one-dimensional system features the insufficientconditions for the realization of BEC.
Such an absence of BEC can be explained bythe Hohenberg-Mermin-Wagner theorem, that states the absence of spontaneous .2 One-dimensional interacting Bose gas symmetry breaking in low dimensions d ≤ I This theorem has, of course, profoundimplications in the occurrence of BEC in low dimensions because the existence of BEC isassociate with the spontaneous breaking of the U (1) gauge symmetry. However, realisticallyspeaking, there is no infinite system in nature, thus it is possible to realize BEC in one-and two-dimensional finite systems. As we have already discussed, the interparticle interactions play a special roletowards the complete description of the one-dimensional system. In this context, thepioneer work on the exact analysis of the 1D Bose gas interacting via a delta-like potentialat zero temperature was done by E. H. Lieb and W. Liniger and its extension tofinite temperature by C. N. Yang and C. P. Yang.
However, in real experiments theatoms are harmonically trapped, thence one has to take its effect in the Hamiltoniansince V ( x ) = 0 → V ( x ) ∝ x . This change breaks down the integrability of the system.Nevertheless, by sufficiently increasing the interactions, it is possible to achieve the so-called Tonks-Girardeau gas , where the particles behave as impenetrable hard-core spheres.In this regime, as first stated by M. Girardeau in 1960, an effect known as fermionization occurs, thus enabling one to reduce the original problem into a much simpler one by meansof the Bose-Fermi mapping .In order to experimentally realize atomic gases in one dimension, one needs totune the trapping frequencies in such a way that ω y = ω z = ω ⊥ (cid:29) ω x , also known as tight confinement aspect. In such a regime, the characteristic energy scale is much greaterthan the thermal energy of the atoms (cid:126) ω ⊥ (cid:29) k B T . Furthermore, it also implies thatthe confinement characteristic length a ⊥ ≡ q (cid:126) /mω ⊥ is much smaller than the three-dimensional scattering length of the atoms a (see App. A), a ⊥ (cid:28) a . Therefore,the consequence of this experimental setting is that all the dynamics of the system occurspredominantly in the x -direction, characterizing a quasi one-dimensional system.1.2.1 State of the artAn accurate description of strongly correlated quantum systems, for an arbitrarynumber of particles, is often a dare without a simple solution. Apart from the veryspecific family of integrable systems, where all observables can, in principle, betheoretically predicted, our knowledge is, in general, limited to simplifications like thetwo-body case, the thermodynamic limit, the low-energy regime, or themean-field approximations. So, it is quite challenging to extract general informationfrom such systems, e.g. , the scaling of physical observables with respect to the number ofparticles.Considering the system composed of a quantum gas where the particles interactwith each other via a delta-like interaction potential, the short-range correlations are I For mathematical details on the Hohenberg-Mermin-Wagner theorem, see App. B.6
Chapter 1 Introduction embedded in the Tan’s contact C , an experimentally relevant quantity that deter-mines the asymptotic behavior of the momentum distribution n ( k ) via C ≡ lim k →∞ k n ( k ).This observable can be measured via time-of-flight techniques, via radio-frequencyspectroscopy, by Bragg spectroscopy, by measuring the energy variation as afunction of the interaction strength, or by analyzing the three-body losses in quantummixtures. Such a quantity depends on many physical aspects, such as the interactionenergy, the density-density correlation functions, the trapping configuration, the tem-perature, and the magnetization, . Thus, it fluctuates in a nontrivial way with thenature and the number of particles N . Therefore, even in one dimension, the behavior of C is not completely clarified, especially in trapped systems, despite of many theoreticalinvestigations. For one-dimensional particles trapped in a harmonic potential offrequency ω , it has been shown that, in the thermodynamic limit, at zero temperature, thecontact rescaled by N / is an universal function of one scaling parameter, the adimensionalinteraction strength ˜ g ≡ − a / √ N a D , where a is the 1D scattering length (App.A) and a ≡ q (cid:126) /mω is the harmonic oscillator length. Such a scaling property also holdsat finite temperatures in the grand-canonical ensemble: the contact rescaled by h N i / is anuniversal function of two scaling parameters, ˜ g and τ ≡ T /T F , where T F = N (cid:126) ω/k B is the Fermi temperature. However, for systems with a small number of particles, the N / -scaling fails. In the zero-temperature limit, it is possible to change the paradigm andto introduce a different scaling form that holds for any number of particles N ≥ Atfinite temperature, considering the grand-canonical ensemble, the h N i / -scaling law holdsfor N >
However, corrections for a small number of particles have, to our knowledge,not yet been studied in 1D, and the question of the relevance of the statistical ensemblehas also not yet been addressed. The latter is, indeed, a crucial point, since ultracold-atomexperiments are more properly described by the canonical ensemble or, more often, by anaverage over canonical ensembles. This ensemble study is also motivated by the fact thatthe scaling properties of the system are strongly affected by the statistical distribution ofthe number of particles.
Now, let us provide a succint summary of each chapter. In Chap. 2, we present abrief description of optical lattices and some basic concepts towards the understandingof the atom-laser interacting potential. Then, we describe the solutions of the respectiveSchödinger equation due to such a potential and argue, due to the periodicity of thepotential, that the solutions can be given in terms of the Bloch functions and also in termsof the convenient Wannier functions. With these concepts, we have the fundamentals inorder to interpret the Bose-Hubbard model and derive its Hamiltonian in terms of theimportant parameters. To finish, we briefly study such parameters in order to check howthey depend on the laser potential depth as well as on the recoil energy. .3 Thesis outline In Chap. 3, we briefly discuss the fundamentals of second-order quantum phasetransitions, and then we introduce the Landau expansion of the thermodynamic potentialtogether with the mean-field approximation in order to evaluate the phase boundaryassociated with the Mott-insulator-superfluid (MI-SF) quantum phase transition of bosonicparticles in optical lattices. In addition, we apply nondegenerate perturbation theory(NDPT) at finite temperature to calculate the Landau coefficients, the condensate density, | Ψ | , and the density of particles, − ∂ F /∂µ . Consequently, we show that the calculationsfrom NDPT lead to nonphysical behaviors for these two physical quantities, which areclear consequences of the incorrect treatment of the degeneracies that happen to occurbetween two consecutive Mott lobes. Finally, we show, as a first degenerate approach, howsuch problems can be fixed within an adequate analysis.Chap. 4 is concerned with the development of the Brillouin-Wigner perturbationtheory (BWPT) applied to the zero-temperature regime of bosons in optical lattices. Webegin by introducing the formulation behind the BWPT, which consists in achieving aSchrödinger-like equation for an effective Hamiltonian so that it can be expanded up tothe desired order in the perturbation parameter. Then we apply the BWPT for the casewhere the degenerate Hilbert subspace is consisted of one state, and check that the resultsfor the condensate density are slightly improved, but still unsatisfactory, leading to thenecessity of considering two states in the degenerate Hilbert subspace. After calculatingthe important physical quantities in our two-state approach, we realize that it producesphysically consistent results for both the condensate and the particle densities. Afterwards,we develop a useful graphical approach for easily calculating higher-order terms in thepertubative expansion. Finally, we consider the effects of a harmonic trap in the systemand calculate how it affects the equation of state.In Chap. 5, we turn our attention to the finite-temperature scenario. We develop afinite-temperature degenerate perturbation theory (FTDPT) based on a projection operatorformalism that, similarly to BWPT, separates the Hilbert space into the degeneratesubspace and the complementary one, which is free from any degeneracy. We then applyour developed FTDPT to the one lattice site mean-field Bose-Hubbard Hamiltonian inorder to get meaningful results for the condensate density as well as for the particle densityin the vicinity of the MI-SF quantum phase transition.In Chap. 6, we study the one-dimensional interacting Bose gas. We begin with thecase with N = 2 particles, which is an integrable system and an instructive example to en-lighten some basic concepts that arise from considering delta-like interparticle interactions.We work out the details behind the calculations in order to get the relative-motion wavefunction and check the discontinuity it presents at the contact point. Then we work outthe asymptotic behavior of the momentum distribution, that leads to a relation whereone can recognize the valuable role of a term that depends on the second-order correla- Chapter 1 Introduction tion function, the so-called
Tan’s contact , C . Then, we exactly calculate the two-bosoncontact. Subsequently, we develop an analytical expression for the N -boson contact in theTonks-Girardeau limit. Following, we analyze the scaling properties of the Tan’s contactwithin some specific temperature regimes, such as the zero- and the large-temperaturescalings. From those, we propose a generalized scaling conjecture for all ranges of tem-perature. Furthermore, from quantum Monte Carlo (QMC) calculations, we investigatethe intermediate-interaction regime, ˜ g ∼
1. Finally, we draw a comparison between thecanonical and the grand-canonical ensembles in the context of the interparticle contact. In this chapter, we discuss the fundamental concepts associated with the descriptionof Bose gases loaded into optical lattices. We begin with the theory behind the atom-laser interaction, focusing on a brief discussion about the atomic energy shift due to itsinteraction with the laser-generated electric field and on the form of the periodic potential,which leads us to describe the solutions of the respective Schödinger equation by applyingthe Bloch theorem, i.e. , the same one used in solid state physics in order to interpretthe solutions of a generic periodic potential in terms of a plane wave times a periodicfunction, which are then named as Bloch functions. After that, we perform a descriptionof the Bose-Hubbard model, introducing its general form in terms of the Hamiltonianparameters. Then, we apply a harmonic approximation in the laser field potential so thatwe can perform a first estimation of the Hamiltonian parameters.
The interaction between the laser-generated electric field and the atom electricdipole within the electric dipole approximation is given by
69, 70 V ext ( r ) = − d · E ( r ) , (2.1)where d ≡ − e P i r i denotes the atomic electric dipole, with e being the electronic chargeand r i its distance from the nucleus, and E ( r ) is the external electric field. Let us consideran atomic transition from the fundamental state | i to any excited state | n i due to theexternal electric field. It is possible to calculate a second-order correction in the atomicground-state energy, which is given by ∆ E = − X n |h n | ˆ V ext | i| E n − E = − α | E | , (2.2)where E and E n are the energies of the fundamental and the excited state, respectively,and α ≡ − ∂ ∆ E∂ E = 2 X n |h n | ˆ d · ˆ (cid:15) | i| E n − E , (2.3)is the atomic polarizability, while ˆ (cid:15) represents the direction of the laser electric field. Ofcourse that this simple static analysis already gives us some insights about the energyshift due to the interaction, but it does not correspond to the more realistic case. So, inorder to approximate the discussion to the real world, let us consider a time-dependentlaser-generated electric field E ( r , t ) = E ( r )e − iωt + c . c . In such a case, the energy shift dueto the interaction is given by ∆ E = X n h | ˆ d · ˆ E | n ih n | ˆ d · ˆ E † | i E n − E + (cid:126) ω + h | ˆ d · ˆ E † | n ih n | ˆ d · ˆ E | i E n − E − (cid:126) ω . (2.4) Chapter 2 Bosons in optical lattices
It is instructive to note that the first term comes from the absorption of a photon bythe atom, while the second one comes from the emission of a photon.
Eq. (2.4) can besimplified to∆ E = X n |h n | ˆ d · ˆ (cid:15) | i| | E | h ( E n − E − (cid:126) ω ) − + ( E n − E + (cid:126) ω ) − i = − α ( ω )2 h E ( r , t ) i , (2.5)where h E ( r , t ) i = 2 | E | represents the time-average of the squared electric field. Also, wehave introduced the dynamical polarizability, that reads a ( ω ) = X n |h n | ˆ d · ˆ (cid:15) | i| h ( E n − E − (cid:126) ω ) − + ( E n − E + (cid:126) ω ) − i = 2 X n ( E n − E ) |h n | ˆ d · ˆ (cid:15) | i| ( E n − E ) − ( (cid:126) ω ) . (2.6)In many cases of interest, where the frequency of the laser field is close to the atomicresonance one, the polarizability can be reduced to I α ( ω ) ≈ |h n | ˆ d · ˆ (cid:15) | i| E n − E − (cid:126) ω . (2.7)Now, let the excited state have a finite lifetime Γ − n , whereΓ n = 43 X m ω n,m πε (cid:126) c |h n | ˆ d · ˆ (cid:15) | m i| (2.8)is the rate of decay by spontaneous emission. In this more realistic scenario, it impliesthat the energy of the excited state must contain an additional term to account for thespontaneous emission, namely E n → E n − i (cid:126) Γ n /
2. Consequently, the energy shift (2.5)results in ∆ E = (cid:126) R δ n + Γ n / δ n − i Γ n ! , (2.9)where we have introduced the Rabi frequency
71, 160, 161 Ω R ≡ |h n | ˆ d · ˆ E | i| / (cid:126) as well as the detuning , given by the difference between the radiation field frequency and the frequencyof the atomic transition δ n ≡ ω − ( E n − E ) / (cid:126) . In the cold atoms literature, δ n > blue detuning , while δ n < red detuning .Let us turn our attention to the form of the potential V ext ( r ) generated by the laser.The profile of a monochromatic Gaussian laser beam is given by
95, 162 V ext ( r ) = − V X ( i,j,l ) e − ( x i + x j ) /b cos ( k L x l ) , (2.10) I For a more profound discussion and algebraic manipulations, see PETHICK; SMITH andPITAEVSKII; STRINGARI. .1 Atom-laser interaction where b is the laser beam waist, while k L = 2 π/λ and λ are the laser wavevector andwavelength, respectively. Here, the sum is performed for the three possible sequence of theindependent coordinate variables that produce different results for the argument of thesum, i.e. , ( i, j, l ) = (1 , , , (1 , , , (2 , , r (cid:28) b . Thisjustifies the approximation e x ≈ x + O ( x ) , | x | (cid:28)
1, leaving us with a simplifiedformula for the external potential V ext ( r ) = V r b − X ( i,j,l ) sin ( k L x l ) " − b (cid:16) x i + x j (cid:17) . (2.11)Again, by the same reasoning as above, we perform another round of simplifications, thatresults in V ext ( r ) = V X i sin ( k L x i ) , (2.12)where we have performed the shift V ext ( r ) + 3 V → V ext ( r ), since V is simply a constant.2.1.1 Solutions for the laser field potentialNow, as the laser potential (2.12) is a periodic one, we know from solid statephysics that the solutions of the Schrödinger equation regarding noninteracting particlesin such a potential, " − (cid:126) m ∇ + V ext ( r ) Ψ n, k ( r ) = E n, k Ψ n, k ( r ) , (2.13)are given by Ψ n, k ( r ) = e i k · r Φ n, k ( r ) , (2.14)where Φ n, k ( r ) are the so-called Bloch functions and they possess the same periodicityof the trapping potential. The wave vector is represented by k and n is the band index.When the potential depth is big enough and the temperature low enough so thatthe tunneling probability between neighboring sites is small, the single-particle wavefunctions can be approximated by a linear combination of localized states in each potentialwell. In order to explore this effect, it is convenient to use the more convenient Wannierfunctions, which are localized functions defined according to W n ( r − r i ) ≡ √ N s X k e − i k · r i Ψ n, k ( r ) , (2.15)where N s is the number of lattice sites, r i is the location of the i -th lattice site, and thesum in k runs over the first Brillouin zone, i.e. , − π/d ≤ k i ≤ π/d , where d = π/k L = λ/ it is possible to derive the respective orthonormality and Chapter 2 Bosons in optical lattices completeness properties for the Wannier functions: Z < d r W ∗ n ( r − r i ) W m ( r − r j ) = 1 N s X k , k e i ( k · r i − k · r j ) Z < d r Ψ ∗ n, k ( r )Ψ m, k ( r )= δ n,m N s X k e i k · ( r i − r j ) = δ n,m δ i,j , (2.16)and X n,i W ∗ n ( r − r i ) W n ( r − r i ) = X n, k , k Ψ ∗ n, k ( r )Ψ n, k ( r ) 1 N s X i e i ( k − k ) · r i = X n, k Ψ ∗ n, k ( r )Ψ n, k ( r )= δ ( r − r ) , (2.17)respectively. Such conditions assure that any wave function can be written as a seriesexpansion of the Wannier functions.Now, as the laser-generated potential (2.12) consists of separated contributionson each coordinate variable, the original three-dimensional Schrödinger equation can beseparated into three identical one-dimensional Schrödinger equations, which read " − (cid:126) m ∇ j + V ext ( x j ) ψ n j ,k j ( x ) = (cid:15) n j ,k j ψ n j ,k j ( x j ) , (2.18)with ψ n, k ( r ) = Y j ψ n j ,k j ( x j ) , (2.19)and E n, k = X j (cid:15) n j ,k j . (2.20)Analogously to the three-dimensional problem, each one-dimensional solution can bewritten in terms of the respective one-dimensional Bloch functions asΨ n j ,k j ( x j ) = e ik j x j φ n j ,k j ( x j ) . (2.21)Consequently, the corresponding one-dimensional Wannier functions read w n j (cid:16) x j − x ( i ) j (cid:17) = √ N s ! / X k j e − ik j x ( i ) j ψ n,k j ( x j ) . (2.22)Therefore, the tridimensional Bloch and Wannier functions becomeΦ n ( r ) = Y j φ n j ,k j ( x j ) (2.23)and W n ( r − r i ) = Y j w n j (cid:16) x j − x ( i ) j (cid:17) , (2.24)respectively. .2 The Bose-Hubbard model The Bose-Hubbard model provides a suitable description of interactingspinless bosonic atoms confined in optical lattices. The fundamental mathematical con-siderations within the model are developed as follows. Let us start with the generalsecond-quantized Hamiltonianˆ H = Z d r ˆΨ † ( r ) " − (cid:126) m ∇ + V ext ( r ) − µ ˆΨ( r )+ 12 Z d r Z d r ˆΨ † ( r ) ˆΨ † ( r ) V int ( r , r ) ˆΨ( r ) ˆΨ( r ) , (2.25)where the first term represents the single-particle Hamiltonian, with V ext ( r ) representingthe atom-laser interaction, and µ is the grand-canonical chemical potential; while thesecond term corresponds to the interparticle interaction term, where V int ( r , r ) is theatomic interaction potential. The bosonic field operators are represented by ˆΨ † ( r ) andˆΨ( r ), and they obey the usual bosonic commutation rules h ˆΨ( r ) , ˆΨ † ( r ) i = δ ( r − r ) , (2.26a) h ˆΨ( r ) , ˆΨ( r ) i = h ˆΨ † ( r ) , ˆΨ † ( r ) i = 0 . (2.26b)Considering gases with low density profiles, the interaction between particles canbe approximated by
20, 71 V int ( r , r ) = 4 π (cid:126) a m δ ( r − r ) , (2.27)where a is the three-dimensional s-wave scattering length (see App. A). Thus, Eq. (2.25)reduces toˆ H = Z d r ˆΨ † ( r ) " − (cid:126) m ∇ + V ext ( r ) − µ ˆΨ( r ) + g Z d r ˆΨ † ( r ) ˆΨ † ( r ) ˆΨ( r ) ˆΨ( r ) , (2.28)where g ≡ π (cid:126) a /m is the coupling constant.Now, taking into account that ultracold atoms confined in deep periodic potentialscan be regarded as occupying only the lowest Bloch band, we can simplify the problem byrestricting ourselves to the Wannier function corresponding to n = 0, W ( r ). Therefore,due to the completeness of the Wannier functions (2.17), the field operators can then beexpanded as ˆΨ( r ) = X i ˆ a i W ( r − r i ) , (2.29a)ˆΨ † ( r ) = X i ˆ a † i W ∗ ( r − r i ) , (2.29b)where ˆ a i and ˆ a † i are, respectively, the annihilation and creation operators of particles at agiven lattice site i . From (2.26), it is also possible to derive the commutation rules for the Chapter 2 Bosons in optical lattices lattice operators, resulting in h ˆ a i , ˆ a † j i = δ i,j , (2.30a)[ˆ a i , ˆ a j ] = h ˆ a † i , ˆ a † j i = 0 . (2.30b)By substituting (2.29) into (2.28), we haveˆ H = 12 X i,j,k,l U ijkl ˆ a † i ˆ a † j ˆ a k ˆ a l + X i,j J ij ˆ a † i ˆ a j − X i,j µ ij ˆ a † i ˆ a j , (2.31)where the parameters read U ijkl = g Z d r W ∗ ( r − r i ) W ∗ ( r − r j ) W ( r − r k ) W ( r − r l ) , (2.32a) J ij = Z d r W ∗ ( r − r i ) " − (cid:126) m ∇ + V ext ( r ) W ( r − r j ) , (2.32b) µ ij = µ Z d r W ∗ ( r − r i ) W ( r − r j ) . (2.32c)Following the discussion from UEDA, in a scenario where the confining potentialis sufficiently deep, the Wannier functions are strongly localized, hence the overlap betweenthe different-site-particle wave functions is small. Therefore, in this model we consideronly nearest neighbors transitions and local interparticle interaction. Such considerationsand the orthonormality of the Wannier functions lead to the Bose-Hubbard Hamiltonianˆ H BH = U X i ˆ a † i ˆ a † i ˆ a i ˆ a i − J X h i,j i ˆ a † i ˆ a j − µ X i ˆ a † i ˆ a i , (2.33)where U = g Z d r | W ( r ) | , (2.34a) J = Z d r W ∗ ( r ) " (cid:126) m ∇ − V ext ( r ) W ( r ) . (2.34b)Such parameters have clear interpretations: U is the on-site interaction parameter betweenparticles and J is the hopping parameter, which describes the tunneling probability ofparticles between its original site to a neighboring one.The Bose-Hubbard model predicts two different phases for the whole systemdepending on the ratio between the on-site interaction and the hopping parameters: ifthe on-site interaction between atoms is much stronger than the hopping parameter, i.e. , U/J (cid:29)
1, the system realizes a Mott insulator phase; on the other hand, if thehopping parameter predominates over the on-site interaction parameter, i.e. , U/J (cid:28)
20, 71 | Ψ MI i = ( n !) − N s / N s Y i =1 (cid:16) ˆ a † i (cid:17) n O N s | i , (2.35) .2 The Bose-Hubbard model where N s is the number of lattice sites and n is the average occupation number per site.In the opposite scenario, i.e. , the ground state of the system deep in the superfluid phasecan be considered as
20, 71 | Ψ SF i = N − N/ s √ N ! N s X i =1 ˆ a † i ! N O N s | i , (2.36)where N is the total number of particles and N N s | i = | i ⊗ | i · · · ⊗ | i is the vacuumstate.2.2.1 The Hamiltonian parametersFor a deep periodic potential, we can consider the lowest-band Wannier functionas a solution of the laser field potential " − (cid:126) m ∇ x + V sin ( k L x ) w ( x ) = E w ( x ) . (2.37)This differential equation has approximate solutions in terms of the Mathieu functions w ( x ) = C − ˜ V , − ˜ V , k L x ! + S − ˜ V , − ˜ V , k L x ! , (2.38)where C( a, q, z ) and S( a, q, z ) are the even and odd Mathieu functions, respectively. Herewe have defined ˜ V ≡ V /E R and introduced the so-called recoil energy E R ≡ (cid:126) k L / m .With such a solution, it is possible to approximately evaluate the hopping energy (2.34b),which is performed in Ref. 175 with the following result J = 4 √ π E R ˜ V / e − V / . (2.39)2.2.1.1 Harmonic approximationA first approximation of the laser field potential is the harmonic approximation sin ( k L x ) ≈ ( k L x ) . Again, let us consider the lowest-band Wannier function as a solutionof the harmonic potential " − (cid:126) m ∇ x + V ( k L x ) w ( x ) = E w ( x ) . (2.40)The solution is the known fundamental state of the harmonic oscillator w ( x ) = k L ˜ V / π / exp − x k L q ˜ V ! , (2.41)with the energy given by E = E R q ˜ V . Chapter 2 Bosons in optical lattices
It follows that we can also find an expression for the the Bose-Hubbard parameters(2.34) from the solution w ( x ) within the harmonic approximation. So, from (2.34a) wehave that the on-site interaction energy reads U = g Z < d r | W ( r ) | = g (cid:18)Z + ∞−∞ dx | w ( x ) | (cid:19) = s π a k L E R ˜ V / . (2.42)Similarly, from (2.34b) J = Z < d r W ∗ ( r ) " (cid:126) m ∇ − V ext ( r ) W ( r )= Z + ∞−∞ dx w ∗ ( x − d ) (cid:126) m ∇ x − V sin ( k L x ) ! w ( x ) . (2.43)Substituting the solution (2.41) into (2.43) and performing the integral, the hopping energyresults in J = E R (cid:16) π ˜ V − V / (cid:17) exp − π q ˜ V ! − E R V (cid:18) ˜ V − / (cid:19) exp − π ˜ V q ˜ V . (2.44)As a comparison, we plot the hopping energy from the Mathieu solution (2.39) andfrom the harmonic approximation (2.44) in Fig. 4 and conclude, by direct observation, thatthe harmonic approximation does not result in a good enough estimation of the hoppingenergy for shallow potentials. V / E R J / E R Figure 4 – Hopping energy from the Mathieu solution (2.39) (blue line) and from theharmonic approximation (2.44) (yellow line).Source: By the author. The purpose of this chapter is the study of the main considerations taken intoaccount in order to investigate the Mott-insulator-superfluid (MI-SF) quantum phasetransition of bosons in optical lattices. We begin by introducing some basic conceptsabout second-order phase transitions. Also, we discuss the Landau assumptions for thethermodynamic potential in the vicinity of the phase transition. Then, we introduce themean-field approximation, which is the main path taken in order to remove the nonlocalitypresent in the Bose-Hubbard Hamiltonian, leading to a great simplification. Following, weperform calculations based on nondegenerate perturbation theory (NDPT), which resultsin the MI-SF phase diagram. Following such calculations, we show that NDPT leads tosome inconsistencies due to degeneracy, which turns out to provide a nonphysical behaviorof the order parameter.
In 1933, Paul Ehrenfest noted that different systems in thermodynamical equilib-rium could present distinct-order discontinuities in their thermodynamic potential: sometransitions were characterized by a discontinuity in the first derivative of the thermody-namic potential with respect to some variable (which we will call order parameter lateron), which he then named first-order phase transitions ; others indicated a discontinuity inthe second derivative of the thermodynamic potential, and those he called second-orderphase transitions . Differently from classical phase transitions, that arise as a result of thermal fluctua-tions, quantum phase transitions can happen even at zero temperature, driven by quantumfluctuations.
This is the case of our considered system constituted of bosons inoptical lattices: the transition from the Mott insulator to the superfluid phase can happenat T = 0 without the effects of thermal fluctuations, thus characterizing a quantum phasetransition. In the Mott insulating phase, the atoms are localized at the minima of thelaser-generated potential, meaning that the condensate density is zero in such a regime. Onthe other hand, in the superfluid phase, the system is characterized by a high delocalizationof the atoms, which means that it has achieved a nonzero condensate density. Due to thisexplicit change from a zero value to a nonzero one of the condensate density, we can regardit as being the order parameter of the quantum phase transition in question.3.1.1 Landau theory of second-order phase transitionsLandau argued that the thermodynamic potential F could be written as a poly-nomial function of the order parameter in the vicinity of a phase transition. In the Chapter 3 Mott-insulator-superfluid quantum phase transition case of BEC, where the order parameter is the condensate wave function Ψ, the Landauexpansion could in principle be F = a + a | Ψ | + a | Ψ | + a | Ψ | + a | Ψ | + · · · . (3.1)However, in the case of the Bose-Hubbard Hamiltonian described by (2.33), which possessesa global U (1) phase invariance, i.e. , the Bose-Hubbard Hamiltonian is invariant under thetransformation ˆ a → e iθ ˆ a , a n will not vanish only for even values of n . Therefore, since weare considering only small values of | Ψ | , further analysis will be held on the even Landauexpansion up to fourth order, F ≈ a + a | Ψ | + a | Ψ | . (3.2)From Fig. 5, for a >
0, it is possible to realize that, for a >
0, the stable state, i.e. , the minimum of F , happens at | Ψ | = 0, which corresponds to the symmetrical phase.On the other hand, when a <
0, the stable state is given by nonvanishing values of theorder parameter, | Ψ | 6 = 0, corresponding to the unsymmetrical phase. Conclusively, thecondition a = 0 defines the phase boundary between the two phases. Also, the solutionfor the unsymmetrical phase is given by ∂ F ∂ | Ψ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Ψ |6 =0 = 0 ⇒ | Ψ | = − a a . (3.3) - - | Ψ | ℱ Figure 5 – Landau expansion of the thermodynamic potential from (3.2). The blue, yellow,and green curves correspond to, respectively, a > a = 0, and a < Due to the non-local character of the hopping term in the Bose-Hubbard Hamilto-nian (2.33), we perform a mean-field approximation,
76, 89, 90, 95 which consists in considering .2 Mean-field approximation the bosonic operators as the contribution of its mean value summed to a fluctuationˆ a i = h ˆ a i i + δ ˆ a i . Thus, the hopping term in (2.33) reads − J X h i,j i ˆ a † i ˆ a j = − J X h i,j i (cid:16) h ˆ a † i i + δ ˆ a † i (cid:17) ( h ˆ a j i + δ ˆ a j ) . (3.4)Neglecting quadratic terms of fluctuations we have − J X h i,j i ˆ a † i ˆ a j = − J X h i,j i (cid:16) h ˆ a † i i ˆ a j + ˆ a † i h ˆ a j i − h ˆ a † i ih ˆ a j i (cid:17) . (3.5)Now, let us consider a homogeneous system so that the average value of the annihilationoperator is site-independent, implying the definition Ψ ≡ h ˆ a i,j i . Moreover, denoting thenumber of nearest neighbors by z , we obtain the mean-field Hamiltonian,ˆ H MF = U X i (cid:16) ˆ n i − ˆ n i (cid:17) − X i µ ˆ n i − J z X i (cid:16) Ψ ∗ ˆ a i + Ψˆ a † i − Ψ ∗ Ψ (cid:17) , (3.6)where ˆ n i = ˆ a † i ˆ a i is the number operator at the lattice site i . Note that we have used thebosonic commutation relations (2.30) in order to rewrite the on-site interaction term interms of the number operator ˆ n .Since (3.6) is a sum of local Hamiltonians, we restrict ourselves to the one latticesite Hamiltonian, ˆ H = U (cid:16) ˆ n − ˆ n (cid:17) − µ ˆ n − J z (cid:16) Ψ ∗ ˆ a + Ψˆ a † − Ψ ∗ Ψ (cid:17) . (3.7)3.2.1 Nondegenerate perturbation theoryAs mentioned before, the transition from Mott-insulator to superfluid is associatedto the breakdown of the U (1) symmetry and can then be characterized by the change inthe order parameter from zero to a non-zero value. Since we are considering our system inthe vicinity of the phase transition, where | Ψ | has a small value, and only the hoppingterm depends explicitly on Ψ in (3.7), we can treat the hopping term as a perturbation.Thus, (3.7) decomposes according to ˆ H = ˆ H + ˆ V into the unperturbed Hamiltonianˆ H = U (cid:16) ˆ n − ˆ n (cid:17) − µ ˆ n + J z Ψ ∗ Ψ , (3.8)and the perturbation ˆ V = − J z (cid:16) Ψ ∗ ˆ a + Ψˆ a † (cid:17) . (3.9)The unperturbed eigenenergies are E n = U (cid:16) n − n (cid:17) − µn + J z | Ψ | , (3.10)where the quantum number n = 0 , , , . . . indicates the number of bosons per site. Chapter 3 Mott-insulator-superfluid quantum phase transition
At this point, we are interested in evaluating how the perturbation changes thefree energy of the system. To this purpose, we must work out the partition function, Z = Tr h e − β ˆ H i , (3.11)in order to obtain the free energy of the system.The quantum-mechanical evolution operator within the imaginary-time formalism, i.e. , ˆ U = e − τ ˆ H , can be factorized according toˆ U = e − τ ˆ H ˆ U I ( τ ) , (3.12)where ˆ U I ( τ ) is the interaction picture imaginary-time evolution operator. Note that weare assuming (cid:126) = 1. The equation for the time evolution of such an operator is d ˆ U I ( τ ) dτ = − ˆ V I ( τ ) ˆ U I ( τ ) , (3.13)with ˆ V I ( τ ) = e τ ˆ H ˆ V e − τ ˆ H . (3.14)Equation (3.13) can be iteratively solved, thus allowing the construction of aperturbative expansion. Performing the expansion, with the initial value ˆ U I (0) = ˆ , up tofourth order, we haveˆ U I ( β ) ≈ ˆ − Z β dτ ˆ V I ( τ ) + Z β dτ Z τ dτ ˆ V I ( τ ) ˆ V I ( τ ) − Z β dτ Z τ dτ Z τ dτ ˆ V I ( τ ) ˆ V I ( τ ) ˆ V I ( τ )+ Z β dτ Z τ dτ Z τ dτ Z τ dτ ˆ V I ( τ ) ˆ V I ( τ ) ˆ V I ( τ ) ˆ V I ( τ ) . (3.15)It is possible to observe, from the perturbative Hamiltonian (3.9), that odd-orderterms in (3.15) will vanish. Therefore, we can restrict ourselves to the calculation of thezeroth-, second-, and fourth-order terms in (3.15).Making use of the time-evolution operator in the interaction picture Z = Tr h e − β ˆ H ˆ U I ( β ) i , we calculate the partition function up to fourth order, Z = ∞ X n =0 e − βE n h n | ˆ U I ( β ) | n i ≈ Z (0) + Z (2) + Z (4) , (3.16)with the single-site eigenstates | n i corresponding to the occupation number in the Mottinsulator state. The zeroth-order term yields Z (0) = ∞ X n =0 e − βE n h n | ˆ | n i = ∞ X n =0 e − βE n . (3.17) .2 Mean-field approximation Now, let us proceed to the detailed calculation of the second- and fourth-orderterms, Z (2) and Z (4) , respectively. The second-order term reads Z (2) = ∞ X n =0 e − βE n Z β dτ Z τ dτ h n | ˆ V I ( τ ) ˆ V I ( τ ) | n i . (3.18)Inserting (3.14) into (3.18), we have Z (2) = ∞ X n =0 e − βE n Z β dτ Z τ dτ h n | e τ ˆ H ˆ V e − τ ˆ H e τ ˆ H ˆ V e − τ ˆ H | n i . (3.19)The exponential of an Hermitian operator ˆ O with eigenstates | φ λ i and respective eigenvalues λ is simply given by e ˆ O = e P λ λ | φ λ ih φ λ | = ∞ X n =0 n ! X λ λ | φ λ ih φ λ | ! n = X λ ∞ X n =0 λn ! ! | φ λ ih φ λ | = X λ e λ | φ λ ih φ λ | . (3.20)As | n i are eigenstates of ˆ H , Eq. (3.19) reduces to Z (2) = ∞ X n =0 e − βE n Z β dτ Z τ dτ e ( τ − τ ) E n h n | ˆ V e − τ ˆ H e τ ˆ H ˆ V | n i . (3.21)According to (3.9), we have Z (2) = J z ∞ X n =0 e − βE n Z β dτ Z τ dτ e ( τ − τ ) E n h n | (cid:16) Ψ ∗ ˆ a + Ψˆ a † (cid:17) e − τ ˆ H × e τ ˆ H (cid:16) Ψ ∗ ˆ a + Ψˆ a † (cid:17) | n i , (3.22)yielding Z (2) = J z ∞ X n =0 e − βE n Z β dτ Z τ dτ e ( τ − τ ) E n (cid:16) Ψ √ n h n − | + Ψ ∗ √ n + 1 h n + 1 | (cid:17) × (cid:16) Ψ ∗ √ n e ( τ − τ ) E n − | n − i + Ψ √ n + 1e ( τ − τ ) E n +1 | n + 1 i (cid:17) . (3.23)The scalar products reduce (3.23) to Z (2) = J z | Ψ | ∞ X n =0 e − βE n Z β dτ Z τ dτ h n e ( τ − τ )∆ n,n − + ( n + 1)e ( τ − τ )∆ n,n +1 i . (3.24)Finally, the integrations yield Z (2) = J z | Ψ | ∞ X n =0 e − βE n n e β ∆ n,n − − n,n − − β ∆ n,n − ! +( n + 1) e β ∆ n,n +1 − n,n +1 − β ∆ n,n +1 ! , (3.25) Chapter 3 Mott-insulator-superfluid quantum phase transition where we have introduced the abbreviation ∆ i,j ≡ E i − E j for the differences between twoconsecutive energies given by (3.10).For the fourth-order term, we have Z (4) = ∞ X n =0 e − βE n Z β dτ Z τ dτ Z τ dτ Z τ dτ h n | ˆ V I ( τ ) ˆ V I ( τ ) ˆ V I ( τ ) ˆ V I ( τ ) | n i . (3.26)Inserting (3.8) and (3.14) into (3.26) gives Z (4) = ∞ X n =0 e − βE n Z β dτ Z τ dτ Z τ dτ Z τ dτ e ( τ − τ ) E n h n | ˆ V e − τ ˆ H ˆ V I ( τ ) ˆ V I ( τ )e τ ˆ H ˆ V | n i . (3.27)According to (3.9), we have Z (4) = J z ∞ X n =0 e − βE n Z β dτ Z τ dτ Z τ dτ Z τ dτ e ( τ − τ ) E n (cid:16) Ψ √ n e ( τ − τ ) E n − h n − | +Ψ ∗ √ n + 1e ( τ − τ ) E n +1 h n + 1 | (cid:17) ˆ V e − τ ˆ H e τ ˆ H ˆ V (cid:16) Ψ ∗ √ n e ( τ − τ ) E n − | n − i +Ψ √ n + 1e ( τ − τ ) E n +1 | n + 1 i (cid:17) . (3.28)Using again (3.8) and (3.14) together with (3.28) results in Z (4) = J z ∞ X n =0 e − βE n Z β dτ Z τ dτ Z τ dτ Z τ dτ e ( τ − τ ) E n × h Ψ √ n e ( τ − τ ) E n − (cid:16) Ψ √ n − − τ E n − h n − | + Ψ ∗ √ n e − τ E n h n | (cid:17) +Ψ ∗ √ n + 1e ( τ − τ ) E n +1 (cid:16) Ψ √ n + 1e − τ E n h n | + Ψ ∗ √ n + 2e − τ E n +2 h n + 2 | (cid:17)i × h Ψ ∗ √ n e ( τ − τ ) E n − (cid:16) Ψ ∗ √ n − τ E n − | n − i + Ψ √ n e τ E n | n i (cid:17) +Ψ √ n + 1e ( τ − τ ) E n +1 (cid:16) Ψ ∗ √ n + 1e τ E n | n i + Ψ √ n + 2e τ E n +2 | n + 2 i (cid:17)i , (3.29)which, after performing all the scalar products, reduces to Z (4) = J z | Ψ | ∞ X n =0 e − βE n Z β dτ Z τ dτ Z τ dτ Z τ dτ × h n ( n − ( τ − τ )∆ n,n − e ( τ − τ )∆ n − ,n − + ( n + 1)( n + 2)e ( τ − τ )∆ n,n +1 e ( τ − τ )∆ n +1 ,n +2 + n e ( τ − τ )∆ n,n − e ( τ − τ )∆ n − ,n + n ( n + 1)e ( τ − τ )∆ n,n − e ( τ − τ )∆ n,n +1 + n ( n + 1)e ( τ − τ )∆ n,n +1 e ( τ − τ )∆ n,n − + ( n + 1) e ( τ − τ )∆ n,n +1 e ( τ − τ )∆ n,n +1 i . (3.30) .2 Mean-field approximation Finally, the integrations result in Z (4) = J z | Ψ | ∞ X n =0 e − βE n ( n ( n −
1) e β ∆ n,n − − n,n − ∆ n − ,n − ∆ n,n − n − ,n − − n,n − ! + n ( n −
1) e β ∆ n,n − − n,n − ∆ n,n − n,n − + 1∆ n − ,n − ! + n ( n −
1) e β ∆ n,n − − n,n − ∆ n − ,n − n,n − − n − ,n − ! − n ( n − β ∆ n,n − e β ∆ n,n − ∆ n − ,n − + 1∆ n,n − ! + ( n + 1) ( n + 2) e β ∆ n,n +2 − n,n +1 ∆ n +1 ,n +2 ∆ n,n +2 n +1 ,n +2 − n,n +2 ! + ( n + 1) ( n + 2) e β ∆ n,n +1 − n,n +1 ∆ n,n +2 n,n +1 + 1∆ n +1 ,n +2 ! + ( n + 1) ( n + 2) e β ∆ n,n +1 − n,n +1 ∆ n +1 ,n +2 n,n +1 − n +1 ,n +2 ! − ( n + 1) ( n + 2) β ∆ n,n +1 e β ∆ n,n +1 ∆ n +1 ,n +2 + 1∆ n,n +2 ! + 3 n − e β ∆ n,n − ∆ n,n − + n β ∆ n,n − (cid:16) β ∆ n,n − (cid:17) + n β n,n − + n ( n + 1)∆ n,n +1 ∆ n − ,n +1 e β ∆ n,n +1 − n,n +1 + 1 − e β ∆ n,n − ∆ n,n − ! + n ( n + 1) 1 − e β ∆ n,n − ∆ n,n − ∆ n,n +1 n,n − + 1∆ n,n +1 ! + n ( n + 1) β ∆ n,n − ∆ n,n +1 n,n − + 1∆ n,n +1 ! + n ( n + 1) β n,n − ∆ n,n +1 + n ( n + 1)∆ n,n − ∆ n +1 ,n − e β ∆ n,n − − n,n − + 1 − e β ∆ n,n +1 ∆ n,n +1 ! + n ( n + 1) 1 − e β ∆ n,n +1 ∆ n,n +1 ∆ n,n − n,n +1 + 1∆ n,n − ! + n ( n + 1) β ∆ n,n +1 ∆ n,n − n,n +1 + 1∆ n,n − ! + n ( n + 1) β n,n +1 ∆ n,n − +3 ( n + 1) − e β ∆ n,n +1 ∆ n,n +1 + ( n + 1) β ∆ n,n +1 (cid:16) β ∆ n,n +1 (cid:17) + ( n + 1) β n,n +1 ) . (3.31)Now that we have an expression for the partition function, we then evaluate thefree energy, F = − β ln Z . (3.32)By considering the natural logarithm expansion at x = 0, ln(1 + x ) ≈ x − x /
2, we get, Chapter 3 Mott-insulator-superfluid quantum phase transition up to fourth order in the hopping parameter,
F ≈ − β ln Z (0) + Z (2) Z (0) + Z (4) Z (0) − Z (2) Z (0) ! . (3.33)Therefore, by comparing (3.1) and (3.33), we read off the Landau expansioncoefficients: a = − β ln Z (0) , (3.34a) a = − β | Ψ | Z (2) Z (0) , (3.34b) a = − β | Ψ | Z (4) Z (0) − Z (2) Z (0) ! . (3.34c)At zero temperature, we obtain results which are equivalent to RSPT. In particular,the Landau expansion coefficients reduce to:lim β →∞ a = E n − J z | Ψ | ≡ E (0) n , (3.35a)lim β →∞ a = J z + (
J z ) n + 1∆ n,n +1 + n ∆ n,n − ! , (3.35b)lim β →∞ a = ( J z ) " n ( n − n,n − ∆ n,n − + ( n + 1) ( n + 2)∆ n +1 ,n ∆ n,n +2 + n ∆ n − ,n + ( n + 1) ∆ n +1 ,n + n ( n + 1)∆ n +1 ,n ∆ n − ,n + n ( n + 1)∆ n,n − ∆ n +1 ,n . (3.35c)As previously discussed, the explicit solution of a = 0 results in the phaseboundaries between the superfluid and the Mott insulator. Such phase boundaries aredepicted in Fig. 6 for four different temperatures.3.2.1.1 Nondegenerate perturbation theory inconsistenciesAs already pointed out, NDPT is expected to exhibit degeneracy-related problems.Indeed, by directly observing the coefficient denominators in (3.35b) and (3.35c), weclearly identify such a degeneracy problem. Whenever µ/U becomes an integer n , thereis an equality between two consecutive energy values, for instance E n and E n +1 , thuscharacterizing a divergence in those expressions.According to (3.1), we consider the Landau expansion up to fourth order for thefree energy in the vicinity of a phase transition. Extremizing (3.1) with respect to theorder parameter leads to ∂ F ∂ | Ψ | = a + 2 a | Ψ | = 0 , (3.36)with the solution in the superfluid phase | Ψ | = − a a . (3.37) .2 Mean-field approximation - μ / U J z / U Figure 6 – Phase diagrams for the inverse temperatures β = 5 /U (black), β = 10 /U (red), β = 30 /U (green), and β → ∞ (blue).Source: SANT’ANA et al. Therefore, in order to explicitly show the degeneracy-related problems, we calculatethe particle density, n = − ∂ F ∂µ , (3.38)and the condensate density | Ψ | via NDPT.The plots of | Ψ | and n as functions of µ/U in Fig. 7 are interesting for ourpurposes since they reveal some nonphysical behaviors, which are consequences of NDPT:the order parameter approaches zero at a point where no phase transition occurs while theparticle density shows strange behaviors, especially at the degeneracy points, presentingdivergences at µ/U ∈ N . Fig. 7 (a) shows equation (3.37) for J z/U = 0 . µ/U ,the order parameter at the zero-temperature limit goes to zero, while for T >
J z/U , the system is in the superfluid phase, far away from the phaseboundary, as the Mott insulator needs low hopping probabilities. Since all of our theory isbased on the assumption of being close to the phase boundary, we cannot obtain reliableresults for values of
J z/U deep in the superfluid phase. Nevertheless, for
J z/U (cid:46) .
35, weassume our model to be valid. While for
J z/U = 0 we have no superfluid phase but onlyMott insulator, it is possible to reach the superfluid phase by increasing
J z/U . Another Chapter 3 Mott-insulator-superfluid quantum phase transition μ / U Ψ ( a ) - - - μ / U n ( b ) Figure 7 – Condensate density (a) from (3.37) and particle density (b) from (3.38) viaNDPT as functions of µ/U for
J z/U = 0 . β = 5 /U (dotted-dashed black), β = 10 /U (dashed red), β = 30 /U (dotted green), and β → ∞ (continuous blue). Source: SANT’ANA et al. way of changing the phase of the system from the Mott insulator to the superfluid phaseis by tuning µ/U at J z/U >
0. If we start in the first Mott lobe and increase µ/U , theordered structure breaks down at some point and the superfluid phase is energeticallymore favorable and thus realized. For µ/U <
0, the system is in the superfluid phase for
J z/U > − µ/U , whereas for J z/U < − µ/U we have no particles at all, as depicted in Fig.6. After obtaining the phase boundary, we take a closer look at the ground stateenergies for increasing n . In the plot of the unperturbed energies from (3.35a) in Fig. 8,we see that the ground state energies have a degeneracy at integer values of µ/U . As, forexample, in between the lobes for n = 1 (line with the smallest slope, red) and n = 2(line with the second smallest slope, blue) at µ/U = 1, we are at the degeneracy pointwhere the energies E (0)1 and E (0)2 coincide. Analogous formulas are valid between everytwo neighboring lobes. Such degeneracies at µ = U n make any algebraic treatment of thesystem quite complex. However, since we always have only two degenerate energies tohandle at a time, a solution to this problem can be found, as it will be shown further on.With this degeneracy in mind, we now discuss the order parameter. Firstly, let usplot Ψ ∗ Ψ = − a / a by using (3.35b) and (3.35c). Since a approaches infinity at µ = U n ,where E (0) n = E (0) n +1 , the condensate density Ψ ∗ Ψ tends to zero at these points, which falselyindicates a phase boundary. This nonphysical behavior is depicted in Fig. 9 through thedashed plots.3.2.3 A first degenerate correctionOne way to improve these results is to apply degenerate perturbation theory, whichwas done up to the first perturbative order in Ref. 89. In the referred work, the corrected .2 Mean-field approximation - 2 - 1 1 2 3 4 5- 10- 5510 Figure 8 – Unperturbed ground state energies E (0) n = E n − J z | Ψ | . Different lines cor-respond to different values for n from smaller to larger slope: n = 1 (red), n = 2 (blue), n = 3 (green), and n = 4 (purple). Vertical dashed black linescorrespond to the points of degeneracy. Solid colored lines represent realizedlowest energies, while dashed colored lines indicate the continuation of theenergy lines. Source: KÜBLER et al. condensate density yields Ψ ∗ Ψ = ( n + 1)4 − ( µ − U n ) J z ( n + 1) . (3.39)Let us now introduce the parameter ε ≡ µ − U n in order to analyze the system inthe vicinity of the degeneracy, according toΨ ∗ Ψ = ( n + 1)4 − ε J z ( n + 1) . (3.40)The resulting condensate densities from (3.40) are depicted by the dotted curves in Fig. 9.By setting Ψ ∗ Ψ = 0 in (3.39), we obtain the phase boundary, which is shown inFig. 10 by the dotted magenta curve. The phase boundary obtained out of the degenerateapproach is linear in µ/U , thus coinciding with the one from NDPT only at the vicinityof the points µ/U ∈ N . Nevertheless, for small values of J z/U , it can be considered agood approximation (see inset in Fig. 10). The tips of the triangular Mott lobes (dottedmagenta) correspond to µ/U = 1 / ≈ . µ/U = 7 / . µ/U = 17 / ≈ . µ/U = 31 / ≈ .
444 for increasing n , which differ from the tips of the curved lobes (dashedorange), that correspond, respectively, to µ/U = √ − ≈ . µ/U = √ − ≈ . µ/U = 2 √ − ≈ . µ/U = 2 √ − ≈ . J z/U = 0 .
02 (red),
J z/U = 0 .
08 (blue), and
J z/U = 5 − √ ≈ .
101 (green), while the latter one hits the second lobe exactly onits tip. These lines allow a better comparison between the dashed orange and the dottedmagenta phase boundaries. Chapter 3 Mott-insulator-superfluid quantum phase transition - - ε U Ψ * Ψ Figure 9 – Condensate densities from nondegenerate perturbation theory (dashed lines) incomparison to the condensate densities from degenerate perturbation theoryaccording to (3.40) (dotted lines) with µ = U n + ε and n = 1 for theleft part (negative ε/U ) and n = 2 for the right part (positive ε/U ). Thehopping strengths are, from the spacing inside to outside, J z/U = 0 .
02 (red),
J z/U = 0 .
08 (blue), and
J z/U = 0 .
101 (green). The dashed plots vanish at themean-field phase boundary, yielding a nonphysical behavior at the degeneracy.Also, they have increasing maxima for increasing
J z/U , and for
J z/U = 0 . ε/U = 0 .
442 the lobe is just touching in one point and goes smoothly tozero. The dotted plots provide a physical behavior at the degeneracies, althoughthey always present the value Ψ ∗ Ψ = 0 . J z/U andclose to the phase boundary, the plots coincide.Source: KÜBLER et al. .2 Mean-field approximation Mottn = = = = = = = μ U U Figure 10 – Zero-temperature phase boundaries for bosons in optical lattices from differenttreatments. The nondegenerate theory yields the dashed orange plot, whilethe degenerate one results in the dotted magenta plot. Inside the lobes thesystem is in the Mott insulator phase, while outside the lobes the superfluidphase takes place. The number of particles per site, n , increases from left toright by one per lobe. The three horizontal continuous lines correspond to, frombottom to top, J z/U = 0 .
02 (red),
J z/U = 0 .
08 (blue), and
J z/U = 0 . J z/U = − µ/U , which indicates n = 0, andend at µ/U = 2 .
15. The inset shows the zoomed region between the first twoMott lobes. Source: KÜBLER et al. This chapter is devoted to the development of the Brillouin-Wigner perturbationtheory (BWPT) method to treat the degeneracy-related problems that artificially arisefrom NDPT for the bosonic lattice at zero temperature, e.g. , the condensate densitywould vanish in a region of the quantum phase boundary where no transition occurs,which is a strong evidence of a nonphysical behavior generated by such an erroneoustreatment, the NDPT. So, firstly we develop the fundamentals of the BWPT, whichconsists in establishing a Schödinger-like equation with an effective Hamiltonian that canbe perturbatively expanded up to the desired order, providing a valuable tool in order tocorrect the degeneracy-related problems. Then, we apply the BWPT in the cases wherethe degenerate Hilbert subspace is composed of both one and two states, namely one- andtwo-state approaches. After the evaluation for the condensate density in both treatments,we conclude that the one-state approach results in better results when compared to theNDPT procedure, but not fully overcoming the nonphysical results from the latter. Thence,we found ourselves the necessity of considering a degenerate Hilbert subspace composed oftwo states, that turns out to entirely correct the degeneracy-related problems from NDPT.In addition, we develop a graphical approach for the BWPT method that allows one toeasily calculate higher-order terms in the perturbative expansion. Finally, we consider theeffects of a harmonic trap in the equation of state of the system, i.e. , how the number ofparticles changes with the chemical potential.
We begin by providing a concise summary of the Brillouin-Wigner perturbationtheory.
It amounts to derive an effective Hamiltonian for an arbitrarily chosen Hilbertsubspace, which is characterized by a projection operator ˆ P . To this purpose, we haveto eliminate the complementary Hilbert subspace, which is spanned by the projectionoperator ˆ Q .4.1.1 General formalismLet us start by reformulating the time-independent Schrödinger equation,ˆ H | Ψ n i = E n | Ψ n i , (4.1)with the help of the projection operators. To this end, we insert the unity operatorˆ = ˆ P + ˆ Q on both sides of (4.1), yieldingˆ H ˆ P| Ψ n i + ˆ H ˆ Q| Ψ n i = E n ˆ P| Ψ n i + E n ˆ Q| Ψ n i . (4.2) Chapter 4 The zero-temperature regime
Multiplying the left side of (4.2) by ˆ P and considering the relations ˆ P = ˆ P and ˆ P ˆ Q = 0,we have that ˆ P ˆ H ˆ P| Ψ n i + ˆ P ˆ H ˆ Q| Ψ n i = E n ˆ P| Ψ n i . (4.3)Analogously, if we multiply the left side of (4.2) by ˆ Q and make use of the correspondingrelations ˆ Q = ˆ Q and ˆ Q ˆ P = 0, we also have thatˆ Q ˆ H ˆ P| Ψ n i + ˆ Q ˆ H ˆ Q| Ψ n i = E n ˆ Q| Ψ n i . (4.4)The next step consists in finding a single equation for ˆ P| Ψ n i in a similar shape tothe time-independent Schrödinger-equation. So, in order to eliminate ˆ Q| Ψ n i from (4.3),let us work out Eq. (4.4), ˆ Q ˆ H ˆ P| Ψ n i + ˆ Q ˆ H ˆ Q | Ψ n i = E n ˆ Q| Ψ n i . (4.5)From rearranging and factoring out, it follows thatˆ Q ˆ H ˆ P| Ψ n i = (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) ˆ Q| Ψ n i . (4.6)Thus, a formal solution with respect to ˆ Q| Ψ n i readsˆ Q| Ψ n i = (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ˆ Q ˆ H ˆ P| Ψ n i . (4.7)A further action of ˆ Q results inˆ Q| Ψ n i = ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ˆ Q ˆ H ˆ P| Ψ n i . (4.8)Note that, at this point, we have succeeded in isolating the therm ˆ Q| Ψ n i . Inserting (4.8)into (4.3), we get a single equation for ˆ P| Ψ n i : (cid:20) ˆ P ˆ H ˆ P + ˆ P ˆ H ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ˆ Q ˆ H ˆ P (cid:21) | Ψ n i = E n ˆ P| Ψ n i . (4.9)Splitting the Hamiltonian regarding the perturbation allows one to rewrite (4.9) asˆ P ˆ H ˆ P| Ψ n i + ˆ P (cid:16) ˆ H + λ ˆ V (cid:17) ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ˆ Q (cid:16) ˆ H + λ ˆ V (cid:17) ˆ P| Ψ n i = E n ˆ P| Ψ n i . (4.10)From the fact that ˆ Q ˆ H ˆ P = 0, we finally obtainˆ P (cid:20) ˆ H + λ ˆ V ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ˆ Q λ ˆ V (cid:21) ˆ P| Ψ n i = E n ˆ P| Ψ n i . (4.11)Equation (4.11) represents a single equation for ˆ P| Ψ n i , which represents the basis of theBrillouin-Wigner perturbation theory. The resulting equation (4.11) has the form of a time-independent Schrödinger-equation ˆ P ˆ H eff ˆ P| Ψ n i = E n ˆ P| Ψ n i , (4.12) .1 Brillouin-Wigner perturbation theory with the effective Hamiltonian defined asˆ H eff ≡ ˆ H + λ ˆ V ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ˆ Q ˆ V . (4.13)Another way to represent ˆ H eff isˆ H eff = ˆ H + λ ˆ V + λ ˆ V ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q − λ ˆ Q ˆ V ˆ Q (cid:17) − ˆ Q ˆ V . (4.14)The resolvent ˆ R ( E n ) ≡ h E n − ˆ Q (cid:16) ˆ H + λ ˆ V (cid:17) ˆ Q i − (4.15)can be written as a series expansion of λ in the following way:ˆ R ( E n ) = (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − ∞ X s =0 (cid:20) λ ˆ Q ˆ V ˆ Q (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − (cid:21) s . (4.16)Note the crucial property of (4.16): instead of the unperturbed energy eigenvalues E (0) n , itcontains the full energy eigenvalues E n .Inserting (4.15) into (4.14), it follows thatˆ H eff = ˆ H + λ ˆ V + λ ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V . (4.17)As λ approaches zero, it reproduces the unperturbed Schrödinger equation. The essentialproperty of (4.17) is, however, that E n appears nonlinearly within the resolvent ˆ R ( E n ),Eq. (4.15).Note that the first perturbative order λ ˆ V in (4.17) corresponds to the originalcontribution of ˆ H . In contrast, all higher orders in (4.17) originate from the resolventterm ˆ R ( E n ). In particular, s = 0 gives the second perturbative order, s = 1 producesthe third perturbative order, and so on. This fundamental difference in the origins ofthe perturbative orders was already evident in (4.2), where the term ˆ H ˆ P gave rise tothe zeroth and the first perturbative orders, while the term ˆ H ˆ Q gave rise to all higherorders. In other words, the zeroth and the first perturbative orders are contained withinthe Hilbert subspace P , while for all higher orders, the Hilbert subspace Q must be takeninto account.Now, let us calculate the correction terms of the effective Hamiltonian up to λ .To do so, we evaluate the sum over s in the resolvent formula (4.16) up to second order, s = 2. This way, Eq. (4.17) readsˆ H eff = ˆ H + λ ˆ V + λ ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V + λ ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V + λ ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V ˆ Q ˆ R ( E n ) ˆ Q ˆ V + · · · . (4.18)Here, we have introduced the unperturbed Hamiltonian resolventˆ R ( E n ) ≡ (cid:16) E n − ˆ Q ˆ H ˆ Q (cid:17) − . (4.19) Chapter 4 The zero-temperature regime
Now, let us represent the projection operators as ˆ P = P k ∈P | Ψ (0) k ih Ψ (0) k | andˆ Q = P l ∈Q | Ψ (0) l ih Ψ (0) l | . Using these relations, the matrix elements of the resolvent, Eq.(4.19), yield h Ψ (0) l | ˆ R ( E n ) | Ψ (0) l i = 1 E n − E (0) l , (4.20)where l ∈ Q . Taking into account (4.20) within Eq. (4.18), we obtainˆ H eff = ˆ H + λ ˆ V + λ X l ∈Q ˆ V | Ψ (0) l ih Ψ (0) l | ˆ VE n − E (0) l + λ X l,l ∈Q ˆ V | Ψ (0) l ih Ψ (0) l | ˆ V | Ψ (0) l ih Ψ (0) l | ˆ V (cid:16) E n − E (0) l (cid:17) (cid:16) E n − E (0) l (cid:17) + λ X l,l ,l ∈Q ˆ V | Ψ (0) l ih Ψ (0) l | ˆ V | Ψ (0) l ih Ψ (0) l | ˆ V | Ψ (0) l ih Ψ (0) l | ˆ V (cid:16) E n − E (0) l (cid:17) (cid:16) E n − E (0) l (cid:17) (cid:16) E n − E (0) l (cid:17) + · · · . (4.21)This representation of the effective Hamiltonian ˆ H eff possesses no operator in the denomi-nators, hence it can be used as a starting point for further calculations.Now, we want to determine an equation for the perturbed energies E n . To this end,let us reformulate (4.12) according to X k,k ∈P | Ψ (0) k ih Ψ (0) k | ˆ H eff | Ψ (0) k ih Ψ (0) k | Ψ n i = E n X k ∈P | Ψ (0) k ih Ψ (0) k | Ψ n i . (4.22)Then, multiplying the left-hand side by h Ψ (0) k | , X k,k ∈P h Ψ (0) k | ˆ H eff | Ψ (0) k ih Ψ (0) k | Ψ n i = E n X k,k ∈P h Ψ (0) k | Ψ (0) k ih Ψ (0) k | Ψ n i , (4.23)yields h Ψ (0) k | Ψ n i X k,k ∈P (cid:16) h Ψ (0) k | ˆ H eff | Ψ (0) k i − E n δ k,k (cid:17) = 0 . (4.24)In order to obtain a nontrivial solution of (4.24), h Ψ (0) k | Ψ n i 6 = 0, we have to demanddet (cid:16) h Ψ (0) k | ˆ H eff | Ψ (0) k i − E n δ k,k (cid:17) = 0 , (4.25)where the determinant in (4.25) is performed with respect to k, k ∈ P .4.1.2 Specific casesIn the following, we specialize in the cases where the projection operator ˆ P consistsof either one or two states.4.1.2.1 One-state approachFirstly, let us consider the special case where ˆ P contains only one state, namelyˆ P ≡ | Ψ (0) k ih Ψ (0) k | . (4.26) .2 Degenerate solutions of the mean-field Bose-Hubbard Hamiltonian In this case, where k = k , and considering that k = n , Eq. (4.25) simplifies to E n = h Ψ (0) n | ˆ H eff | Ψ (0) n i . (4.27)Inserting (4.21) into (4.27), we have that E n = E (0) n + λV n,n + λ X l = n V n,l V l,n E n − E (0) l + λ X l,l = n V n,l V l,l V l ,n (cid:16) E n − E (0) l (cid:17) (cid:16) E n − E (0) l (cid:17) + λ X l,l ,l = n V n,l V l,l V l ,l V l ,n (cid:16) E n − E (0) l (cid:17) (cid:16) E n − E (0) l (cid:17) (cid:16) E n − E (0) l (cid:17) + · · · , (4.28)where we have taken into account that h Ψ (0) n | ˆ H | Ψ (0) n i = E (0) n and defined the matrixelements according to V i,j ≡ h Ψ (0) i | ˆ V | Ψ (0) j i .Note that, due to the nonlinear appearance of E n , (4.28) represents a self-consistencyequation for the energies E n . Furthermore, we observe that, up to third order, every powerof λ consists of one single term. Since n = l, l , l , the denominator does not vanish in anysituation, thus no divergence occurs in this perturbative representation of the perturbedenergies E n .4.1.2.2 Two-state approachNow, let us consider the case where the projection operator ˆ P is constituted of twostates: ˆ P ≡ | Ψ (0) k ih Ψ (0) k | + | Ψ (0) k ih Ψ (0) k | . (4.29)Thus, (4.25) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H eff ,k,k − E n H eff ,k,k H eff ,k ,k H eff ,k ,k − E n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (4.30)with the matrix elements defined as H eff ,i,j ≡ h Ψ (0) i | ˆ H eff | Ψ (0) j i . Note thatΓ ≡ H eff ,k,k H eff ,k,k H eff ,k ,k H eff ,k ,k (4.31)represents a 2 × P in (4.29) is composed of twostates. At the end of Chap. 3, by comparing Fig. 10 to Fig. 9, we have concluded that thenondegenerate approach (dashed lines) yields a reasonable quantum phase boundary, butan inconsistent condensate density, while the degenerate approach (dotted lines) yields animproved result for the order parameter, but a worse quantum phase boundary. Therefore,in order to handle both adequately, another approach is necessary. To this end, we stayin the perturbative picture, which already succeeded in reproducing the quantum phaseboundary. So, in order to accurately calculate the order parameter, we will apply theBrillouin-Wigner perturbation theory developed in Sec. 4.1 in the following. Chapter 4 The zero-temperature regime | n i . Hence,its projection operator reads ˆ P = | n ih n | . (4.32)The ground state energy is then identified as E n = h Ψ (0) n | ˆ H eff | Ψ (0) n i . From (4.28), up tothird order in λ , we have that E n = E (0) n + λJ z Ψ ∗ Ψ + λ J z Ψ ∗ Ψ nE n − E (0) n − + n + 1 E n − E (0) n +1 + λ J z (Ψ ∗ Ψ) n (cid:16) E n − E (0) n − (cid:17) + n + 1 (cid:16) E n − E (0) n +1 (cid:17) . (4.33)As already pointed out, let us emphasize that (4.33) represents a self-consistency equationfor E n .4.2.1.1 Quantum phase boundaryNow, we work out the mean-field quantum phase boundary within the one-state approach of the Brillouin-Wigner perturbation theory. To this end, we evaluate ∂E n (Ψ ∗ Ψ) / (Ψ ∂ Ψ ∗ ), with E n being the energy formula from the one-state approach upto third order in λ according to (4.33).We proceed by showing, in a general manner, that we can neglect all terms withpower higher than three in λ . To such a purpose, we can write down a generic structure of E n ( E n , Ψ ∗ Ψ) from Eq. (4.33): E n ( E n , Ψ ∗ Ψ) = α + Ψ ∗ Ψ β + Ψ ∗ Ψ γ γ + Ψ ∗ Ψ γ + ∞ X m =2 (Ψ ∗ Ψ) m k m P ( E n , Ψ ∗ Ψ) . (4.34)The coefficients α , β , γ , γ , γ , and k m are independent of Ψ ∗ Ψ, but they depend on E n ,and P ( E n , Ψ ∗ Ψ) is a polynomial. Performing the differentiation of (4.34), we have that1Ψ ∂E n ( E n , Ψ ∗ Ψ) ∂ Ψ ∗ = β + γ γ ( γ + Ψ ∗ Ψ γ ) + ∞ X m =2 m (Ψ ∗ Ψ) m − k m PP − (Ψ ∗ Ψ) m k m P ∂P∂ Ψ ∗ ! . (4.35)Therefore, the quantum phase boundary yields1Ψ ∂E n ( E n , Ψ ∗ Ψ) ∂ Ψ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ ∗ Ψ=0 = β + γ γ = 0 . (4.36) .2 Degenerate solutions of the mean-field Bose-Hubbard Hamiltonian Here, we see that all corrections from higher-order terms, λ >
2, can be neglected.Consequently, the phase boundary does not change even if higher orders in λ are takeninto account.Comparing (4.36) to the derivative of (4.33), we identify the relevant coefficients: β = λJ z, (4.37a) γ = λ J z h (2 n + 1) E n + ( n − E (0) n − − nE (0) n +1 i , (4.37b) γ = (cid:16) E n − E (0) n +1 (cid:17) (cid:16) E n − E (0) n − (cid:17) . (4.37c)Inserting them into (4.36), we obtain1Ψ ∂E n (Ψ ∗ Ψ) ∂ Ψ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ ∗ Ψ=0 = λJ z + λ z E n − E (0) n − + 2 nE n − nE (0) n +1 + nE (0) n − (cid:16) E n − E (0) n +1 (cid:17) (cid:16) E n − E (0) n − (cid:17) . (4.38)Thus, from (4 .
38) = 0, we achieve the mean-field quantum phase boundary condition
J zU = − λU (cid:16) E n − E (0) n +1 (cid:17) (cid:16) E n − E (0) n − (cid:17) E n − E (0) n − + 2 nE n − nE (0) n +1 − nE (0) n − , (4.39)which turns out to be identical to the one obtained from RSPT in Fig. 10 (dashed orangeline).4.2.1.2 Energy and condensate densityIn order to calculate the energy and the condensate density within the one-stateapproach, we make use of ∂E n / (Ψ ∂ Ψ ∗ ) = 0 from (4.33),0 = (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n +1 (cid:17) + λJ z h n (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n +1 (cid:17) + ( n + 1) (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n +1 (cid:17)(cid:21) + 2 λ J z Ψ ∗ Ψ (cid:20) n (cid:16) E n − E (0) n +1 (cid:17) + ( n + 1) (cid:16) E n − E (0) n − (cid:17) (cid:21) , (4.40)and Eq. (4.33) itself up to second order in λ ,0 = (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n +1 (cid:17) (cid:16) E (0) n − E n + λJ z Ψ ∗ Ψ (cid:17) + λ J z Ψ ∗ Ψ h n (cid:16) E n − E (0) n +1 (cid:17) + ( n + 1) (cid:16) E n − E (0) n − (cid:17)i . (4.41)Both (4.40) and (4.41) are now used to calculate the ground state energy E n and thecondensate density Ψ ∗ Ψ.The corrections on the energy are obtained by subtracting the unperturbed energyfrom the perturbed energy. From zeroth to second order, the corrections amount to +1 . − . − . J z/U theconvergence turns out to be slower. Chapter 4 The zero-temperature regime ◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼ - - ε U Ψ * Ψ Figure 11 – Condensate density from the one-state approach for n = 1 (negative ε/U ,purple squares) and n = 2 (positive ε/U , red circles), with the hoppingstrength of J z/U = 0 . et al. The condensate density Ψ ∗ Ψ follows from iteratively solving both (4.40) and (4.41).The results are plotted in Fig. 11 for µ = U n + ε , λ = 1, and J z/U = 0 .
08. We observethat the order parameter obtained from the Brillouin-Wigner perturbation theory for theone-state approach according to Fig. 11 is better than the one obtained from Rayleigh-Schrödinger perturbation theory, where the order parameter vanishes at the degeneracy,as in Fig. 9. Nevertheless, the order parameter depicted in Fig. 11 is discontinuous at ε/U = 0. Therefore, we conclude that it does not represent a physically acceptable result.4.2.2 Two-state approachIn the following, we will consider the degenerate Hilbert subspace composed of twostates, | Ψ (0) n i and | Ψ (0) n +1 i . This choice is motivated due to the degeneracy present betweentwo consecutive Mott lobes in the zero-temperature phase diagram of the Bose-Hubbardmodel. Any state vector is projected into the referred subspace through the projectionoperator ˆ P = | n ih n | + | n + 1 ih n + 1 | . (4.42)4.2.2.1 Quantum phase boundaryIn order to calculate the mean-field quantum phase boundary via the two-stateapproach, we start by evaluating the entries of the matrix (4.31). Up to fourth order, they .2 Degenerate solutions of the mean-field Bose-Hubbard Hamiltonian read Γ , = E (0) n + λJ z Ψ ∗ Ψ + λ J z Ψ ∗ Ψ nE n − E (0) n − − λJ z Ψ ∗ Ψ (4.43a)+ λ J z (Ψ ∗ Ψ) n ( n − (cid:16) E n − E (0) n − − λJ z Ψ ∗ Ψ (cid:17) (cid:16) E n − E (0) n − − λJ z Ψ ∗ Ψ (cid:17) , (4.43b)Γ , = Γ ∗ , = − λJ z Ψ ∗ √ n + 1 , (4.43c)Γ , = E (0) n +1 + λJ z Ψ ∗ Ψ + λ J z Ψ ∗ Ψ ( n + 2) E n − E (0) n +2 − λJ z Ψ ∗ Ψ (4.43d)+ λ J z (Ψ ∗ Ψ) ( n + 2) ( n + 3) (cid:16) E n − E (0) n +2 − λJ z Ψ ∗ Ψ (cid:17) (cid:16) E n − E (0) n +3 − λJ z Ψ ∗ Ψ (cid:17) . (4.43e)To calculate the phase boundary, we perform1Ψ ∂ | Γ − I E n | ∂ Ψ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ ∗ Ψ=0 = λJ z h(cid:16) E (0) n − E n (cid:17) + (cid:16) E (0) n +1 − E n (cid:17) − λJ z ( n + 1) i + λ J z ( n + 2) (cid:16) E (0) n − E n (cid:17) E n − E (0) n +2 + n (cid:16) E (0) n +1 − E n (cid:17) E n − E (0) n − = 0 , (4.44)resulting in J zU = − (cid:16) E n − E (0) n − E (0) n +1 (cid:17) (cid:16) E n − E (0) n +2 (cid:17) (cid:16) E n − E (0) n − (cid:17) λnU (cid:16) E n − E (0) n +1 (cid:17) (cid:16) E n − E (0) n +2 (cid:17) + λU h ( n + 1) (cid:16) E n − E (0) n +2 (cid:17) + ( n + 2) (cid:16) E n − E (0) n (cid:17)i , (4.45)which is the mean-field phase boundary. All higher-order corrections drop out of theformula if we set Ψ ∗ Ψ = 0. Thus, the phase boundary does not change even if higherorders in λ are taken into account. To determine the perturbed energies E n , we calculatethe determinant of Γ − I E n and set Ψ ∗ Ψ = 0, which is effectively equivalent to calculatethe matrix up to zeroth order. Hence, the roots ofdet (Γ − I E n ) = (cid:16) E (0) n − E n (cid:17) (cid:16) E (0) n +1 − E n (cid:17) = 0 (4.46)are given by E n = E (0) n and E n = E (0) n +1 . Consequently, the mean-field phase boundary(4.45) calculated with λ = 1 agrees with the previous result from Eq. (4.39). By usingthe explicit forms of the unperturbed energies (3.35a) together with µ = U + ε , the twopossible solutions for the perturbed energies are given by E U = E (0)1 = − (cid:18) εU (cid:19) , (4.47)and E U = E (0)2 = − (cid:18) εU (cid:19) . (4.48)These two energies are depicted in Fig. 8 and they yield the corresponding lowest energieswithin the first and the second Mott lobes, i.e. , for − < ε/U < E is the minimal Chapter 4 The zero-temperature regime energy, while for 0 < ε/U < E turns into the lowest one. Therefore, in order toevaluate the phase boundary, we insert (4.47) and (4.48) into (4.45). According to Fig.8, we conclude that E gives rise to the first lobe, while E gives rise to the second one,originating the Mott-lobe structure from Fig. 10.4.2.2.2 Energy and particle densityNow, we proceed to numerically calculating the perturbed ground state energies E n from the two conditions det (Γ − I E n ) = 0 , (4.49a)1Ψ ∂ | Γ − I E n | ∂ Ψ ∗ = 0 , (4.49b)with the Γ entries given by (4.43). The perturbed ground state energy E n is then determinedby iteratively solving both (4.49a) and (4.49b), resulting in Fig. 12, where the groundstate energy E n is depicted as a function of the chemical potential. The calculationcorresponds to λ = 1. Note that, as we are evaluating the superfluid energy, the missingdata corresponds to Mott insulating regions. ○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ××××× ××××××××××× ×××××××××××× μ U E n U + + μ U (a) ×××××××××××××××××××××××××××××××××××××××××× ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - ε U E n U + + ε U (b) Figure 12 – Perturbed ground state energies E n /U up to O ( λ ) between the Mott lobes,inside the superfluid regions, for three different hopping values: J z/U = 0 . J z/U = 0 .
08 (blue crosses), and
J z/U = 5 − √ ≈ .
101 (greenrings). At
J z/U = 5 − √
6, the second lobe achieves its tip. (a) Superfluidenergies between the first two Mott lobes. For a better visualization, thelinear equation 0 .
15 + 1 . µ/U , which scales the outmost points of the greenplot to zero, is added to the energy. (b) Zoomed region centered around thedegeneracy by introducing µ = U + ε . For a better visualization, the linearequation 1 .
15 + 1 . ε/U , which scales the outmost points of the green plot tozero, is added to the energy.Source: KÜBLER et al. Now, we proceed to calculating the particle densities regarding both the superfluidand the Mott insulator. As we have already explained, a fundamental feature of the Mott .2 Degenerate solutions of the mean-field Bose-Hubbard Hamiltonian insulator phase is its integer occupation number, i.e. , the particle densities for the Mottinsulator regions are n = 1 within the first lobe, n = 2 within the second one, and so on.Within the superfluid regions, the particle densities must be evaluated from the previouslycalculated energies via − ∂E n /∂µ . Therefore, by doing so, we achieve the results depictedin Fig. 13 for two different hopping values. We can observe the effect that the increase ofthe hopping produces on the curves, they become smoother. Such an outcome has a clearinterpretation: the increase of the hopping has a direct consequence on the single-particleenergies, also increasing them, thus boosting the probabilities of the particles to hop fromone site to a neighboring one. Consequently, the on-site characteristic occupation numberchanges from an integer value to a real one. μ U - ∂ E n ∂μ - - (a) J z/U = 0 . μ U - ∂ E n ∂μ - - (b) J z/U = 0 . Figure 13 – Particle densities − ∂E n /∂µ as functions of the chemical potential µ/U ac-cording to the corresponding hopping values. Horizontal lines correspondto Mott-insulating regions, while ascending curves correspond to superfluidregions. The higher the hopping, the rounder the curves become.Source: KÜBLER et al. ∗ Ψ are presented in Figs.14 and 15. The data start at the phase boundary on the first Mott lobe, n = 1, and end atthe phase boundary on the second Mott lobe, n = 2. Note that these different values ofthe occupation number n are taken into account by the structure of the matrix entries(4.43). Thus, evaluating the matrix elements with n = 1, we get the physical results forthe right half of the Mott lobe, while for the left half of the Mott lobe we must performthe calculation with n = 2. Also, we observe that every corresponding condensate densitydata has a maximum at ε/U > ∗ Ψ over ε/U for two different hoppingvalues and for four different orders in λ . As we can observe, the relative error between thecondensate densities from O ( λ ) and O ( λ ) is about 0 . λ . Chapter 4 The zero-temperature regime ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ - - ε U Ψ * Ψ (a) J z/U = 0 . ◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ - - ε U Ψ * Ψ (b) J z/U = 0 . Figure 14 – Condensate densities as functions of ε/U = µ/U − O ( λ ) (red circles), O ( λ ) (blue squares), O ( λ ) (green rings), and O ( λ ) (purple triangles). For small values of J z/U ,and thus close to the degeneracy, the third- (green rings) and fourth-order(purple triangles) data coincide.Source: KÜBLER et al. - - - ε U Ψ * Ψ Figure 15 – Condensate densities Ψ ∗ Ψ as functions of ε/U = µ/U − O ( λ )between the first and the second Mott lobes for different hopping values: from J z/U = 0 .
01 (innermost points) until
J z/U = 0 .
20 (outermost points) with astep size of 0.01. Source: KÜBLER et al.
Fig. 15 illustrates the condensate densities Ψ ∗ Ψ as functions of ε/U for twentydifferent hopping strengths. Considering the hopping values from
J z/U = 0 .
01 (pinkdots) to
J z/U = 0 .
09 (purple dots), the data behave very similarly to parabolas. For
J z/U = 5 − √ ≈ .
101 (blue dots), the second Mott lobe achieves its tip, and the datatouches the ε/U axis at positive ε/U . From
J z/U = 0 .
11 (pink dots) up to
J z/U = 0 . ε/U , the data present a minimum, while forthe negative ε/U region, the data intersect the corresponding axis. For J z/U = 3 − √ ≈ .3 Graphical approach .
172 (orange dots), which corresponds to the tip of the first lobe, the data touches the ε/U axis at negative ε /U. From J z/U = 0 .
18 (red dots) up to
J z/U = 0 .
20 (blue dots),for which the system is found to be deeply in the superfluid phase, the whole graph ismonotonically increasing. Finally, note that this is a representation of the condensatedensity Ψ ∗ Ψ that provides nonzero and continuous results at the degeneracy, an outcomethat was not obtained by the Rayleigh-Schrödinger perturbation theory (see Fig. 9) norby the Brillouin-Wigner one-state approach (see Fig. 11). Therefore, we conclude that thecondensate density out of the Brillouin-Wigner two-state approach is the most appropriatechoice and is the one that should be used for further calculations.4.2.2.4 ComparisonBy comparing our developed BWPT to the numerical diagonalization of the Bose-Hubbard Hamiltonian performed in M. Kübler et al. (2019) , we find a good convergenceat small hoppings. In Fig. 16, the uppermost curve (blue line) stems from the numericalcalculation, while the remaining curves correspond to different orders from the BWPT.As we can observe, the one-state energy is quasi-exact at small hopping values. Moreover,as the energies from the one-state and the two-state approaches coincide, the two-stateapproach energy can also be considered as quasi-exact within the small hopping regime. Ψ U - - - - - - - FN S - Jz Ψ Figure 16 – Ground-state energy E out of the one-state approach for µ = 0 . U . Fromtop to bottom, the curves represent the numerical diagonalization calculation(blue line) as well as the perturbative analytical calculations up to O ( λ )(yellow line), O ( λ ) (red line), and O ( λ ) (green line). Here, N s representsthe number of lattice sites and F stands for the zero-temperature free energy.Source: KÜBLER et al. In order to evaluate (4.30), it is mandatory to evaluate the matrix elements (4.31).It is possible to observe from the effective Hamiltonian form in Eq. (4.21) that, for higherorders in λ , there is an increase in the algebraic difficulty in calculating such terms. So, Chapter 4 The zero-temperature regime for the sake of simplifying the evaluation of higher-order terms of (4.21), we work out anefficient graphical approach.In particular, for the mean-field Hamiltonian (3.7), we work out, within the two-state approach, a graphical representation for the j th-order terms H ( j )eff ,k,k according tothe expansion H eff ,k,k = X j H ( j )eff ,k,k , (4.50)which is depicted in Fig. 17. The first row of Fig. 17 represents the orders in λ for therespective correction terms. In the first column we have different states ranging from n − n + 4. Within the two-state matrix approach we choose ˆ P = ˆ P n + ˆ P n +1 , once there is adegeneracy between two consecutive Mott lobes in the zero-temperature phase diagram ofthe Bose-Hubbard model.Figure 17 – Graphical approach for the matrix elements (4.31) from the effective Hamil-tonian (4.21) for the Bose-Hubbard mean-field Hamiltonian (3.7) up to fifthorder in the hopping term for the two-state approach.Source: KÜBLER et al. In order to obtain all possible graphs in Fig. 17, we have to take into account thefollowing empirical rules:1. Since ˆ V is linear in ˆ a and ˆ a † in Eq. (3.9), we can only go from one state to one of itsnearest-neighbor states;2. Because the effective Hamiltonian ˆ H eff in (4.13) contains only the projection oper-ator ˆ Q , but it is sandwiched by the projection operator ˆ P according to (4.12), itfollows that only the first and the last states are allowed to be within P , while theintermediate states must be contained in Q .We interpret each graph according to the following rules:• The starting point of every graph corresponds to S ( m ) = E n − E (0) m , (4.51)with m being the state we start the graph in. .3 Graphical approach • Every line in the graph corresponds to the following terms. Ascending lines correspondto L A ( m ) = − λJ z Ψ √ m + 1 E n − E (0) m , (4.52)with m being the state the line started in.• Descending lines correspond to L D ( m ) = − λJ z Ψ ∗ √ mE n − E (0) m . (4.53)• Horizontal lines correspond to L H ( m ) = λJ z Ψ ∗ Ψ E n − E (0) m . (4.54)The off-diagonal matrix elements vanish for all orders except for O ( λ ): H (1)eff ,n,n +1 = S ( n + 1) L D ( n + 1) = − λJ z Ψ ∗ √ n + 1 , (4.55) H (1)eff ,n +1 ,n = S ( n ) L A ( n ) = − λJ z Ψ √ n + 1 . (4.56)Now we proceed to evaluating the diagonal matrix elements for ascending orders of λ . For O ( λ ) we have H (1)eff ,n,n = S ( n + 1) L H ( n + 1) = λJ z Ψ ∗ Ψ , (4.57) H (1)eff ,n +1 ,n +1 = S ( n ) L H ( n ) = λJ z Ψ ∗ Ψ . (4.58)For O ( λ ) we have, correspondingly, H (2)eff ,n,n = S ( n + 1) L A ( n + 1) L D ( n + 2) = λ J z Ψ ∗ Ψ n + 2 E n − E (0) n +2 (4.59)and H (2)eff ,n +1 ,n +1 = S ( n ) L D ( n ) L A ( n −
1) = λ J z Ψ ∗ Ψ nE n − E (0) n − . (4.60)For O ( λ ) one obtains H (3)eff ,n,n = S ( n + 1) L A ( n + 1) L H ( n + 2) L D ( n + 2) = λ J z Ψ ∗ Ψ n + 2 (cid:16) E n − E (0) n +2 (cid:17) (4.61)together with H (3)eff ,n +1 ,n +1 = S ( n ) L D ( n ) L H ( n − L A ( n −
1) = λ J z Ψ ∗ Ψ n (cid:16) E n − E (0) n − (cid:17) . (4.62)For O ( λ ) we find H (4)eff ,n,n = S ( n + 1) L A ( n + 1) [ L A ( n + 2) L D ( n + 3) + L H ( n + 2) L H ( n + 2)] L D ( n + 2)= λ J z Ψ ∗ Ψ ( n + 2) ( n + 3) (cid:16) E n − E (0) n +2 (cid:17) (cid:16) E n − E (0) n +3 (cid:17) + λ J z Ψ ∗ Ψ n + 2 (cid:16) E n − E (0) n +2 (cid:17) (4.63) Chapter 4 The zero-temperature regime and H (4)eff ,n +1 ,n +1 = S ( n ) L D ( n ) [ L D ( n − L A ( n −
2) + L H ( n − L H ( n − L A ( n − λ J z Ψ ∗ Ψ n ( n − (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n − (cid:17) + λ J z Ψ ∗ Ψ n (cid:16) E n − E (0) n − (cid:17) . (4.64)Finally, the fifth column, corresponding to O ( λ ), results in H (5)eff ,n,n = S ( n + 1) L A ( n + 1) [ L A ( n + 2) L H ( n + 3) L D ( n + 3)+ L H ( n + 2) L H ( n + 2) L H ( n + 2)+ 2 L A ( n + 2) L D ( n + 3) L H ( n + 2)] L D ( n + 2)= λ J z Ψ ∗ Ψ ( n + 2) ( n + 3) (cid:16) E n − E (0) n +2 (cid:17) (cid:16) E n − E (0) n +3 (cid:17) + 2 λ J z Ψ ∗ Ψ ( n + 2) ( n + 3) (cid:16) E n − E (0) n +2 (cid:17) (cid:16) E n − E (0) n +3 (cid:17) + λ J z Ψ ∗ Ψ n + 2 (cid:16) E n − E (0) n +2 (cid:17) , (4.65)together with H (5)eff ,n +1 ,n +1 = S ( n ) L D ( n ) [ L D ( n − L H ( n − L A ( n − L H ( n − L H ( n − L H ( n − L D ( n − L A ( n − L H ( n − L A ( n − λ J z Ψ ∗ Ψ n ( n − (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n − (cid:17) + 2 λ J z Ψ ∗ Ψ n ( n − (cid:16) E n − E (0) n − (cid:17) (cid:16) E n − E (0) n − (cid:17) + λ J z Ψ ∗ Ψ n (cid:16) E n − E (0) n − (cid:17) . (4.66) In view of actual experiments, we consider now the impact of a harmonic confinementupon the equation of state. Although most traps in experiments have an ellipsoidalshape, for simplicity we perform calculations regarding the case of a spherical trap.
Inorder to add a trap to our calculations, we have to perform the so-called Thomas-Fermiapproximation µ = ˜ µ − mω r . (4.67)Here, m denotes the mass of the particles and ω stands for the trap frequency. Thus, thechemical potential is now consisting of a trap term and the original chemical potential ˜ µ . .4 Harmonic trap This strategy effectively gives rise to the same curves as in Fig. 13. We identify˜ µ max as being the center of the trap, while its borders are identified by the vanishing pointsof the condensate density. In between, we have Mott insulating and superfluid regions,which produce, in a three-dimensional trap, a wedding-cake structure with alternatingMott insulating and superfluid shells.
86, 89, 190
In order to identify the curves from Fig. 13 with a real experimental setting, we haveto determine ˜ µ . This can done by integrating the curves from Fig. 13. Such a procedureresults in a curve for the total number of particles, which allows one to determine thecorresponding value of ˜ µ . Thus, the number of particles reads N µ i ,µ o = − a Z ∂E n ∂µ d r = − πa Z R o R i r ∂E n ∂µ dr, (4.68)where the radii R i and R o are the inner and the outer radius of the shell that we wantto compute, respectively. Here, a is the lattice spacing. The following calculation is donefor J z/U = 0 .
101 and 2 ≤ n ≤ . ≤ µ/U ≤ . E n , we have that theintegration argument yields N . , . = − πa Z R R r " − µU + 37 (cid:18) µU (cid:19) − (cid:18) µU (cid:19) + 4 (cid:18) µU (cid:19) − . (cid:18) µU (cid:19) dr, (4.69)with R = s µ − . U ) mω , (4.70a) R = s µ − . U ) mω . (4.70b)The last step consists of inserting (4.67) into (4.69) and perform the integration.The same procedure must be repeated for all other regions of Fig. 13, namely, N . , . , N . , . , N . , . , and N − . , . , which represent the remaining superfluid and Mottinsulating shells, respectively. Then, the total number of particles is obtained by summingall these contributions, N = N − . , . + N . , . + N . , . + N . , . + N . , . . (4.71)The resulting equation of state N = N (˜ µ ) is shown in Fig. 18. For small values of ˜ µ ,the particle number rapidly vanishes. Thus, we conclude that, for a given ˜ µ , the minimalparticle number is not achieved for J z/U = 0, where all particles are in the Mott insulatorphase, nor for
J z/U > . J z/U , which can be determinedby the methods introduced here. Chapter 4 The zero-temperature regime μ ˜ × N Figure 18 – Equations of state N = N (˜ µ ) for the following parameters: m = 87 u , a =400nm, and ω = 48 π Hz. From left to right, the hopping values are:
J z/U =0 .
02 (dashed red line),
J z/U = 0 .
101 (dotted green line), and
J z/U = 0 . et al. In this chapter, we introduce the finite-temperature degenerate perturbation theory(FTDPT) method, which consists of a degenerate perturbative calculation making use ofprojection operators. From the developed FTDPT, we evaluate the condensate densitiesfor different temperatures and hopping values. Following, we turn our attention to aregion between two consecutive Mott lobes, which is a region in the phase diagram wherethe superfluid clearly dominates and also a region where the NDPT fails at very lowtemperatures, i.e. , it predicts a phase transition, even though there should be none. Then,we compare the results from NDPT and FTDPT in order to corroborate the results fromour developed method. Finally, we calculate the particle densities for different temperaturesand hopping values, observing the existence of the melting of the structure due to boththe temperature and the hopping increase.
We begin by considering two adjacent degenerate states | n i and | n + 1 i . Firstly,we let us define the subspace of the Hilbert space in which those two state are located, P .Then, the projection operator that allows us to access the respective Hilbert subspace isgiven by ˆ P ≡ | n ih n | + | n + 1 ih n + 1 | . (5.1)Likewise, we define the complementary Hilbert subspace, Q , in which all the remainingstates are located. Thus, the corresponding complementary operator is defined asˆ Q ≡ X m/ ∈P | m ih m | . (5.2)Let us begin our analysis by considering the one-site mean-field Hamiltonian (3.7)and regard, as in Sec. 3.2.1, the hopping term (3.9) as a perturbation in (3.8). Multiplyingboth sides of the perturbation by the identity operator, ˆ = ˆ P + ˆ Q , we have thatˆ H = ˆ H + (cid:16) ˆ P + ˆ Q (cid:17) ˆ V (cid:16) ˆ P + ˆ Q (cid:17) = ˆ H + ˆ P ˆ V ˆ P + ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q . (5.3)This procedure allows us to define a new unperturbed Hamiltonian and a new perturbationaccording to ˆ H ≡ ˆ H + ˆ P ˆ V ˆ P , (5.4a)ˆ V ≡ ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q . (5.4b) Chapter 5 The system at finite temperature
The Hamiltonian (5.4a), written in the basis of the unperturbed eigenstates, is ablock diagonal matrix, whose only non-diagonal block isˆ H (nd)0 = E n − J z Ψ √ n + 1 − J z Ψ ∗ √ n + 1 E n +1 . (5.5)Its eigenvalues and eigenstates are respectively given by E ± = E n + E n +1 ± h ( E n − E n +1 ) + 4 J z | Ψ | ( n + 1) i / , (5.6a) | Φ ± i = " |E ± − E n | J z | Ψ | ( n + 1) − / | n i + E n − E ± J z q | Ψ | ( n + 1) | n + 1 i . (5.6b)Note that we have dropped out the index (0) for the unperturbed eigenenergies that wereused all along Chap. 4 for the sake of simplicity. Therefore, keep in mind that we use E n instead of E (0) n throughout the entire current chapter.As pointed out in Sec. 3.2.1, we must evaluate the partition function (3.11) in orderto calculate the free energy (3.32). The only difference is that now we are working withthe new unperturbed Hamiltonian (5.4a) and the new perturbation (5.4b). With this, thetime-evolution operator now reads ˆ U = e − β ˆ H ˆ U I . (5.7)The respective Schrödinger equation for the time-evolution operator in the interactionpicture is given by d ˆ U I ( τ ) dτ = − ˆ V I ( τ ) ˆ U I ( τ ) , (5.8)where ˆ V I ( τ ) = e τ ˆ H (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) e − τ ˆ H . (5.9)The solution of equation (5.8) with the initial condition ˆ U I (0) = ˆ , is given by, upto second order, ˆ U I ( β ) ≈ ˆ − Z β dτ ˆ V I ( τ ) + Z β dτ Z τ dτ ˆ V I ( τ ) ˆ V I ( τ ) . (5.10)Evaluating the partition function Z = Tr h e − β ˆ H ˆ U I ( β ) i , we have Z = e − β E + h Φ + | ˆ U I ( β ) | Φ + i + e − β E − h Φ − | ˆ U I ( β ) | Φ − i + X m ∈Q e − βE m h m | ˆ U I ( β ) | m i . (5.11)Considering only the zeroth-order term from (5.10) into (5.11) yields Z (0) = e − β E + + e − β E − + X m ∈Q e − βE m . (5.12)Furthermore, from (3.9), (5.9), and (5.4b), we read off that the first-order contribution in(5.11) must vanish. .1 The projection operators method Now we proceed to performing the calculation of the second-order term, Z (2) = e − β E + Z β dτ Z τ dτ h Φ + | ˆ V I ( τ ) ˆ V I ( τ ) | Φ + i + e − β E − Z β dτ Z τ dτ h Φ − | ˆ V I ( τ ) ˆ V I ( τ ) | Φ − i + X m ∈Q e − βE m Z β dτ Z τ dτ h m | ˆ V I ( τ ) ˆ V I ( τ ) | m i . (5.13)We shall calculate each term separately and identify them as Z (2) = Z (2)+ + Z (2) − + Z (2) m . Asthe evaluation of Z (2)+ and Z (2) − are completely equivalent, we perform a generic calculationfor both contributions. Inserting the expression of the interaction-picture perturbation,(5.9), into the first term, we have Z (2) ± = e − β E ± Z β dτ Z τ dτ h Φ ± | e τ ˆ H (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) e − τ ˆ H × e τ ˆ H (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) e − τ ˆ H | Φ ± i . (5.14)As | Φ ± i are eigenstates of ˆ H , we get Z (2) ± = e − β E ± Z β dτ Z τ dτ e ( τ − τ ) E ± h Φ ± | (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) e − τ ˆ H × e τ ˆ H (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) | Φ ± i . (5.15)Also, from the projection relations ˆ Q| Φ ± i = 0 and ˆ P| Φ ± i = | Φ ± i , and having in mindthat ˆ Q and ˆ P are hermitian operators, (5.15) reduces to Z (2) ± = e − β E ± Z β dτ Z τ dτ e ( τ − τ ) E ± h Φ ± | ˆ V ˆ Q e − τ ˆ H e τ ˆ H ˆ Q ˆ V | Φ ± i . (5.16)From (3.9), (5.6b), and the scalar products h n | Φ ± i = " |E ± − E n | J z | Ψ | ( n + 1) − / , (5.17a) h n + 1 | Φ ± i = " |E ± − E n | J z | Ψ | ( n + 1) − / E n − E ± J z q | Ψ | ( n + 1) , (5.17b)we have that Z (2) ± = e − β E ± Z β dτ Z τ dτ e ( τ − τ ) E ± J z (cid:16) Ψ h Φ ± | n i√ n h n − | +Ψ ∗ h Φ ± | n + 1 i√ n + 2 h n + 2 | (cid:17) e − τ ˆ H e τ ˆ H × (cid:16) Ψ ∗ h n | Φ ± i√ n | n − i + Ψ h n + 1 | Φ ± i√ n + 2 | n + 2 i (cid:17) . (5.18)Thus, evaluating (5.18) leads to Z (2) ± = J z | Ψ | e − β E ± Z β dτ Z τ dτ (cid:18) e ( τ − τ )∆ ± ,n − n (cid:12)(cid:12)(cid:12) h Φ ± | n i (cid:12)(cid:12)(cid:12) +e ( τ − τ )∆ ± ,n +2 ( n + 2) (cid:12)(cid:12)(cid:12) h Φ ± | n + 1 i (cid:12)(cid:12)(cid:12) (cid:19) , (5.19) Chapter 5 The system at finite temperature where we have introduced the abbreviation ∆ i, ± ≡ E i − E ± . Finally, performing theintegrations in (5.19), the term Z (2) ± results in Z (2) ± = J z | Ψ | e − β E ± " n (cid:12)(cid:12)(cid:12) h Φ ± | n i (cid:12)(cid:12)(cid:12) e β ∆ ± ,n − − ± ,n − − β ∆ ± ,n − ! +( n + 2) (cid:12)(cid:12)(cid:12) h Φ ± | n + 1 i (cid:12)(cid:12)(cid:12) e β ∆ ± ,n +2 − ± ,n +2 − β ∆ ± ,n +2 ! . (5.20)The last term to be calculated is Z (2) m . The first steps of this calculation are similarto those from the evaluation of Z (2) ± . Therefore, we have Z (2) m = X m ∈Q e − βE m Z β dτ Z τ dτ h m | e τ ˆ H (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) e − τ ˆ H × e τ ˆ H (cid:16) ˆ P ˆ V ˆ Q + ˆ Q ˆ V ˆ P + ˆ Q ˆ V ˆ Q (cid:17) e − τ ˆ H | m i = X m ∈Q e − βE m Z β dτ Z τ dτ e ( τ − τ ) E m h m | ˆ V e − τ ˆ H e τ ˆ H ˆ V | m i = J z X m ∈Q e − βE m Z β dτ Z τ dτ e ( τ − τ ) E m (cid:16) Ψ √ m h m − | + Ψ ∗ √ m + 1 h m + 1 | (cid:17) × e − τ ˆ H e τ ˆ H (cid:16) Ψ ∗ √ m | m − i + Ψ √ m + 1 | m + 1 i (cid:17) . (5.21)Applying the exponential operators to the eigenstates, we are left with Z (2) m = J z X m ∈Q e − βE m Z β dτ Z τ dτ e ( τ − τ ) E m × (cid:20) Ψ √ m (cid:18) e − τ E + h m − | Φ + ih Φ + | + e − τ E − h m − | Φ − ih Φ − | + X m ∈Q e − τ E m h m − | m ih m | (cid:19) + Ψ ∗ √ m + 1 (cid:18) e − τ E + h m + 1 | Φ + ih Φ + | + e − τ E − h m + 1 | Φ − ih Φ − | + X m ∈Q e − τ E m h m + 1 | m ih m | (cid:19)(cid:21) × (cid:20) Ψ ∗ √ m (cid:18) e τ E + h Φ + | m − i| Φ + i + e τ E − h Φ − | m − i| Φ − i + X m ∈Q e τ E m h m | m − i| m i (cid:19) + Ψ √ m + 1 (cid:18) e τ E + h Φ + | m + 1 i| Φ + i + e τ E − h Φ − | m + 1 i| Φ − i + X m ∈Q e τ E m h m | m + 1 i| m i (cid:19)(cid:21) . (5.22)When we evaluate the multiplication among the terms between brackets, we must beaware of the fact that the cross terms, i.e. , those that contain Ψ or Ψ ∗ vanish since theycontain the products h m − | Φ ± i and h m + 1 | Φ ± i , which cannot be both non-zero becauseit is not possible that m + 1 and m − n or n + 1. With this, .1 The projection operators method we have that Z (2) m = J z | Ψ | X m ∈Q e − βE m Z β dτ Z τ dτ m e ( τ − τ )∆ m, + (cid:12)(cid:12)(cid:12) h Φ + | m − i (cid:12)(cid:12)(cid:12) + m e ( τ − τ )∆ m, − (cid:12)(cid:12)(cid:12) h Φ − | m − i (cid:12)(cid:12)(cid:12) + m X m ∈Q e ( τ − τ )∆ m,m (cid:12)(cid:12)(cid:12) h m − | m i (cid:12)(cid:12)(cid:12) + ( m + 1)e ( τ − τ )∆ m, + (cid:12)(cid:12)(cid:12) h Φ + | m + 1 i (cid:12)(cid:12)(cid:12) + ( m + 1)e ( τ − τ )∆ m, − (cid:12)(cid:12)(cid:12) h Φ − | m + 1 i (cid:12)(cid:12)(cid:12) + ( m + 1) X m ∈Q e ( τ − τ )∆ m,m (cid:12)(cid:12)(cid:12) h m + 1 | m i (cid:12)(cid:12)(cid:12) . (5.23)Finally, the integrations lead to Z (2) m = J z | Ψ | X m ∈Q e − βE m m (cid:12)(cid:12)(cid:12) h Φ + | m − i (cid:12)(cid:12)(cid:12) e β ∆ m, + − m, + − β ∆ m, + + m (cid:12)(cid:12)(cid:12) h Φ − | m − i (cid:12)(cid:12)(cid:12) e β ∆ m, − − m, − − β ∆ m, − + m X m ∈Q e β ∆ m,m − m,m − β ∆ m,m (cid:12)(cid:12)(cid:12) h m − | m i (cid:12)(cid:12)(cid:12) + ( m + 1) (cid:12)(cid:12)(cid:12) h Φ + | m + 1 i (cid:12)(cid:12)(cid:12) e β ∆ m, + − m, + − β ∆ m, + + ( m + 1) (cid:12)(cid:12)(cid:12) h Φ − | m + 1 i (cid:12)(cid:12)(cid:12) e β ∆ m, − − m, − − β ∆ m, − + ( m + 1) X m ∈Q e β ∆ m,m − m,m − β ∆ m,m (cid:12)(cid:12)(cid:12) h m + 1 | m i (cid:12)(cid:12)(cid:12) . (5.24)Combining the contributions (5.20) and (5.24), the second-order term of the Chapter 5 The system at finite temperature partition function reads Z (2) = J z | Ψ | e − β E + n (cid:12)(cid:12)(cid:12) h Φ + | n i (cid:12)(cid:12)(cid:12) e β ∆ + ,n − − ,n − − β ∆ + ,n − + ( n + 2) (cid:12)(cid:12)(cid:12) h Φ + | n + 1 i (cid:12)(cid:12)(cid:12) e β ∆ + ,n +2 − ,n +2 − β ∆ + ,n +2 + J z | Ψ | e − β E − n (cid:12)(cid:12)(cid:12) h Φ − | n i (cid:12)(cid:12)(cid:12) e β ∆ − ,n − − − ,n − − β ∆ − ,n − + ( n + 2) (cid:12)(cid:12)(cid:12) h Φ − | n + 1 i (cid:12)(cid:12)(cid:12) e β ∆ − ,n +2 − − ,n +2 − β ∆ − ,n +2 + J z | Ψ | X m ∈Q e − βE m m (cid:12)(cid:12)(cid:12) h Φ + | m − i (cid:12)(cid:12)(cid:12) e β ∆ m, + − m, + − β ∆ m, + + m (cid:12)(cid:12)(cid:12) h Φ − | m − i (cid:12)(cid:12)(cid:12) e β ∆ m, − − m, − − β ∆ m, − + m X m ∈Q e β ∆ m,m − m,m − β ∆ m,m (cid:12)(cid:12)(cid:12) h m − | m i (cid:12)(cid:12)(cid:12) + ( m + 1) (cid:12)(cid:12)(cid:12) h Φ + | m + 1 i (cid:12)(cid:12)(cid:12) e β ∆ m, + − m, + − β ∆ m, + + ( m + 1) (cid:12)(cid:12)(cid:12) h Φ − | m + 1 i (cid:12)(cid:12)(cid:12) e β ∆ m, − − m, − − β ∆ m, − + ( m + 1) X m ∈Q e β ∆ m,m − m,m − β ∆ m,m (cid:12)(cid:12)(cid:12) h m + 1 | m i (cid:12)(cid:12)(cid:12) . (5.25)Now, taking into account that the scalar products h m − | m i and h m + 1 | m i lead to onefurther restriction each in the summations, we finally obtain Z (2) = J z | Ψ | ( n + 2) β (cid:12)(cid:12)(cid:12) h Φ + | n + 1 i (cid:12)(cid:12)(cid:12) e − β E + − e − βE n +2 ∆ n +2 , + + (cid:12)(cid:12)(cid:12) h Φ − | n + 1 i (cid:12)(cid:12)(cid:12) e − β E − − e − βE n +2 ∆ n +2 , − + nβ (cid:12)(cid:12)(cid:12) h Φ + | n i (cid:12)(cid:12)(cid:12) e − β E + − e − βE n − ∆ n − , + + (cid:12)(cid:12)(cid:12) h Φ − | n i (cid:12)(cid:12)(cid:12) e − β E − − e − βE n − ∆ n − , − + X m ∈Q m = n − ( m + 1) e − βE m +1 − e − βE m ∆ m,m +1 − β e − βE m ∆ m,m +1 + X m ∈Q m = n +2 m e − βE m − − e − βE m ∆ m,m − − β e − βE m ∆ m,m − . (5.26)From Eq. (5.26) we observe that the differences between the degenerate energies .2 Condensate density E n and E n +1 will no longer appear in the denominator of the free energy as it did in theNDPT treatment, thus solving the degeneracy-related problems. Now we turn our attention to the calculation of the condensate density, whichturns out to coincide to the superfluid density within the mean-field approximation.
Our degenerate approach, up to second order, results in the partition function given by Z = Z (0) + Z (2) with (5.12) and (5.26), which is free from any divergence despite of thedegeneracies. Thus, from the partition function, the free energy of the system reads, up tosecond order, F = − β " ln Z (0) + Z (2) Z (0) . (5.27)Hence, we calculate the condensate density | Ψ | by evaluating ∂ F ∂ | Ψ | = 0 . (5.28)Now, applying the above-mentioned procedure for different temperatures andhopping values, the resulting condensate densities are depicted in Fig. 19. In order to checkthe fidelity of the calculated condensate densities, we must observe the phase boundariesevaluated by FTDPT, which emerge from ∂ F ∂ | Ψ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ=0 = 0 . (5.29)Such an operation leads to the same phase diagrams evaluated by NDPT. From Fig. 6, weread off that for small values of J z/U there are bigger portions of values of µ/U where thecondensate density can be evaluated, since we regard the Landau expansion of the orderparameter being only valid in the vicinity of the phase transition, i.e. , the smaller thehopping the bigger the region of the calculated condensate density. Therefore, we concludethat we are able to reliably calculate | Ψ | via FTDPT near the phase boundary in Fig. 19.Furthermore, we observe that for µ/U ∈ N the condensate densities no longer vanish orapproach zero as they do when calculated from NDPT.Let us remark that the results for the zero-temperature condensate density fromFTDPT, which is depicted in Fig. 19(d), are similar to those obtained from BWPT in Fig.15. Also, it is important to note that we have restricted ourselves to the second-order term.This can be explained by the following: in the NDPT approach, the fourth-order term isnecessary for the calculation of the condensate density since it corresponds to the firstnontrivial solution of the extremization equation for the free energy, see Eqs. (3.36) and(3.37). However, this is not the case for our FTDPT due to fact that the exact solution ofthe problem in the projected Hilbert space automatically generates higher-order terms. Chapter 5 The system at finite temperature μ / U Ψ ( a ) μ / U Ψ ( b ) μ / U Ψ ( c ) μ / U Ψ ( d ) Figure 19 – Condensate densities as functions of µ/U evaluated from FTDPT via (5.28)for four different temperatures: (a) β = 5 /U , (b) β = 10 /U , (c) β = 30 /U , and(d) T = 0. Different data styles correspond to different hoppings: J z/U = 0 . J z/U = 0 .
15 (orange squares),
J z/U = 0 . J z/U = 0 .
05 (red triangles), and
J z/U = 0 .
01 (purple inverted triangles).Source: SANT’ANA et al.
Therefore, our FTDPT is an effective resummation of the power series generated by NDPT.In the FTDPT, the second- and higher-order calculations only include extra effects due tothe non-projected Hilbert space. Indeed, it is even possible to calculate the condensatedensity from the zeroth-order term, as one can observe from Eq. (5.12), since it has animplicit dependency on the OP. Therefore, we have restricted ourselves to the second-ordercorrection. In order to check how important the fourth-order corrections would be, wecompared the zero-temperature results from FTDPT to the results from BWPT up tothe fourth-order term in the perturbation. The analogous results, T = 0, are displayed inFigs 15 and 19(d). The errors between the BWPT- and FTDPT-calculated | Ψ | , for thehopping strengths J z/U = 0 .
2, 0 .
15, 0 .
1, 0 .
05, and 0 .
01, are 4 . . . . . .2 Condensate density and FTDPT between the Mott lobes n = 0 and 1, and n = 1 and 2, as shown in Fig. 20.We observe that the NDPT gives rise to condensate densities that approach zero or have adecreasing behavior at the degeneracy point, which correspond to µ/U = 0 for the regionbetween n = 0 and n = 1 and are depicted in Figs. 20(a) and 20(b); while for the regionbetween the first and the second Mott lobes, i.e. , n = 1 and 2, the degeneracy occursat µ/U = 1 and the corresponding condensate densities are depicted in Figs. 20(c) and20(d). Such behavior indicates an inaccuracy of the theory, since it mimics the nonphysicalvanishing of the OP typical of RSPT, which is a direct consequence of not taking intoaccount the degeneracies that happen in between two consecutive Mott lobes. WhileNDPT presents such a nonphysical behavior due to the incorrect treatment of degeneracies,FTDPT gives consistent results for the condensate density between two consecutive Mottlobes. - - μ / U Ψ ( a ) - - μ / U Ψ ( b ) μ / U Ψ ( c ) μ / U Ψ ( d ) Figure 20 – Comparison between the condensate densities calculated via FTDPT (dots)and NDPT (lines) for the temperatures β = 30 /U (left panel) and T = 0 (rightpanel), and for the hoppings J z/U = 0 . J z/U = 0 .
15 (orange squares and dashed orange lines), and
J z/U = 0 . n = 0 and 1, while (c) and (d) correspond to the region between thefirst and second lobes.Source: SANT’ANA et al. We observe from Fig. 20 that the condensate densities calculated via FTDPT,which are represented by the data, do not present any decreasing behavior in the vicinityof the degeneracy, concluding that they are consistent in all considered regions of the phase Chapter 5 The system at finite temperature diagram. In particular, at integer µ/U the condensate densities no longer vanish or presenta decreasing behavior as they do when calculated from NDPT. The decreasing behaviorpresented by the condensate densities calculated via NDPT can clearly be observed bythe curves in Fig. 20. Such decreasing behavior is a direct consequence of the incorrecttreatment of degeneracies by NDPT, which happens to occur between two consecutiveMott lobes.
Now, let us calculate the particle density (3.38) by making use of our developedFTDPT. We consider different temperatures and different hopping values for the purposeof analyzing their effects on the density of particles. We plot the resulting equation ofstate for two different values of the hopping parameter and four different values of thetemperature, thus observing the melting of the structure, as shown in Fig. 21. - - μ / U n ( a ) - - μ / U n ( b ) Figure 21 – Equation of state for the hopping strengths (a)
J z/U = 0 .
05 and (b)
J z/U =0 . T = 0 (continuous blue), β = 30 /U (dotted green), β = 10 /U (dashed red), and β = 5 /U (dotted-dashed black).Source: SANT’ANA et al. We observe the effects that the change of both the temperature and the hoppinghave upon the particle density in Fig. 21. First, we conclude that increasing the temperaturemakes the particle density to vary more smoothly when compared to those particle densitieswith lower temperatures. This fact is due to thermal fluctuations, which make the systemmore feasible to exist in the Mott insulator phase. Moreover, by comparing the left panelto the right one we observe the melting of the Mott lobes due to an increased hopping,which is also very intuitive: the particles, having more kinetic energy, are more likely tohop from one site to another, which is a characteristic of the SF phase. Another factorresponsible for making the curves smoother is the increase of the chemical potential, µ/U .The reason for this relies on the fact that the bigger µ/U becomes, the smaller the Mottlobes are, as can be seen in Fig. 6. Thus, the system is more likely to exist in the superfluidphase for bigger values of µ/U . .3 Particle density Now we must turn our attention to the points of the figures where the degeneracieshappen, which correspond to µ/U = 0 , i.e. , µ/U ∈ N , we also concludethat FTDPT provides reliable results for the particle density since there is no decreasingbehavior or discontinuities in the vicinity of the degeneracies in Fig. 21. The purpose of this chapter is the description of one-dimensional repulsivelyinteracting bosonic particles, also known as Lieb-Liniger gas, trapped in a harmonicconfinement at finite temperature. We begin by briefly discussing the model and thesolutions for the homogeneous gas and the role of the interactions. Then, we study thedetails behind the solutions for the two-particle case, followed by a development of theasymptotic behavior of the momentum distribution. In such an asymptotic context, weintroduce an important physical quantity that gives us valuable short-range insightsabout the system, the so-called
Tan’s contact . Moreover, we take our studies beyond thetwo-particle scenario up to the N -particle system. As our knowledge and abilities to solvethe problem are reduced in such a N > i.e. , for
N >
Tonks-Girardeau gas (TG gas). In this limit, we are able to exactly solve the problem due to the fermionization of the bosons. In the last part of this chapter, we study the scaling properties of theTan’s contact in all ranges of temperatures and in the intermediate- and strong-interactionregimes. To finalize, we compare our analytical results to quantum Monte Carlo (QMC)simulations.
To begin with, let us consider the one-dimensional system consisted of N bosonsof mass m repulsively interacting via a delta potential and confined in a generic potentialof the form V ( x ). The Hamiltonian describing such a system is given byˆ H = N X i =1 − (cid:126) m ∂ ∂x i + V ( x i ) ! + g X i
0. Then, the respectivestationary Schrödinger equation reads N X i =1 − (cid:126) m ∂ ∂x i + V ( x i ) ! + g X i Chapter 6 One-dimensional interacting Bose gas Considering any pair of particles ( i, j ), let us integrate Eq. (6.2) in the vicinity of theinteraction x ij ≡ x i − x j = 0, i.e. , inside the small interval ( − ε, + ε ): Z + ε − ε " − (cid:126) m ∂ ∂x ij + V ( x ij ) + gδ ( x ij ) − E ! Ψ ( x , x , . . . , x N ) dx ij = 0 ⇒ − (cid:126) m ∂ Ψ ∂x ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ε − ε + g Ψ ( x ij = 0) = 0 . (6.3)Therefore, as ε → 0, the contact interaction generates a condition given by ∂ Ψ ∂x i − ∂ Ψ ∂x j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i − x j → + − ∂ Ψ ∂x i − ∂ Ψ ∂x j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i − x j → − = 2 mg (cid:126) Ψ( x i = x j ) , (6.4)which can be interpreted as the following: whenever two particles meet, there happens adiscontinuity in the many-body wave function and it abruptly falls to zero.6.1.1 The Lieb-Liniger gasThe case of free Bosons, V ( x ) = 0, interacting via a delta-like potential is knownas Lieb-Liniger gas and its solutions are given by Ψ( x , . . . , x N ) = X P a ( P ) exp i N X j =1 k j x j , (6.5)where k j ∈ < with k < k < · · · < k N , and the sum in P is taken over all the N !permutations of 1 , , . . . , N . The coefficients read a ( P ) = Y ≤ i 2. With thesechanges the Hamiltonian readsˆ H = − (cid:126) M ∂ ∂x cm + 12 M ω x cm − (cid:126) µ ∂ ∂x r + 12 µω x r + gδ ( x r ) . (6.10)The solutions for the center of mass coordinate equation are the known solutions for theharmonic oscillator, Ψ ( cm ) n ( x cm ) = e − ( x cm /a ) / H n ( x cm /a ) π / √ a n n ! , (6.11)where a = q (cid:126) /M ω is the harmonic oscillator length and H n are the Hermite polynomials, with n ∈ N . The energies are given by E ( cm ) n = ( n + 1 / (cid:126) ω . n = n = n = = - - - - - x cm Ψ n ( c m ) Figure 22 – The fundamental and the first three excited states of the harmonic oscillatorwave function (6.11).Source: By the author.Regarding the equation for the relative coordinate, the solutions for the analoguethree-dimensional system was given by BUSCH et al. Here, we work out the solutionfor the one-dimensional system. Chapter 6 One-dimensional interacting Bose gas Firstly, let us consider the relative motion Hamiltonian given byˆ H ( r ) = − (cid:126) µ ∂ ∂x + 12 µω x + gδ ( x ) . (6.12)Now let us write the wave functions, which are solutions of the respective stationarySchrödinger equation ˆ H ( r ) Ψ ( r ) ν = E ( r ) ν Ψ ( r ) ν , (6.13)as an expansion of the complete set of the known solutions of the harmonic oscillator φ n ( x ): Ψ ( r ) ν ( x ) = ∞ X n =0 c n φ n ( x ) . (6.14)Inserting (6.14) into (6.13) we have ∞ X n =0 c n ( E n − E ν ) φ n ( x ) + gδ ( x ) ∞ X m =0 c m φ m ( x ) = 0 . (6.15)Nota that we have omitted, and will continue omitting during the following analyticaldemonstrations, the identification indexes ( r ) and ( cm ) for the sake of simplicity.Now, multiplying (6.15) by φ ∗ j ( x ) and integrating it all over the real space < , wehave that c n ( E n − E ν ) + gφ ∗ n (0) X m c m φ m (0) = 0 , (6.16)where we have used the orthogonality of the φ ’s Z + ∞−∞ dx φ m ( x ) φ n ( x ) = δ m,n . (6.17)We can observe that the coefficients c n possess the following form: c n = A φ ∗ n (0) E n − E ν , (6.18)with A being a proportionality constant. Therefore, the solutions we are seeking for reduceto Ψ ( r ) ν ( x ) = e − x / ∞ X n =0 φ ∗ n (0) E n − E ν H n ( x ) . (6.19)As the possible energies are given by E ν = ( ν + 1 / (cid:126) ω , we have thatΨ ( r ) ν ( x ) = e − x / ∞ X n =0 φ ∗ n (0)( n − ν ) (cid:126) ω H n ( x ) . (6.20)Now we transform the Hermite polynomials into Laguerre ones through their relationships H n ( x ) = ( − n n n ! L ( − / n ( x ) , (6.21a) H n +1 ( x ) = ( − n n +1 n ! xL (1 / n ( x ) , (6.21b) .2 The two particles case and make use of the integral representation1 n − ν = Z ∞ dy (1 + y ) y y ! n − ν − , (6.22)so that we arrive atΨ ( r ) ν ( x ) = e − x / ∞ X n =0 Z ∞ dy (1 + y ) y y ! n − ν − h L ( − / n ( x ) + xL (1 / n ( x ) i , (6.23)where we have embedded all the constants into the normalization of the wave functionthat we shall deal later.From the generating function of the Laguerre polynomials ∞ X n =0 L ( α ) n ( x ) t n = e − xt/ (1 − t ) (1 − t ) α +1 , (6.24)(6.23) readsΨ ( r ) ν ( x ) = e − x / Z ∞ dy (1 + y ) y y ! − ν − e − yx h (1 + y ) / + x (1 + y ) / i . (6.25)At this point, we are able to recognize the integral representation of the Tricomi hypergeo-metric function U ( a, b, z ) = 1Γ( a ) Z ∞ e − zt t a − (1 + t ) b − a − dt, (6.26)so that Ψ ( r ) ν ( x ) = e − x / Γ( − ν ) h U (cid:16) − ν, / , x (cid:17) + x U (cid:16) − ν, / , x (cid:17)i , (6.27)where Γ( x ) is the Euler gamma function.Before assuming that Eq. (6.27) is our final form of Ψ ( r ) ν , we must remind ourselvesthat we got two solutions from the relations between the Hermite and the Laguerrepolynomials. Hence, let us test these solutions with the help of the condition at x r = 0,Eq. (6.4), which, for the N = 2 case, reduces to ∂ Ψ ( r ) ν ∂x r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x r → + − ∂ Ψ ( r ) ν ∂x r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x r → − = 2 µg (cid:126) Ψ ( r ) ν (0) . (6.28)The first solution unrestrictedly satisfies (6.28), while the second one only satisfies it forinteger values of ν , which is a restriction that we never imposed. This means that at x r = 0, x r U ( − ν, / , x r ) → ∞ , ∀ ν / ∈ N . Therefore, it can not be a solution because thewave function must vanish when two particles meet, Ψ( x = x ) = 0. In conclusion, wehave that the final solution for the relative motion problem is given byΨ ( r ) ν ( x r ) = s N ( ν ) e − ( x r /a ) / U − ν , , x r a ! , (6.29) Chapter 6 One-dimensional interacting Bose gas where now the oscillator length reads a = q (cid:126) /µω . Also, it is worth noting that werescaled the factor ν by 2, and this can be done without loss of generality. We did it inorder to keep the solutions more similar to the solutions of the harmonic oscillator, once H n ( x ) = 2 n U ( − n/ , / , x ). After all, the interacting system is simply the same as theharmonic oscillator except at the contact point x r = 0.The normalization is given by N ( ν ) = 12 Z dx e − x /a (cid:12)(cid:12)(cid:12)(cid:12) U − ν , , x a ! (cid:12)(cid:12)(cid:12)(cid:12) = a √ π − ν − Γ( − ν ) (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) , (6.30)where ψ ( x ) = Γ ( x ) / Γ( x ) is the digamma function. - - - x r Ψ ν ( r ) (a) ˜ g = 0 . - - - x r Ψ ν ( r ) (b) ˜ g = 0 . - - - x r Ψ ν ( r ) (c) ˜ g = 1. - - - x r Ψ ν ( r ) (d) ˜ g = 100. Figure 23 – Relative motion wave functions for different interaction strength ˜ g . Differentcolors correspond to different states, from the fundamental state to the thirdexcited one. In growing order of excitation they correspond to the blue, yellow,green, and red curves.Source: By the author.The behavior of the relative motion wave function is depicted in Fig. 23 for differentvalues of the adimensional interaction parameter ˜ g ≡ − a / √ a . We can observe itsdiscontinuity happening at x = x . The higher the interaction, the more pronounced itbecomes. Also, its expansion around x = 0 isΨ ( r ) ν ( x ) ∼ s π N ( ν ) " Γ (cid:18) − ν (cid:19) − − (cid:18) − ν (cid:19) − | x | a + O ( x ) . (6.31) .3 The momentum distribution asymptotic behavior Conclusively, it becomes clear, not only from Fig. 23, but also from Eq. (6.31), the | x | -behavior of Ψ ( r ) ν ( x ) around x = 0.Inserting (6.29) into (6.28) we obtain the following relation: f ( ν ) ≡ Γ (cid:16) − ν (cid:17) Γ (cid:16) − ν (cid:17) = − g . (6.32)Thus, we are able to find the ν ’s, which are the quantum numbers with respect to therelative motion wave function (6.29), by solving (6.32).In order to have an idea about the behavior of the function f ( ν ) + 1 / ˜ g , we plot itfor different values of ˜ g in Fig. 24. We observe that the solutions of (6.32) for the weaklyand the strongly interacting limits are given by ν ( n ) = n, for ˜ g (cid:28) n + 1 , for ˜ g (cid:29) , ∀ n ∈ N . (6.33) -40-20 0 20 40 0 2 4 6 8 10 12 f( ν ) + / g ~ ν g~ = 0.05g~ = 0.1g~ = 1g~ = 100 -2-1 0 1 2 0 1 2 Figure 24 – f ( ν ) + 1 / ˜ g as a function of ν for different adimensional interaction strengths˜ g . The zoomed inset helps us recognize the vanishing of the function between ν = 0 (weakly interacting limit) and ν = 1 (strongly interacting limit).Source: By the author. Let us begin this section with the N -body wave function solution of (6.2). As the N = 2 case is solved, we can choose any pair of particles in the respective N -body problem Chapter 6 One-dimensional interacting Bose gas so that x ( cm ) ij = ( x i + x j ) / x ( r ) ij = x i − x j , so that the solution is given byΨ( x , . . . , x N ) = Ψ( x , . . . , x N )Ψ ( cm ) (cid:16) x ( cm ) ij (cid:17) Ψ ( r ) (cid:16) x ( r ) ij (cid:17) . (6.34)Also, from the behavior of the relative motion wave function near x i = x j (6.31), we havethat Ψ( x , . . . , x N ) ≈ Ψ (cid:16) x , . . . , x ( cm ) ij , . . . , x N (cid:17) − √ | x ( r ) ij | a + O (cid:18) x ( r ) ij (cid:19) , (6.35)where we have left the normalization factor and other constants out for the sake ofsimplicity. Now, following the developments from Refs. 130, 195, the Fourier transform ofΨ, i.e. , its representation in the momentum space, is given by˜Ψ( k, x , . . . , x N ) = 1 √ π Z dx e − ikx Ψ( x , . . . , x N ) ≈ √ π Z dx e − ikx N X j =2 Ψ (cid:16) x = x ( cm )1 j , . . . , x j = x ( cm )1 j , . . . , x N (cid:17) − √ | x ( r )1 j | a . (6.36)In addition, making use of the asymptotic behavior of the Fourier transform of f ( x ) = f ( x ) | x − x | , with f ( x ) being a smooth function, Z dx e − ikx f ( x ) ∼ k →∞ − k f ( x )e − ikx , (6.37)and that R dx e − ikx f ( x ) falls to zero as O ( k − ), (6.36) reduces to˜Ψ( k, x , . . . , x N ) ∼ k →∞ k − √ πa N X j =2 e − ikx j Ψ ( x = x j , x , . . . , x j , . . . , x N ) . (6.38)Now we proceed to the evaluation of n ( k ) itself: n ( k ) = N Z dx . . . dx N | ˜Ψ( k, x , . . . , x N ) | ∼ k →∞ Nπa k − Z dx . . . dx N N X j,l =2 e − ik ( x j − x l ) Ψ ∗ ( x = x j , x , . . . , x j , . . . , x N ) × Ψ( x = x l , x , . . . , x l , . . . , x N ) . (6.39)Noting that the terms j = l all cancel out, we have n ( k ) ∼ k →∞ Nπa k − Z dx . . . dx N N X j =2 | Ψ( x = x j , x , . . . , x j , . . . , x N ) | . (6.40)Moreover, because of the indistinguishability nature of quantum particles, it is possible towrite the two-body correlation function as % (2) ( x, x ) = Z dx . . . dx N | Ψ( x , . . . , x N ) | X i = j δ ( x − x i ) δ ( x − x j ) . (6.41)Finally, we conclude the asymptotic behavior of the momentum distribution: n ( k ) ∼ k →∞ πa k − Z dx % (2) ( x, x ) . (6.42) .3 The momentum distribution asymptotic behavior C ≡ lim k →∞ k n ( k ) (6.43)together with (6.42) we have that C = 2 πa Z dx % (2) ( x, x ) . (6.44)Now we want to relate the Tan’s contact to the slope of the energy with respect to theinverse of the interaction strength g − . From the Hellmann-Feynman theorem we havethat ∂E∂g − = − g Z dx . . . dx N | Ψ( x , . . . , x N ) | X i 0. We begin by making use of C = − m π (cid:126) ∂ F ∂g − , (6.47)where F = − k B T ln Z is the free energy of the system, with Z = P n,ν e − β ( E ( cm ) n + E ( r ) ν ) beingthe respective partition function and β = 1 /k B T . We must note that only the energiesrelated to the relative motion coordinate depend on g (6.32). Also, making use of ∂ν∂ ˜ g = 1˜ g ∂f ( ν ) ∂ν ! − , (6.48)and evaluating ∂f ( ν ) /∂ν , ∂f ( ν ) ∂ν = 12 Γ (cid:16) − ν (cid:17) Γ (cid:16) − ν + (cid:17) (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) , (6.49)the contact, as a function of ˜ g and β , reads C (˜ g, β ) = 2 / ˜ gπa Z X n,ν e − β ( E ( cm ) n + E ( r ) ν ) (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) − . (6.50) Chapter 6 One-dimensional interacting Bose gas It is straightforward to observe that the contact is independent of the center-of-massenergy E ( cm ) n , which is a direct consequence of the Kohn’s theorem : differently from therelative-motion energy E ( r ) ν , E ( cm ) n is independent of the interatomic interactions. Thence,the contact reduces to C (˜ g, β ) = 2 / ˜ gπa Z r X ν e − βE ( r ) ν (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) − , (6.51)where we have defined the relative-motion partition function Z r ≡ P ν e − βE ( r ) ν . As E ( r ) ν =( ν + 1 / (cid:126) ω , we can perform one more reduction in the contact expression, C (˜ g, β ) = 2 / ˜ gπa Z r X ν e − β (cid:126) ων (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) − . (6.52)Here, for the sake of avoiding the introduction of unnecessary terms, we simply transformthe relative-motion partition function according to Z r → e β (cid:126) ω/ Z r .The two-boson contact from (6.52) is depicted in Fig. 25 for different values ofthe adimensional interaction parameter ˜ g . From the referred curves, we observe that thecontact increases with both the temperature and the interaction strength, an effect thatwas to be expected once both variables contribute to the increase of the interaction energy. C a π k B T/h ω g~ = 0.05g~ = 1g~ = 2.5g~ = 5g~ = 100 Figure 25 – Tan’s contact from (6.52) as a function of the adimensional temperature k B T / (cid:126) ω for different values of the adimensional interaction parameter ˜ g .Source: By the author. .3 The momentum distribution asymptotic behavior Tonks-Girardeau limit (TG limit). II From the two-boson contact of Eq. (6.52) and fromthe fact that ν = 2 j − C (˜ g → ∞ , β ) = 2 / πa Z r X j> e − β (cid:126) ω (2 j − Γ(1 − j )Γ (cid:16) − j (cid:17) (cid:20) ψ (1 − j ) − ψ (cid:18) − j (cid:19)(cid:21) − . (6.53)Because of Γ(1 − j ) → ∞ , ∀ j ∈ N ∗ , as well as ψ (1 − j ) → ∞ , ∀ j ∈ N ∗ , we can performthe approximation ψ (1 − j ) − ψ (cid:16) − j (cid:17) ≈ ψ (1 − j ), so that we are able to restrict ourselvesto evaluating Γ(1 − j ) /ψ (1 − j ). Let us begin with the definition of the gamma functionΓ( z ) ≡ ( z − z ) = Γ( z + n + 1) Q ni =0 ( z + i ) , (6.54)and its derivativeΓ ( z ) = Γ ( z + n + 1) Q ni =0 ( z + i ) − Γ( z + n + 1) n X l =0 z + l ) Q ni =0 ( z + i ) . (6.55)Hence, the digamma function readsΓ ( z )Γ( z ) = Γ ( z + n + 1)Γ( z + n + 1) − n X ‘ =0 z + ‘ . (6.56)Therefore, we have for our inverted-desired ratio the following:Γ ( z )Γ ( z ) = Γ ( z + n + 1)Γ ( z + n + 1) n Y i =0 ( z + i ) − n X ‘ =0 z + ‘ Q ni =0 ( z + i )Γ( z + n + 1) . (6.57)For z = − n , we have that Γ ( − n )Γ ( − n ) = Γ (1)Γ( − n ) − n X ‘ =0 Q ni =0 ( i − n ) ‘ − n . (6.58)Here, all terms vanish except when ‘ = n ,Γ ( − n )Γ ( − n ) = − n − Y i =0 ( i − n ) . (6.59)Therefore, we finally arrive at our desired result:Γ (1 − j )Γ (1 − j ) = ( − j ( j − . (6.60)Consequently, Eq. (6.53) yields C (˜ g → ∞ , β ) = 2 / πa Z r X j> e − β (cid:126) ω (2 j − ( − j Γ (cid:16) − j (cid:17) ( j − . (6.61)The plot of (6.61) as a function of the temperature reproduces the yellow curve (˜ g = 100)in Fig. 25. II Thanks to Patrizia Vignolo for working out these steps.02 Chapter 6 One-dimensional interacting Bose gas C (˜ g, β → ∞ ) = 2 / ˜ gπa (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) − , (6.62)which is depicted in Fig. 26. C a g~ Figure 26 – Zero-temperature Tan’s contact from (6.62) as a function of the adimensionalinteraction parameter ˜ g .Source: By the author.6.3.1.1.3 The Tonks-Girardeau limit in the zero-temperature regimeMoreover, we can observe in Fig. 26 the known zero-temperature strongly interactinglimit as ˜ g → ∞ , lim ˜ g,β →∞ C (˜ g, β ) a = (cid:18) π (cid:19) / . (6.63) N particles In this section we regard the strongly interacting scenario g → ∞ , also known asTonks-Girardeau gas. In such a case, it is known that the relationship between themany-body wave functions of such a bosonic system and a fermionic one is given byΨ ( b ) α ( x , . . . , x N ) = Θ( x , . . . , x N )Ψ ( f ) α ( x , . . . , x N ) , (6.64) .4 The Tonks-Girardeau regime for N particles where Θ( x , . . . , x N ) ≡ Q i>j sgn( x i − x j ) is either +1 or − 1, in order to compensate theanti-symmetrization of the fermionic wave function Ψ ( f ) α , and α is the quantum numberdescribing the particles in a respective set of individual quantum numbers { n , n , . . . , n N } .In the strongly interacting scenario g → ∞ , also called Tonks-Girardeau gas, the systembehaves the same way as the trapped ideal Fermi gas, whose many-body wave function isgiven by the Slater determinantΨ ( f ) α ( x , . . . , x N ) = ( N !) − / det[ φ n i ( x j )] n i ∈{ n ,...,n N } ; x j ∈{ x ,...,x N } , (6.65)where φ n ( x ) are the solutions of the single-particle harmonic oscillator Hamiltonian, φ n ( x ) = e − ( x/a ) / H n ( x/a ) π / √ a n n ! . (6.66)This correspondence between the strongly interacting bosonic system and the noninteract-ing fermionic one is known as fermionization of bosons. The reason behind such aneffect comes from the Pauli exclusion principle , which states that two fermions cannotoccupy the same quantum state. In our scenario, the strong repulsion is the equivalentof such a postulate. Therefore, all these considerations culminate in the fact that, forobservables depending on | Ψ | , strongly interacting bosons behave as the ideal Fermi gas.6.4.1 The zero-temperature caseThe solutions for the Bose gas consisted of N -impenetrable particles (Tonks-Girardeau gas) subject to the periodic boundary condition on a length L at T = 0, wherethe solutions are given by plane waves φ n ( x ) ∝ e i πnx/L instead of (6.66), were carried outin Ref. 114 and are given byΨ ( b )0 ( x , . . . , x N ) = s N ( N − N ! L N Y i>j sin (cid:18) πL | x i − x j | (cid:19) , (6.67)together with its associated ground state energy E = (cid:18) N − N (cid:19) (cid:126) π N mL . (6.68)Now, considering the effects of a harmonic trap, whose solutions are given by (6.66),the ground state many-body wave function is given by Ψ ( b )0 ( x , . . . , x N ) = 2 N ( N − / a N/ N ! N − Y n =0 n ! √ π ! − / N Y i =0 e − x i / a Y ≤ j 17, 20, 71, 207 which, for the system consisted of N interacting particlesat temperature T , is given by % ( j ) ( x , . . . , x j ; x , . . . , x j ) = N !( N − j )! Z − X α e − βE α Z < dx j +1 . . . dx N Ψ ( b ) ∗ α ( x , . . . , x N ) × Ψ ( b ) α ( x , . . . , x j , x j +1 , . . . , x N ) , (6.71)where Z = P α e − βE α is the partition function of the system and its total energy is simplythe summation of all the individual single-particle energies, i.e. , E α = P Ni =1 (cid:15) n i , with (cid:15) n i = ( n i + 1 / (cid:126) ω .From the Bose-Fermi mapping relation (6.64), Eq. (6.71) reads % ( j ) ( x , . . . , x j ; x , . . . , x j ) = N !( N − j )! Z − X α e − βE α Z < dx j +1 . . . dx N Θ( x , . . . , x N ) × Ψ ( f ) α ( x , . . . , x N )Θ( x , . . . , x j , x j +1 , . . . , x N )Ψ ( f ) α ( x , . . . , x j , x j +1 , . . . , x N ) . (6.72)Now we turn our focus to the integrand. It is possible to rewrite the product of the Θ’sas Θ( x , . . . , x N )Θ( x , . . . , x j , x j +1 , . . . , x N ) = Θ( x , . . . , x j )Θ( x , . . . , x j ) × N Y i = j +1 2 j Y l =1 sgn( x i − y l ) , (6.73)with y = x < y = x < . . . < y j = x j < y j +1 = x < . . . < y j = x j . Now let us considerthe union of the disjoint intervals ( y i , y j ), S = { ( y , y ) ∪ ( y , y ) ∪ · · · ∪ ( y j − , y j ) } . It isstraightforward to observe that j Y i =1 sgn( x − y i ) = − , x ∈ S +1 , x / ∈ S . (6.74)Denoting the number of variables among x j +1 , . . . , x N which are in S by M S , we havethatΘ( x , . . . , x N )Θ( x , . . . , x j , x j +1 , . . . , x N ) = Θ( x , . . . , x j )Θ( x , . . . , x j )( − M S . (6.75)Consequently, Eq. (6.72) results in % ( j ) ( x , . . . , x j ; x , . . . , x j ) = N !( N − j )! Z − Θ( x , . . . , x j )Θ( x , . . . , x j ) X α e − βE α × Z < dx j +1 . . . dx N ( − M S Ψ ( f ) α ( x , . . . , x N )Ψ ( f ) α ( x , . . . , x j , x j +1 , . . . , x N ) . (6.76) .4 The Tonks-Girardeau regime for N particles Now, considering any integral of the form I = Z < dx . . . Z < ( − M S f ( x , . . . , x j ) , (6.77)where M S is the number of integration variables inside the subdomain S and f is asymmetric function, it is possible to write I = j X m =0 jm ! ( − m Z S dx . . . dx m Z <− S dx m +1 . . . dx j f ( x , . . . , x j ) . (6.78)Making use of R <− S dx = R < dx − R S dx , we have I = j X m =0 jm ! ( − m j − m X n =0 j − mn ! ( − n Z S dx . . . dx m + n Z < dx m + n +1 . . . dx j f ( x , . . . , x j ) . (6.79)Performing the summation for m + n = i , (6.79) reduces to I = j X i =0 ji ! ( − i Z S dx . . . dx i Z < dx i +1 . . . dx j f ( x , . . . , x j ) . (6.80)Thence, we have that the j -body density matrix (6.76) can be written as % ( j ) ( x , . . . , x j ; x , . . . , x j ) = N !( N − j )! Z − Θ( x , . . . , x j )Θ( x , . . . , x j ) X α e − βE α × N − j X i =0 N − ji ! ( − i Z S dx j +1 . . . dx j + i Z < dx j + i +1 . . . dx N × Ψ ( f ) α ( x , . . . , x N )Ψ ( f ) α ( x , . . . , x j , x j +1 , . . . , x N ) . (6.81)The one-body density matrix is given by % (1) ( x, x ) = N Z X α e − βE α N − X j =1 N − j ! ( − j [sgn( x − x )] j Z x x dx . . . dx j +1 × Z < dx j +2 . . . dx N Ψ ( f ) α ( x, x , . . . , x N )Ψ ( f ) α ( x , x , . . . , x N ) . (6.82)Here it is possible to recognize the j -body fermionic correlator as % (1) ( x, x ) = N − X j =1 ( − j j ! [sgn( x − x )] j Z x x dx . . . dx j +1 % ( j +1) f ( x, x , . . . , x j +1 ; x , x , . . . , x j +1 ) , (6.83)where % ( j ) f ( x , . . . , x j ; x , . . . , x j ) = N !( N − j )! Z − X α e − βE α × Z < dx j +1 . . . dx N Ψ ( f ) α ( x , . . . , x N )Ψ ( f ) α ( x , . . . , x j , x j +1 , . . . , x N ) . (6.84) Chapter 6 One-dimensional interacting Bose gas As we are interested in the contact, we are going to restrict ourselves to smalldistances, | x − x | (cid:28) 1. Therefore, we consider only the term j = 1, because the terms j > % (1) ( x, x ) ∼ x → x x − x ) Z x x dx % (2) f ( x, x ; x , x ) ≈ x − x ) % (2) f ( x, R ; x , R ) δx, (6.85)where R ≡ ( x + x ) / δx ≡ x − x .Now we proceed to the explicit evaluation of % (2) making use of (6.84) togetherwith (6.65). N=2 particles % (2) f ( x, R ; x , R ) = Z − X n ,n e − β ( (cid:15) n + (cid:15) n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ n ( x ) φ n ( x ) φ n ( R ) φ n ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ n ( x ) φ n ( x ) φ n ( x ) φ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z − X n ,n e − β ( (cid:15) n + (cid:15) n ) [ φ n ( R − δx/ φ n ( R ) − φ n ( R − δx/ φ n ( R )] × [ φ n ( R + δx/ φ n ( R ) − φ n ( R + δx/ φ n ( R )]= Z − X n ,n e − β ( (cid:15) n + (cid:15) n ) " φ n − δx ∂ R φ n ! φ n − φ n − δx ∂ R φ n ! φ n × " φ n + δx ∂ R φ n ! φ n − φ n + δx ∂ R φ n ! φ n = Z − X n ,n e − β ( (cid:15) n + (cid:15) n ) δx × h ( φ n ∂ R φ n ) + ( φ n ∂ R φ n ) − φ n φ n ∂ R φ n ∂ R φ n i . (6.86) N=3 particles % (2) f ( x, R ; x , R ) = Z − X n ,n ,n e − β ( (cid:15) n + (cid:15) n + (cid:15) n ) × Z dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ n ( x ) φ n ( x ) φ n ( x ) φ n ( R ) φ n ( R ) φ n ( R ) φ n ( x ) φ n ( x ) φ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ n ( x ) φ n ( x ) φ n ( x ) φ n ( R ) φ n ( R ) φ n ( R ) φ n ( x ) φ n ( x ) φ n ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z − X n ,n ,n e − β ( (cid:15) n + (cid:15) n + (cid:15) n ) φ n δx ∂ R φ n ! + φ n δx ∂ R φ n ! + φ n δx ∂ R φ n ! + φ n δx ∂ R φ n ! + φ n δx ∂ R φ n ! + φ n δx ∂ R φ n ! − φ n φ n δx ∂ R φ n ∂ R φ n − φ n φ n δx ∂ R φ n ∂ R φ n − φ n φ n δx ∂ R φ n ∂ R φ n . (6.87) .4 The Tonks-Girardeau regime for N particles Note that in the steps above we have used the differentiation relation ∂ R φ ( R ) = φ ( R ) − φ ( R − δx/ δx/ φ ’s Z + ∞−∞ dx φ m ( x ) φ n ( x ) = δ m,n . (6.89)Therefore, from the explicit evaluations for N = 2 and 3 particles, we can generalizethe fermionic two-body density matrix for N particles as % (2) f ( x, R ; x , R ) = ( x − x ) Z − X n ,n ,...,n N e − β P Ni =1 (cid:15) ni × X j = k (cid:26)h φ n j ( R ) ∂ R φ n k ( R ) i − φ n j ( R ) φ n k ( R ) ∂ R φ n j ( R ) ∂ R φ n k ( R ) (cid:27) . (6.90)Consequently, we have that % (1) ( x, x ) ≈ | x − x | F ( R ) , (6.91)with the definition F ( R ) ≡Z − X n ,n ,...,n N e − β P Ni =1 (cid:15) ni × X j = k (cid:26)h φ n j ( R ) ∂ R φ n k ( R ) i − φ n j ( R ) φ n k ( R ) ∂ R φ n j ( R ) ∂ R φ n k ( R ) (cid:27) . (6.92)We shall note that the limits of sums in the set n , n , . . . , n N were omitted, althoughthey are not obvious. We must remember that in the Tonks-Girardeau gas, the bosons fermionize due to the strong interaction between them. This phenomena implies that allparticles must be "found" in different states with respect to each other. For example, ifparticle 1 is in the fundamental state n = 0, particle 2 must be in any excited state, n ∈ N − { } . If particle 2 realizes the state n = 1, then the possible set of states forparticle 3 is n ∈ N − { , } . And this reasoning continues for all particles. Therefore thepossible states particle i can be found in is the set { i − , i, . . . , ∞} , which are the sets forthe sums that we fix here.6.4.2.1 Momentum distributionAs our main interest in taking the above simplification steps is the study of theTan’s contact, we still need to go through the momentum distribution in order to achieveour desired goal. Therefore, the momentum distribution, in terms of the one-body densitymatrix, reads n ( k ) = 12 π Z dx Z dx e ik ( x − x ) % (1) ( x, x ) , (6.93)and is depicted in Fig. 27 for the number of particles from N = 2 up to N = 5 as well asfor the low-, intermediate-, and high-temperature regimes. Chapter 6 One-dimensional interacting Bose gas n ( k ) a k a T/T F = 10T/T F = 1T/T F = 0.1 0 1 2 3 4 (a) N = 2. n ( k ) a k a T/T F = 10T/T F = 1T/T F = 0.1 0 1 2 3 4 (b) N = 3. n ( k ) a k a T/T F = 10T/T F = 1T/T F = 0.1 0 1 2 3 4 (c) N = 4. n ( k ) a k a T/T F = 10T/T F = 1T/T F = 0.1 0 1 2 3 4 (d) N = 5. Figure 27 – Momentum distributions for different number of particles as well as differenttemperatures. The insets show the tails of the curves.Source: By the author.6.4.2.2 Tan’s contactMaking use of the asymptotic behavior of the Fourier transform of | x − x | a − f ( x ), Z dx e − ik ( x − x ) | x − x | a − f ( x ) = 2 k a f ( x ) cos( πa/ a ) , (6.94)and the definition of the contact C ≡ k n ( k ) as k → ∞ we arrive at C = 2 π Z + ∞−∞ dx F ( x ) . (6.95)The contact from (6.95) is depicted in Fig. 28 for the number of particles rangingfrom N = 2 to 5 in terms of the temperature, together with the analogous results withinthe grand-canonical ensemble from VIGNOLO, P.; MINGUZZI, A, which we shalldiscuss later. N bosons at T = 0 can be expressedas a function of the two-boson contact, C N = C N ( C ). Also, it was verified that the scalingrelation f N (˜ g, T = 0) ≡ C N (˜ g, T = 0) C N (˜ g → ∞ , T = 0) , (6.96) .5 The contact scaling properties C c , g c N ( ∞ , T ) a h o τ Figure 28 – Tonks-Girardeau contact in the canonical ensemble (empty symbols) from Eq.(6.95) and in the grand-canonical ensemble (filled symbols) from Ref. 209 for N = 2 (violet squares), N = 3 (green circles), N = 4 (light-blue triangles),and N = 5 (orange inverted triangles). Here, τ ≡ T /T F is the adimensionaltemperature and a ho ≡ q (cid:126) /mω is the harmonic oscillator length.Source: SANT’ANA et al. where ˜ g ≡ − a a − / √ N and C N (˜ g, T = 0) ∝ N / − γN η , establishes the following: f N (˜ g, T = 0) ’ f (˜ g, T = 0) . (6.97)In particular, in the Tonks-Girardeau limit, γ ≈ η = 3 / f (˜ g, T = 0) = 2 √ π ˜ g (cid:20) ψ (cid:18) − ν (cid:19) − ψ (cid:18) − ν (cid:19)(cid:21) − . (6.98)6.5.2 Large-temperature scalingWhen the temperature is large enough, T (cid:29) T F , where T F = N (cid:126) ω/k B is the Fermitemperature, quantum correlations become negligible in the system, so that the contactfor N particles is given by the two-particle contact times the number of pairs, C N (˜ g, T (cid:29) T F ) = N ( N − C (˜ g, T (cid:29) T F ) . (6.99)Following the development at high temperatures from Ref. 131 and making use of theEuler reflection formula Γ( x )Γ(1 − x ) = π sin( πx ) , (6.100)it is possible to rewrite Eq. (6.32) as f ( ν ) = − cot( πν/ 2) Γ (1 / ν/ ν/ . (6.101) Chapter 6 One-dimensional interacting Bose gas From the asymptotic behavior of the gamma functionΓ( x ) ∼ x →∞ e x (log( x ) − O ( x − ) s πx + O (cid:16) x − / (cid:17) , (6.102)we obtain the asymptotic formula of (6.101) f ( ν ) ∼ ν →∞ − r ν πν/ . (6.103)By employing the fact that the solutions of (6.32) as ˜ g → ∞ are given by 2 n + 1 , n ∈ N ,we get ν = 2 π cot − s n + 12 ˜ g − + 2 n. (6.104)Now that we have a formula for the ν ’s in the strongly interacting limit for largetemperatures, let us insert it in the contact expression for N = 2. Recalling Eq. (6.47), wehave that C (˜ g → ∞ , T (cid:29) T F ) = 2 / ˜ g πa Z − r X n e − βE ( r ) νn ∂ν n ∂ ˜ g , (6.105)where Z r ≡ P ν e − βE ( r ) ν is the relative motion partition function. By performing thederivative ∂ν n /∂ ˜ g , (6.105) yields C (˜ g → ∞ , T (cid:29) T F ) = 2 / ˜ g πa Z − r X n e − βE ( r ) νn q n + 1) π ˜ g (cid:16) n +12˜ g (cid:17) . (6.106)As we are interested in the strongly interacting limit ˜ g → ∞ , the term n/ ˜ g in thedenominator of the series can be disregarded, leaving us with C (˜ g → ∞ , T (cid:29) T F ) = 2 / a Z − r e − β (cid:126) ω/ X n e − β (cid:126) ω (2 n +1) q n + 1) . (6.107)By exploiting the fact that X n e − α (2 n +1) √ n + 1 = X n e − αn √ n − X n e − α n √ n, (6.108)and that the sums are given by X n e − α (2 n +1) = e α e α − , (6.109a) X n e − αn √ n = Li − / (e − α ) , (6.109b) X n e − α n √ n = √ − / (e − α ) , (6.109c)with α ≡ β (cid:126) ω , (6.107) yields C (˜ g → ∞ , T (cid:29) T F ) = 2 a e − α (cid:16) e α − (cid:17) h Li − / (e − α ) − √ − / (e − α ) i , (6.110) .5 The contact scaling properties where Li n ( z ) is the polylog function. The expasions of the polylog functions around α = 0 yieldLi − / (e − α ) − √ − / (e − α ) ≈ π − / α / + O (cid:16) α / (cid:17) + (cid:16) − √ (cid:17) ζ ( − / 2) + (cid:16) √ − (cid:17) αζ ( − / 2) + O ( α ) , (6.111)with ζ ( x ) being the Riemann zeta function. Therefore, as α approaches zero, the contactreduces to C (˜ g → ∞ , T (cid:29) T F ) = 2 π − / a s k B T (cid:126) ω = (cid:18) π (cid:19) / a − s TT F . (6.112)Hence, the expression for the N -particle contact (6.99) in the strongly interactinglimit is given by C N (˜ g → ∞ , T (cid:29) T F ) = N ( N − a π / s N TT F . (6.113)Therefore, we have the scaling function in the high-temperature regime for the TG limitas being C N (˜ g → ∞ , T (cid:29) T F ) = h N (˜ g → ∞ , T (cid:29) T F ) (cid:16) N / − N / (cid:17) ⇒ h N (˜ g → ∞ , T (cid:29) T F ) = h (˜ g → ∞ , T (cid:29) T F ) = q T /T F a π / . (6.114)6.5.3 Generalized scaling conjectureHaving studied the behavior of the contact in terms of the number of particles forboth high ( T (cid:29) T F ) and low temperatures ( T (cid:28) T F ) in the strongly interacting limit˜ g → ∞ , we now propose a conjecture for the entire range of temperatures in the aforesaidregime.At large temperatures, where quantum correlations play a minor role towards theproperties of the system, the contact dependency on the number of particles is given bythe number of pairs N ( N − 1) times a √ N factor that comes from the Fermi temperature.As the temperature decreases, there happens an intensification on the N -dependency ofthe contact, from N / − N / to N / − N / . Therefore, following such a reasoning, wehave proposed the following scaling hypothesis: C N (˜ g → ∞ , τ ) ∝ N / − N / − /τ )] , (6.115)where τ ≡ T /T F .Fig. 29 displays the Tan’s contacts in the strongly interacting limit from Eq.(6.95) scaled by our proposed generalized conjecture from (6.115) for different numberof particles ranging from 2 to 5. Hence, we can observe the collapse of all data on thesame curve. Moreover, we compare it to the blue dashed line, which corresponds to the Chapter 6 One-dimensional interacting Bose gas . . . . C c N ( ∞ , T ) a h o / s ( N ) τ Figure 29 – Tan’s contacts in the Tonks-Girardeau limit from Eq. (6.95) in adimensionalunits as functions of the reduced temperature τ scaled by the generalized con-jecture s ( N ) ≡ N / − N / − /τ )] for the respective number of particles: N = 2 (violet squares), N = 3 (green circles), N = 4 (blue triangles), and N = 5 (orange inverted triangles). The black cross corresponds to the zero-temperature Tonks-Girardeau two-boson contact from Eq. (6.63) rescaled by s ( N ) for τ = 0: (2 / − / ) − C (˜ g → ∞ , T → 0) = (2 / − / ) − (2 /π ) / =0 . √ τ /π / , while the black con-tinuous line is simply the contact rescaled by the generalized scaling factor, i.e. , the implicit proportionality factor in Eq. (6.115).Source: SANT’ANA et al. high-temperature behavior in the TG limit from Eq. (6.112). We observe that all data, inthe high-temperature regime, also collapse over the curve of the high-temperature TG-limittwo-boson contact C (˜ g → ∞ , T (cid:29) T F ).6.5.4 Intermediate interaction strength scalingWe now turn our attention to the intermediate interaction strength scenario ˜ g ∼ III We analyze the QMC data for ˜ g = 1in Fig. 30 and for ˜ g = 2 . N / − N / (panel (a)), thelarge-temperature scaling N / − N / (panel (b)), and the generalized scaling conjecture N / − N / − /τ )] (panel (c)). In panel (d) of both Figs. 30 and 31, we rescale theQMC data by the TG-limit contact from Eq. (6.95). We observe that, the zero-temperaturescaling factor in Fig. 30(a) makes the data approach each other at small temperatures, III Thanks to Frédéric Hébert for performing the QMC simulations. For the details on theQMC simulations, see SANT’ANA et al. .5 The contact scaling properties . . . 07 0 . C c N ( z , τ ) a h o / ( N / − N / ) τ . . . 09 0 . C c N ( z , τ ) a h o / ( N / − N / ) τ . . . 08 0 . C c N ( z , τ ) a h o / s ( N ) τ . . . . . C c N ( z , τ ) / C c N ( ∞ , τ ) τ Figure 30 – Tan’s contacts from QMC simulations as functions of the adimensional tem-perature τ for z = ˜ g = 1. The panels (a), (b), and (c) correspond to therescaling of the contact regarding the low-temperature factor N / − N / , thehigh-temperature factor N / − N / , and the all-range-temperature factor s ( N ) ≡ N / − N / − /τ )] , respectively. In panel (d), the QMC data isrescaled by the TG-limit contact from Eq. (6.95). The symbol styles correspondto: N = 2 (violet squares), N = 3 (green circles), N = 4 (blue triangles), and N = 5 (orange inverted triangles). The continuous yellow line correspondsto the two-boson contact obtained by Eq. (6.52). The QMC error bars aresmaller than the symbol sizes.Source: SANT’ANA et al. while we observe a collapse of the whole data in Fig. 31(a) at low temperatures. Thesame occurs at high temperatures: the data approach each other in Fig. 30(b), while theycollapse over each other in Fig. 31(b). Differently, the generalized scaling conjecture workswell within the whole temperature range, with an incertitude of 5% for ˜ g = 1 (Fig. 30(c))and of 1% for ˜ g = 2 . f N (˜ g, T ) ≡ C N (˜ g, T ) C N (˜ g → ∞ , T ) (6.116)and certify ourselves that, from Figs. 30(d) and 31(d), the relation f N (˜ g (cid:38) , T (cid:29) T F ) = f (˜ g (cid:38) , T (cid:29) T F ) (6.117)holds for the whole temperature range —not only for the low-temperature regime ( τ (cid:28) τ (cid:29) 1) as previously stated, but also for the intermediate-temperature regime ( τ ∼ Chapter 6 One-dimensional interacting Bose gas . . . . . . 18 0 . C c N ( z , τ ) a h o / ( N / − N / ) τ . . . . 22 0 . C c N ( z , τ ) a h o / ( N / − N / ) τ . . . . 22 0 . C c N ( z , τ ) a h o / s ( N ) τ . . . . . . C c N ( z , τ ) / C c N ( ∞ , τ ) τ Figure 31 – Tan’s contacts from QMC simulations as functions of the adimensionaltemperature τ for z = ˜ g = 2 . 5. The panels (a), (b), and (c) correspond to therescaling of the contact regarding the low-temperature factor N / − N / , thehigh-temperature factor N / − N / , and the all-range-temperature factor s ( N ) ≡ N / − N / − /τ )] , respectively. In panel (d), the QMC data isrescaled by the TG-limit contact from Eq. (6.95). The symbol styles correspondto: N = 2 (violet squares), N = 3 (green circles), N = 4 (blue triangles), and N = 5 (orange inverted triangles). The continuous yellow line correspondsto the two-boson contact obtained by Eq. (6.52). The QMC error bars aresmaller than the symbol sizes.Source: SANT’ANA et al. Now, let us summarize the results of this section. An important consequence of thescaling results is that the contact for N bosons at temperature T with repulsive interactioncharacterized by the adimensional interaction strength ˜ g rescaled by the contact for N strongly interacting bosons at temperature T , i.e. , C N (˜ g, T ) / C N (˜ g → ∞ , T ), is a universalfunction of the adimensional interaction strength ˜ g ≡ − a a − / √ N and the adimensionaltemperature τ ≡ T /T F . Furthermore, another important result comes from the generalizedscaling function. Namely, the ratio between the contact for N bosons at temperature T with repulsive interaction characterized by an interaction strength g and the generalizedscaling function s ( N ) ≡ N / − N / − /τ )] , i.e. , C N (˜ g, T ) /s ( N ), is also a universalfunction of ˜ g and τ . In this section we draw a comparison between the contact calculated from thecanonical ensemble evaluated in this thesis and the grand-canonical one from Ref. 209 at .6 Comparison between ensembles finite temperature. The motivation for comparing both ensembles comes from the fact thatthe scaling properties of the contact are strongly affected by the statistical distribution.Considering that in most ultracold atom experiments the number of particles is fixed,performing the calculations within the canonical ensemble is a more appropriate choice. Onthe other hand, the grand-canonical ensemble is advantageous when dealing with systemswhere the number of particles vary, such as open systems. In the latter, the average numberof particles h N i is then determined by fixing the temperature and the chemical potential,while the most remarkable feature of the former is that the particles can have any valuefor its energy and the average energy h E i of the whole system is then determined bythe temperature. For the sake of clarity, in this section we introduce the index ( gc ) thatrepresents the respective physical quantity in the grand-canonical ensemble. Moreover, weanalyze the differences from both ensembles in the QMC simulations.6.6.1 Analytical formulaIn the zero-temperature limit, there is no physical distinction between the grand-canonical and the canonical ensembles. Thus, the contacts from both calculations scale as N / − N / . However, by increasing the temperature, the distinction between ensemblesis enhanced. Such differences can be observed from Fig. 28, where the grand-canonicalcontact increases more rapidly when compared to the canonical one as the temperaturerises. In fact, in the large-temperature scenario, the term corresponding to the number ofpairs N ( N − 1) in Eq. (6.99) has to be replaced by its average value in the grand-canonicalcalculations: h N ( N − i = h N i − h N i = h N i . (6.118)Note that this last step follows from the fact that, at large T , h ∆ N i ’ h N i . So, analogouslyto the canonical scaling (6.99), we have for the grand-canonical contact that C ( gc ) N (˜ g, T (cid:29) T F ) = h N i C (˜ g, T (cid:29) T F ) . (6.119)In the TG limit, the above relation together with (6.113) yields, by defining T F = h N i (cid:126) ω/k B , the h N i / -dependency, C ( gc ) N (˜ g → ∞ , τ (cid:29) 1) = h N i / π / a √ τ , (6.120)corroborating the result from Ref. 131. This result is shown in Fig. 32.6.6.2 Quantum Monte Carlo simulationsNow, let us analyze the differences between the canonical and the grand-canonicalQMC simulations. Fig. 33 displays the contact from both ensembles calculated via QMCconsidering the weak-intermediate interaction regime, ˜ g = 0 . 5. As expect and alreadydiscussed, the grand-canonical contact presents a steeper rise as the temperature increases Chapter 6 One-dimensional interacting Bose gas . . . . C g c N a h o / N / , C c N a h o / ( N / − N / ) τ Figure 32 – Canonical (empty symbols) contact in the TG limit from (6.95) rescaledby N / − N / and grand-canonical (filled symbols) contact from Ref. 209rescaled by N / as functions of τ for the respective number of particles: N = 2 (violet squares), N = 3 (green circles), N = 4 (blue triangles), and N = 5 (orange inverted triangles). The black continuous curve corresponds to √ τ /π / . Source: SANT’ANA et al. when compared to the canonical one. This difference can be explained by the fundamentalsof the grand-canonical ensemble: there is a probability that the system contains any numberof particles; and such contributions, especially for N > h N i , produces an initial growth ofthe contact at low temperatures. This explanation is better understood by mathematicalmeans: consider the fugacity term P N e βµN , it is then straightforward to see that such acontribution becomes considerable at large values of N as well as small values of T ∝ β − .Then, at some point, the contact reaches its maximum, which was recently explained asthe mark of the crossover between a quasicondensate and an ideal Bose gas. Afterwards,it begins to decrease. This decline is explained by the very nature of the quantum realm:as T increases, the de Broglie wavelength decreases, then the overlap between individualparticle waves also decreases, resulting in the overall drop of the total contact.Now, let us test the scaling hypothesis (6.117) by plotting the ratio C ( gc ) N (˜ g = 1 , T ) C ( gc ) N (˜ g → ∞ , T ) , (6.121)with C ( gc ) N (˜ g = 1 , T ) being the data from QMC calculation and C ( gc ) N (˜ g → ∞ , T ) being theanalytical formula from Ref. 209. These results are displayed in Fig. 34. We observe thatthe curves do not collapse over each other. Thence, we conclude that the scaling hypothesis .6 Comparison between ensembles . . . . C c , g c N ( z = . , τ ) a h o τ Figure 33 – Tan’s contacts evaluated from canonical QMC simulations (empty symbols)and from grand-canonical QMC simulations (filled symbols) as functions of τ in the weak-intermediate regime z = ˜ g = 0 . N = 2 (violet squares), N = 3 (green circles), N = 4 (blue triangles).QMC error bars in the canonical-ensemble calculation are smaller than thesymbols size. Source: SANT’ANA et al. (6.121) fails in the grand-canonical ensemble and that the grand-canonical TG contactdoes not embed the full N -dependency as it does in the canonical ensemble, at least inthe low- and intermediate-temperature regimes.It is worth remarking that the QMC simulations are limited within the weak- andintermediate-interaction regimes, ˜ g (cid:28) g ∼ 1, and also for low and intermediatetemperatures, τ (cid:28) τ ∼ 1. Moreover, by increasing the number of particles, QMCsimulations become very difficult, and the relative errors increase together with the increaseof τ , ˜ g , and N . For more details on the QMC calculations see SANT’ANA et al. Chapter 6 One-dimensional interacting Bose gas . . . . . . C g c N ( z = , τ ) / C g c N ( ∞ , τ ) τ Figure 34 – QMC grand-canonical contact rescaled by the TG grand-canonical contactfrom Ref. 209 as a function of τ for the respective number of particles: N = 2(violet squares), N = 3 (green circles), and N = 4 (blue triangles).Source: SANT’ANA et al. In this thesis we have studied the system formed by bosonic atoms loaded intooptical lattices and the one-dimensional repulsively interacting Bose gas under a harmonicconfinement. Regarding the first part, bosons in optical lattices, we focused on the MI-SFquantum phase transition. We introduced the general basis behind the Bose-Hubbardmodel in order to construct the BH Hamiltonian, which is the main mathematical quantityfor the evaluation of important physical properties of the system. Another cornerstone ofour theory is the mean-field approximation, which neglects some quantum fluctuationsover the so-called mean-field, which is simply the average of the bosonic lattice operator,throughout the whole lattice. Within the mean-field theory, which turns out to give rise toa Hamiltonian which is a summation of the one-site independent Hamiltonians, we areable to consider such separated single lattice sites, considerably simplifying all calculations.Then, we introduced the general expansion of the thermodynamic potential in the vicinityof a second-order phase transition using the analyticity assumptions suggested by Landau.This expansion provides valuable insights about the phase in which the system is interms of the series coefficients of the thermodynamic potential. By applying perturbationtheory, it is then possible to calculate the MI-SF quantum phase diagram. However, it hasbeen noted that, between every adjacent Mott lobes in the MI-SF phase boundary, therehappens to occur degeneracies between the respective energies of these Mott lobes. Whenusual perturbation theory is applied to this system, such degeneracies lead to nonphysicalresults, e.g. , the condensate density, which is the order parameter in this system, vanishesin a point of the phase diagram where actually no quantum phase transition occurs. Thisresult can be considered as an inconsistency that artificially arises from such an erroneoustreatment. Therefore, in order to correct this problem, we developed two methods basedon degenerate perturbation theory to correctly calculate meaningful physical quantitiesof the system. Both methods are based on a projection-operator formalism, that enablesus to separate the Hilbert subspace where the degeneracies are contained in from thecomplementary subspace, which is free from any degeneracy. With this we are able tosolve the degeneracy problems which are typical of nondegenerate perturbation theories.Firstly, we developed the Brillouin-Wigner perturbation theory to tackle the zero-temperature system composed of bosons in optical lattices. Such results were published inRef. 185. After that, in order to generalize for finite temperature scenarios, we developedthe finite-temperature degenerate perturbation theory approach and published the resultsin Ref. 184. These two methods presented in this thesis are fundamentally different: theBWPT is developed in the context of the time-independent Schrödinger equation, whilethe FTDPT is developed in the context of the time-dependent Schrödinger equation. In Chapter 7 Conclusions the latter, the relation between imaginary time and temperature arises naturally by meansof the correspondence between the time-evolution operator and the partition function,which is simply the trace of the former after performing a Wick rotation. By correctlytreating the degeneracies between the Mott lobes, we were in possession of a reliablemethod to calculate important physical quantities, with special attention to the condensatedensity due to its role as the order parameter. Finally, it is important to remark thatboth methods developed in the first part of the thesis, the BWPT and FTDPT, notonly provide relatively simple frameworks for calculating the condensate density, but areactually very generic approaches in the sense that they can also be applied to a wide rangeof optical-lattice systems, e.g. , out-of-equilibrium systems, 92, 93 bosonic optical latticeswith three-body constraint, and different-geometry lattices such as Kagomé lattice, triangular and hexagonal lattices, as well as the Jaynes-Cummings lattice. 99, 100 Regarding the one-dimensional repulsively interacting Bose gas under harmonicconfinement, we have studied the asymptotic behavior of the momentum distributionat both zero and finite temperatures, with special attention to the intermediate- andstrong-interaction regimes, since the weakly interacting regime simply reduces to theknown ideal Bose gas. We began our study by solving the two-particle problem, which is anintegrable model and provides valuable insights regarding the behavior of the many-bodywave function at the contact point: the contact interaction originates a cusp conditionin the wave function, implying its vanishing whenever two particles meet. Beyond twoparticles, the integrability breaks down and we have to restrict ourselves to specific regimes,such as the strongly interacting limit, also known as Tonks-Girardeau limit, where therehappens the so-called fermionization of the bosons, i.e. , the bosons behave as the idealFermi gas, enabling the mapping of the bosonic system into a simpler noninteractingfermionic one. Then, we were able to derive an analytical formula for the Tan’s contactof N particles repulsively interacting under a harmonic confinement in one dimension interms of the single-particle orbitals of the harmonic oscillator in the TG limit. Afterwards,we investigated the scaling properties of the contact in terms of the number of particles N .Firstly, we argued that, at large temperatures, the N -particle contact can be written asthe two-particle contact times the number of pairs. This comes from the fact that, as thetemperature increases, quantum correlations can be neglected, thence we simply need toconsider the effects of every pair of particles contacting with each other. More specifically,we showed that the ratio between the N -particle contact at finite interaction strengthand its correspondent in the TG limit is a universal N -independent scaling function.Following such advances in the high temperature scenario, we proposed a generalizedscaling conjecture for the entire range of temperatures in the TG limit. Then, we properlydemonstrated that such a conjecture works well for any considered temperature.Furthermore, we also inspected the intermediate interaction regime, ˜ g ∼ 1, withthe help of QMC simulations. In such a regime, we found analogous results to the Tonks- Girardeau limit, i.e. , meaning that all our previously stated conjectures regarding therange of temperatures also works when ˜ g ∼ 1. Consequently, we conclude that thecontact of N bosons at temperature T with repulsive interaction characterized by theadimensional interaction strength ˜ g rescaled by the contact of N strongly interactingbosons at temperature T , i.e. , C N (˜ g, T ) / C N (˜ g → ∞ , T ), is a universal function of theadimensional interaction strength ˜ g and the reduced temperature τ for the whole range oftemperatures. Likewise, the ratio between the contact of N bosons at temperature T withfinite repulsive interaction and the generalized scaling function s ( N ), i.e. , C N (˜ g, T ) /s ( N ),is also a universal function of ˜ g and τ for all temperatures.Finally, as our main calculations were performed within the canonical ensemble, wecompared our results to the grand-canonical calculations performed in Ref. 209. We havechecked that, at large temperatures, both the canonical and the grand-canonical contactsare proportional to the two-boson contact, with the proportionality factor depending onthe number of pairs in the canonical ensemble as well as the average number of pairsin the grand-canonical one. On the other hand, at low and intermediate temperatures,the grand-canonical contact of an average number of h N i bosons cannot be written as afunction of the h i -boson contact or of the h N i -TG-boson contact, to the extent of theQMC simulations, which is the intermediate-interaction regime.The results obtained in the referred part of the thesis were published in Ref.210. Such developments are properly suited for probing the contact as well as the role ofcorrelations and interactions in experiments with a small number of bosonic particles. Moreover, it is worth noting that the canonical ensemble calculations realized in this thesiscorrespond to most real experimental conditions, since the number of atoms is fixed. From a conceptual point of view, it is an important step forward to the understanding ofthe correlation and interaction effects on the harmonically trapped interacting Bose gasin one dimension at finite temperature, as well as to the enlightenment of the role of theparticle-number fluctuations. REFERENCES A course of lectures on natural philosophy and the mechanical arts .London: J. Johnson, 1807. 2 v.2 KIRCHHOFF, G. Über die Fraunhofer’schen Linien. Monatsberichte der KöniglichePreussische Akademie der Wissenschaften zu Berlin , p. 662-665, 1859.3 KIRCHHOFF, G. Über den Zusammenhang zwischen emission und absorption vonLicht und Wärme. Monatsberichte der Akademie der Wissenschaften zu , p. 783-789, 1859.4 KIRCHHOFF, G. Über das Verhältniss zwischen dem emissionsvermögen und demabsorptionsvermögen der Körper für Wärme and Licht. Annalen der Physik , v. 185, n. 2,p. 275-301, 1860.5 BOLTZMANN, L. Vorlesungen über Gastheorie . Leipzig: Barth, 1896. 2 v.6 PLANCK, M. Über eine Verbesserung der Wienschen Spektralgleichung. Verhandlungender Deutschen Physikalischen Gesellschaft , v. 2, p. 202-204, 1900.7 PLANCK, M. Zur Theorie des gesetzes der energieverteilung im normalspectrum. Verhandlungen der Deutschen Physikalischen Gesellschaft. , v. 2, p. 237, 1900.8 PLANCK, M. Entropie und Temperatur strahlender Wärme. Annalen der Physik , v.306, n. 4, p. 719-737, 1900.9 PLANCK, M. Über irreversible Strahlungsvorgänge. Annalen der Physik , v. 306, n. 1,p. 69-122, 1900.10 PLANCK, M. Über das Gesetz der Energieverteilung im Normalspektrum. Annalender Physik , v. 309, n. 3, p. 553-563, 1901.11 NEWTON, I. Philosophiæ naturalis principia mathematica . London: BenjaminMotte, 1687.12 MAXWELL, J. C. A treatise on electricity and magnetism . Oxford: Clarendon Press,1873. 2 v.13 BOSE, S. N. Plancks Gesetz und Lichtquantenhypothese. Zeitschrift für Physik , v.26, p. 178-181, 1924.14 EINSTEIN, A. Quantentheorie des einatomigen idealen Gases. Sitzungsberichte derPreussischen Akademie der Wissenschaften , v. 1, p. 3, 1925.15 LONDON, F. The λ -phenomenon of liquid helium and the Bose-Einstein degeneracy. Nature , v. 141, p. 643, 1938.16 LONDON, F. Superfluids: macroscopic theory of superconductivity . Hoboken: JohnWiley & Sons, 1950. v. 1.17 HUANG, K. Statistical mechanics . Hoboken: John Wiley & Sons, 1987. References 18 REIF, F. Fundamentals of statistical and thermal physics . New York: McGraw-Hill,1965.19 KETTERLE, W. When atoms behave as waves: Bose-Einstein condensation and theatom laser. Reviews of Modern Physis , v. 74, n. 4, p. 1131-1151, 2002.20 PETHICK, C. J.; SMITH, H. Bose-Einstein condensation in dilute gases . 2nd ed.Cambridge: Cambridge University Press, 2008.21 HÄNSCH, T. W.; SHAWLOW, A. L. Cooling of gases by laser radiation. OpticsCommunications , v. 13, n. 1, p. 68, 1975.22 WINELAND, D. J.; DRULLINGER, R. E.; WALLS, F. L. Radiation-pressure coolingof bound resonant absorbers. Physical Review Letters , v. 40, n. 25, p. 1639-1642, 1978.23 ASHKIN, A. Trapping of atoms by resonance radiation pressure. Physical ReviewLetters , v. 40, n. 12, p. 729-732, 1978.24 CHU, S. et al. Three-dimensional viscous confinement and cooling of atoms byresonance radiation pressure. Physical Review Letters , v. 55, n. 1, p. 48-51, 1985.25 ASPECT, A. et al. Laser cooling below the one-photon recoil energy byvelocity-selective coherent population trapping. Physical Review Letters , v. 61, n. 7, p.826-829, 1988.26 CHU, S. The manipulation of neutral particles. Reviews of Modern Physics , v. 70, n.3, p. 685-706, 1988.27 COHEN-TANNOUDJI, C. N. Manipulating atoms with photons. Reviews of ModernPhysics , v. 70, n. 3, p. 707-719, 1988.28 PHILLIPS, W. D. Laser cooling and trapping of neutral atoms. Reviews of ModernPhysics , v. 70, n. 3, p. 721-741, 1998.29 DALIBARD, J.; COHEN-TANNOUDJI, C. N. Laser cooling below the Doppler limitby polarization gradients: simple theoretical models. Journal of the Optical Society ofAmerica B , v. 6, n. 11, p. 2023-2045, 1989.30 LETT, P. D. et al. Observation of atoms laser cooled below the Doppler limit. Physical Review Letters , v. 61, n. 2, p. 169-172, 1998.31 NEUHAUSER, W. et al. Optical-sideband cooling of visible atom cloud confined inparabolic well. Physical Review Letters , v. 41, n. 4, p. 233-236, 1978.32 DIEDRICH, F. et al. Laser cooling to the zero-point energy of motion. PhysicalReview Letters , v. 62, n. 4, p. 403-406, 1989.33 MONROE, C. et al. Resolved-sideband Raman cooling of a bound atom to the 3dzero-point energy. Physical Review Letters , v. 75, n. 22, p. 4011-4014, 1995.34 HAMANN, S. E. et al. Resolved-sideband Raman cooling to the ground state of anoptical lattice. Physical Review Letters , v. 80, n. 19, p. 4149-4152, 1998.35 ESCHNER, J. et al. Laser cooling of trapped ions. Journal of the Optical Society ofAmerica B , v. 20, n. 5, p. 1003, 2003. eferences 36 SCHLIESSER, A. et al. Resolved-sideband cooling of a micromechanical oscillator. Nature Physics , v. 4, p. 415, 2008. DOI: 10.1038/nphys939.37 KASEVICH, M.; CHU, S. Laser cooling below a photon recoil with three-level atoms. Physical Review Letters , v. 69, n. 12, p. 1741-1744, 1992.38 KERMMAN, A. J. et al. Beyond optical molasses: 3D Raman sideband cooling ofatomic cesium to high phase-space density. Physical Review Letters , v. 84, n. 3, p. 439-442,2000.39 LETT, P. D. et al. Optical molasses. Journal of the Optical Society of America B , v.6, n. 11, p. 2084, 1989.40 WEIDEMÜLLER, M. et al. A novel scheme for efficient cooling below the photonrecoil limit. EuroPhysics Letters , v.27, n. 2, p. 109-114, 1994.41 BOIRON, D. et al. Three-dimensional cooling of cesium atoms in four-beam grayoptical molasses. Physical Review A , v. 52, n. 5, p. 3425-3428, 1995.42 NATH, D. et al. Quantum-interference-enhanced deep sub-Doppler cooling of Katoms in gray molasses. Physical Review A , v. 88, n. 5, p. 053407, 2013.43 SIEVERS, F. et al. Simultaneous sub-Doppler laser cooling of fermionic Li and Kon the D line: theory and experiment. Physical Review A , v. 91, n. 2, p. 023426, 2015.44 BRUCE, G. D. et al. Sub-Doppler laser cooling of K with Raman gray molasses onthe D line. Journal of Physics B: atomic, molecular and optical physics, v. 50, n. 9, p.095002, 2017.45 HORAK, P. et al. Cavity-nduced atom cooling in the strong coupling regime. PhysicalReview Letters , v. 79, n. 25, p. 4974-4977, 1997.46 CHEINEY, P. et al. A Zeeman slower design with permanent magnets in a Halbachconfiguration. Review of Scientific Instruments , v. 82, n. 6, p. 063115, 2011.47 OHAYON, B.; RON, G. New approaches in designing a Zeeman slower. Journal ofInstrumentation , v. 8, p. 02016, 2013. DOI: 10.1088/1748-0221/8/02/P02016.48 BUDKER, D. et al. Nonlinear magneto-optics and reduced group velocity of light inatomic vapor with slow ground state relaxation. Physical Review Letters , v. 83, n. 9, p.1767-1770, 1999.49 MORIGI, G.; ESCHNER, J.; KEITEL, C. H. Ground state laser cooling usingelectromagnetically induced transparency. Physical Review Letters , v. 85, n. 21, p.4458-4461, 2000.50 LIU, C. et al. Observation of coherent optical information storage in an atomicmedium using halted light pulses. Nature , v. 409, n. 6819, p. 490, 2001.51 SAFAVI-NAEINI, A. H. et al. Electromagnetically induced transparency and slowlight with optomechanics. Nature , v. 472, p. 69, 2011. DOI: 10.1038/nature09933.52 HALLER, E. et al. Single-atom imaging of fermions in a quantum-gas microscope. Nature Physics , v. 11, p. 738, 2015. DOI: 10.1038/nphys3403 References 53 COHEN-TANNOUDJI, C. N.; PHILLIPS, W. D. New mechanisms for laser cooling. Physics Today , v. 43, n. 10, p. 33, 1990.54 ANDERSON, M. H. et al. Observation of Bose-Einstein condensation in a diluteatomic vapor. Science , v. 269, n. 5221, p. 198-201, 1995.55 DAVIS, K. B. et al. Bose-Einstein condensation in a gas of sodium atoms. PhysicalReview Letters , v. 75, n. 22, p. 3969-3973, 1995.56 MASUHARA, N. et al. Evaporative cooling of spin-polarized atomic hydrogen. Physical Review Letters , v. 61, n. 8, p. 935-938, 1988.57 KETTERLE, W.; DRUTEN, N. J. V. Evaporative cooling of trapped atoms. Advances in Atomic, Molecular, and Optical Physics , v. 37, p. 181-236, 1996. DOI:10.1016/S1049-250X(08)60101-958 CORNELL, E. A.; WIEMAN, C. E. Bose-Einstein condensation in a dilute gas, thefirst 70 years and some recent experiments. Reviews of Modern Physics , v. 74, n. 3, p.875-893, 2002.59 PAULI, W. Über den Zusammenhang des Abschlusses der Elektronengruppen imAtom mit der Komplexstruktur der Spektren. Zeitschrift für Physik , v. 31, n. 1, p. 765,1925.60 COOPER, L. N. Bound electron pairs in a degenerate Fermi gas. Physical Review , v.104, n. 4, p. 1189-1190, 1956.61 GREINER, M.; REGAL, C. A.; JIN, D. S. Emergence of a molecular Bose-Einsteincondensate from a Fermi gas. Nature , v. 426, n. 6966, p. 537, 2003.62 ZWIERLEIN, M. W. et al. Observation of Bose-Einstein condensation of molecules. Physical Review Letters , v. 91, n. 25, p. 250401, 2003.63 BOURDEL, T. et al. Measurement of the interaction energy near a Feshbachresonance in a Li Fermi gas. Physical Review Letters , v. 91, n. 2, p. 020402, 2003.64 BARTENSTEIN, M. et al. Crossover from a molecular Bose-Einstein condensate to adegenerate Fermi gas. Physical Review Letters , v. 92, n. 12, p. 120401, 2004.65 PARTRIDGE, G. B. et al. Molecular Probe of Pairing in the BEC-BCS Crossover. Physical Review Letters , v. 95, n. 2, p. 020404, 2005.66 BARDEEN, J.; COOPER, L. N.; SCHRIEFFER, J. R. Microscopic theory ofsuperconductivity. Physical Review , v. 106, n. 1, p. 162-164, 1957.67 BARDEEN, J.; COOPER, L. N.; SCHRIEFFER, J. R. Theory of superconductivity. Physical Review , v. 108, n. 5, p. 1175-1204, 1957.68 JAKSCH, D. et al. Cold bosonic atoms in optical lattices. Physical Review Letters , v.81, n. 15, p. 3108, 1998.69 BLOCH, I. Ultracold quantum gases in optical lattices. Nature Physics , v. 1, n. 1, p.23, 2005. eferences 70 LEWENSTEIN, M.; SANPERA, A.; AHUFINGER, V. Ultracold atoms in opticallattices: simulating quantum many-body systems. Oxford: Oxford University Press, 2012.71 PITAEVSKII, L.; STRINGARI, S. Bose-Einstein condensation and superfluidity . 2nded. Oxford: Oxford University Press, 2016.72 FEYNMAN, R. P. Simulating physics with computers. International Journal ofTheoretical Physics , v. 21, p. 467-488, 1982. DOI: 10.1007/BF02650179.73 FEYNMAN, R. P. Quantum mechanical computers. Foundations of Physics , v. 16, p.507-531, 1986. DOI: 10.1007/BF01886518.74 GREINER, M.; FÖLLING, S. Optical lattices. Nature , v. 453, p. 736, 2008. DOI:10.1038/453736a.75 UEDA, M. Fundamentals and new frontiers of Bose-Einstein condensation . Singapore:World Scientific Publishing, 2010.76 FISHER, M. P. A. et al. Boson localization and the superfluid-insulator transition. Physical Review B , v. 40, n. 1, p. 546-570, 1989.77 SHESHADRI, K. et al. Superfluid and insulating phases in an interacting-bosonmodel: mean-field theory and the RPA. Europhysics Letters , v. 22, n. 4, p. 257-263, 1993.78 GREINER, M. et al. Quantum phase transition from a superfluid to a Mott insulatorin a gas of ultracold atoms. Nature , v. 415, n. 6867, p. 39, 2002.79 GREINER, M. et al. Collapse and revival of the matter wave field of a Bose–Einsteincondensate. Nature , v. 419, n. 6902, p. 51, 2002.80 GREINER, M. et al. Exploring phase coherence in a 2d lattice of Bose-Einsteincondensates. Physical Review Letters , v. 87, n. 16, p. 160405, 2001.81 WIDERA, A. et al. Coherent collisional spin dynamics in optical lattices. PhysicalReview Letters , v. 95, n. 19, p. 190405, 2005.82 GÜNTER, K. et al. Bose-Fermi mixtures in a three-dimensional optical lattice. Physical Review Letters , v. 96, n. 18, p. 180402, 2006.83 OSPELKAUS, S. et al. Localization of bosonic atoms by fermionic impurities in athree-dimensional optical lattice. Physical Review Letters , v. 96, n. 18, p. 180403, 2006.84 LEWENSTEIN, M. et al. Ultracold atomic gases in optical lattices: mimickingcondensed matter physics and beyond. Advances in Physics , v. 56, n. 2, p. 243-379, 2007.85 GERBIER, F. et al. Interference pattern and visibility of a mott insulator. PhysicalReview A , v. 72, n. 5, p. 053606, 2014.86 BLOCH, I.; DALIBARD, J.; ZWERGER, W. Many-body physics with ultracoldgases. Reviews of Modern Physics , v. 80, n. 3, p. 885-964, 2008.87 HOFFMANN, A.; PELSTER, A. Visibility of cold atomic gases in optical lattices forfinite temperatures. Physical Review A , v. 79, n. 5, p. 053623, 2009. References 88 SACHDEV, S. Quantum phase transitions . 2nd ed. Cambridge: Cambridge UniversityPress, 2011.89 MITRA, K. et al. Superfluid and Mott-insulating shells of bosons in harmonicallyconfined optical lattices. Physical Review A , v. 77, n. 3, p. 033607, 2008.90 SANTOS, F. E. A.; PELSTER, A. Quantum phase diagram of bosons in opticallattices. Physical Review A , v. 79, n. 1, p. 013614, 2009.91 BRADLYN, B.; SANTOS, F. E. A.; PELSTER, A. Effective action approach forquantum phase transitions in bosonic lattices. Physical Review A , v. 79, n. 1, p. 013615,2009.92 GRAß, T. D.; SANTOS, F. E. A.; PELSTER, A. Real-time Ginzburg-Landau theoryfor bosons in optical lattices. Laser Physics , v. 21, n. 8, p. 1459-1463, 2011.93 GRAß, T. D.; SANTOS, F. E. A.; PELSTER, A. Excitation spectra of bosons inoptical lattices from Schwinger-Keldysh calculation. Physical Review A , v. 84, n. 1, p.013613, 2011.94 WANG, T. et al. Tuning the quantum phase transition of bosons in optical latticesvia periodic modulation of the s-wave scattering length. Physical Review A , v. 90, n. 1, p.013633, 2014.95 SANTOS, F. E. A. Ginzburg-Landau theory for bosonic gases in optical lattices . 2011.123p Ph. D. thesis (Physik) - FachbereichPhysik, Freie Universität Berlin, Berlin, 2011.96 LEE, Y.-W.; YANG, M.-F. Superfluid-insulator transitions in attractive Bose-Hubbardmodel with three-body constraint. Physical Review A , v. 81, n. 6, p. 061604, 2010.97 SANTOS, L. et al. Atomic quantum gases in Kagomé lattices. Physical ReviewLetters , v. 93, n. 3, p. 030601, 2004.98 TEICHMANN, N.; HINRICHS, D., HOLTHAUS, M. Reference data for phasediagrams of triangular and hexagonal bosonic lattices. EuroPhysics Letters , v. 91, n. 1, p.10004, 2010.99 NIETNER, C. Quantum phase transition of light in the Jaynes-Cummings lattice .2010. 141 p. (Diploma thesis) - Department of Physics, Free University of Berlin, Berlin2010.100 NIETNER, C.; PELSTER, A. Ginzburg-Landau theory for the Jaynes-Cummings-Hubbard model. Physical Review A , v. 85, n. 4, p. 043831, 2012.101 LANDAU, L. D. The theory of a Fermi liquid. Journal of Experimental andTheoretical Physics , v. 30, n. 6, p. 920, 1956.102 TOMONAGA, S. Remarks on Bloch’s method of sound waves applied tomany-fermion problems. Progress of Theoretical Physics , v. 5, n. 4, p. 544-569, 1950.103 LUTTINGER, J. M. An exactly soluble model of a many-fermion system. Journalof Mathematical Physics , v. 4, n. 9, p. 1154-1162, 1963.104 MATTIS, D. C.; LIEB, E. H. Exact solution of a many-fermion system and itsassociated boson field. Journal of Mathematical Physics , v. 6, n. 2, p. 304, 1965. eferences 105 HOHENBERG, P. C. Existence of long-range order in one and two dimensions. Physical Review , v. 158, n. 2, p. 383-386, 1967.106 PITAEVSKII, L.; STRINGARI, S. Uncertainty principle, quantum fluctuations, andbroken symmetries. Journal of Low Temperature Physics , v. 85, n. 5, p. 337-388, 1991.107 MERMIN, N. D.; WAGNER, H. Absence of ferromagnetism or antiferromagnetismin one- or two-dimensional isotropic Heisenberg models. Physical Review Letters , v. 17, n.22, p. 1133-1136, 1966.108 WIDOM, A. Statistical mechanics of rotating quantum liquids. Physical Review , v.168, n. 1, p. 150-155, 1968.109 BAGNATO, V.; KLEPPNER, D. Bose-Einstein condensation in low-dimensionaltraps. Physical Review A , v. 44, n. 11, p. 7439-7441, 1991.110 LIEB, E. H.; LINIGER, W. Exact analysis of an interacting Bose gas. I. The generalsolution and the ground state. Physical Review , v. 130, n. 4, p. 1605-1616, 1963.111 LIEB, E. H. Exact analysis of an interacting Bose gas. II. The excitation spectrum. Physical Review , v. 130, n. 4, p. 1616-1624, 1963.112 YANG, C. N.; YANG, C. P. Thermodynamics of a one-dimensional system of bosonswith repulsive delta-function interaction. Journal of Mathematical Physics , v. 10, n. 7, p.1115, 1969.113 TONKS, L. The complete equation of state of one, two and three-dimensional gasesof hard elastic spheres. Physical Review , v. 50, n. 10, p. 955-963, 1936.114 GIRARDEAU, M. Relationship between systems of impenetrable bosons andfermions in one dimension. Journal of Mathematical Physics , v. 1, n. 6, p. 516, 1960.115 MORITZ, H. et al. Exciting collective oscillations in a trapped 1d gas. PhysicalReview Letters , v. 91, n. 25, p. 250402, 2003.116 PAGANO, G. et al. A one-dimensional liquid of fermions with tunable spin. NaturePhysics , v. 10, n. 3, p. 198, 2014.117 SALCES-CARCOBA, F. et al. Equations of state from individual one-dimensionalBose gases. New Journal of Physics , v. 20, p. 113032, 2018. DOI: 10.1088/1367-2630/aaef9b.118 MCGUIRE, J. B. Study of exactly soluble one-dimensional N-body problems. Journal of Mathematical Physics , v. 5, n. 5, p. 622, 1964.119 YANG, C. N. Some exact results for the many-body problem in one dimension withrepulsive delta-function interaction. Physical Review Letters , v. 19, n. 23, p. 1313-1315,1967.120 GAUDIN, M. Un systeme a une dimension de fermions en interaction. PhysicsLetter A , v. 24, n. 1, p. 55-56, 1967.121 SUTHERLAND, B. Further results for the many-body problem in one dimension. Physical Review Letters , v. 20, n. 3, p. 98-100, 1968. References 122 LAI, C. K.; YANG, C. N. Ground-state energy of a mixture of fermions and bosonsin one dimension with a repulsive δ -function interaction. Physical Review A , v. 3, n. 1, p.393-399, 1971.123 MCGUIRE, J. B. Interacting fermions in one dimension. I. repulsive potential. Journal of Mathematical Physics , v. 6, n. 3, p. 432, 1965.124 MCGUIRE, J. B. Interacting fermions in one dimension. II. attractive potential. Journal of Mathematical Physics , v. 7, n. 1, p. 123, 1966.125 LUTHER, A.; EMERY, V. J. Backward scattering in the one-dimensional electrongas. Physical Review Letters , v. 33, n. 10, p. 589-592, 1974.126 FUCHS, J. N.; RECATI, A.; ZWERGER, W. Exactly solvable model of theBCS-BEC crossover. Physical Review Letters , v. 93, n. 9, p. 090408, 2004.127 BUSCH, T. et al. Two cold atoms in a harmonic trap. Foundations of Physics , v. 28,p. 549-559, 1998. DOI: 10.1007/s00601-004-0071-1128 AHARONY, A.; ENTIN-WOHLMAN, O.; IMRY, Y. Exact solution for twointeracting electrons on artificial atoms and molecules in solids. Physical Review B , v. 61,n. 8, p. 5452-5456, 2000.129 ABAD, J.; ESTEVE, J. G. Exact two-particle spectrum of the Heisenberg-Peierlsspin chain. Journal of Few-Body Systems , v. 37, p. 107, 2005.130 OLSHANII, M.; DUNJKO, V. Short-distance correlation properties of theLieb-Liniger system and momentum distributions of trapped one-dimensional atomicgases. Physical Review Letters , v. 91, n. 9, p. 090401, 2003.131 YAO, H. et al. Tan’s contact for trapped Lieb-Liniger bosons at finite temperature. Physical Review Letters , v. 121, n. 22, p. 220402, 2018.132 GIAMARCHI, T. Quantum physics in one dimension . Oxford: Clarendon Press,2003.133 GROSS, E. P. Structure of a quantized vortex in boson systems. Il Nuovo Cimento ,v. 20, p. 454, 1961. DOI: 10.1007/BF02731494.134 PITAEVSKII, L. P. Vortex lines in an imperfect Bose gas. Journal of Experimentaland Theoretical Physics , v. 13, n. 2, p. 451, 1961.135 TAN, S. Energetics of a strongly correlated Fermi gas. Annals of Physics , v. 323, n.12, p. 2952-2970, 2008.136 TAN, S. Large momentum part of a strongly correlated Fermi gas. Annals ofPhysics , v. 323, n. 12, p. 2971-2986, 2008.137 TAN, S. Generalized virial theorem and pressure relation for a strongly correlatedFermi gas. Annals of Physics , v. 323, n. 12, p. 2987-2990, 2008.138 BARTH, M.; ZWERGER, W. Tan relations in one dimension. Annals of Physics , v.326, n. 10, p. 2544, 2011. eferences 139 STEWART, J. T. et al. Verification of universal relations in a strongly interactingFermi gas. Physical Review Letters , v. 104, n. 23, p. 235301, 2010.140 SAGI, Y. et al. Measurement of the homogeneous contact of a unitary Fermi gas. Physical Review Letters , v. 109, n. 22, p. 220402, 2012.141 CHANG, R. et al. Momentum-resolved observation of thermal and quantumdepletion in a bose gas. Physical Review Letters , v. 117, n. 23, p. 235303, 2016.142 WILD, R. J. et al. Measurements of Tan’s contact in an atomic Bose-Einsteincondensate. Physical Review Letters , v. 108, n. 14, p. 145305, 2012.143 YAN, Z. et al. Boiling a unitary Fermi liquid. Physical Review Letters , v. 112, n. 9,p. 093401, 2019.144 HOINKA, S. et al. Precise determination of the structure factor and contact in aunitary Fermi gas. Physical Review Letters , v. 110, n. 5, p. 055305, 2013.145 LAURENT, S. et al. Connecting few-body inelastic decay to quantum correlationsin a many-body system: a weakly coupled impurity in a resonant Fermi gas. PhysicalReview Letters , v. 118, n. 10, p. 103403, 2017.146 DECAMP, J. et al. High-momentum tails as magnetic-structure probes for stronglycorrelated SU ( κ ) fermionic mixtures in one-dimensional traps. Physical Review A , v. 94,n. 5, p. 053614, 2016.147 DECAMP, J. et al. Strongly correlated one-dimensional Bose–Fermi quantummixtures: symmetry and correlations. New Journal of Physics , v. 19, n. 12, p. 125001,2017.148 MINGUZZI, A.; VIGNOLO, P.; TOSI, M. P. High-momentum tail in the Tonks gasunder harmonic confinement. Physics Letters A , v. 294, n. 3-4, p. 222-226, 2002.149 GIRARDEAU, M. D.; MINGUZZI, A. Soluble models of strongly interactingultracold gas mixtures in tight waveguides. Physical Review Letters , v. 99, n. 23, p.230402, 2007.150 GRINING, T. et al. Crossover between few and many fermions in a harmonic trap. Physical Review A , v. 92, n. 6, p. 061601, 2015.151 GRINING, T. et al. Many interacting fermions in a one-dimensional harmonic trap:a quantum-chemical treatment. New Journal of Physics , v. 17, n. 11, p. 115001, 2015.152 MATVEEA, N.; ASTRAKHARCHIK, G. E. One-dimensional multicomponentFermi gas in a trap: quantum Monte Carlo study. New Journal of Physics , v. 18, n. 6, p.065009, 2016.153 PÂT , U O. I.; KLÜMPER, A. Thermodynamics, contact, and density profiles of therepulsive Gaudin-Yang model. Physical Review A , v. 93, n. 3, p. 033616, 2016.154 DECAMP, J. et al. Exact density profiles and symmetry classification for stronglyinteracting multi-component Fermi gas in tight waveguides. New Journal of Physics , v.18, n. 5, p. 055011, 2016. References 155 DECAMP, J.; ALBERT, M.; VIGNOLO, P. Tan’s contact is a cigar-shaped diluteBose gas. Physical Review A , v. 97, n. 3, p. 033611, 2018.156 XU, W.; RIGOL, M. Universal scaling of density and momentum distributions inLieb-Liniger gases. Physical Review A , v. 92, n. 6, p. 063623, 2015.157 RIZZI, M. et al. Scaling behavior of Tan’s contact for trapped Lieb-Liniger bosons:from two to many. Physical Review A , v. 98, n. 4, p. 043607, 2018.158 SAKURAI, J. J. Advanced quantum mechanics . Boston: Addison-Wesley, 1967.159 BRANSDEN, B. H.; JOACHAIN, C. J. Physics of atoms and molecules . New York:Longman, 1983.160 COHEN-TANNOUDJI, C. N.; DIU, B.; LALOE, F. Quantum mechanics . New York:Wiley, 1991.161 SAKURAI, J. J.; NAPOLITANO, J. Modern quantum mechanics . Boston:Addison-Wesley, 2011.162 MANDEL, L.; WOLF, E. Optical coherence and quantum optics . Cambridge:Cambridge University Press, 1995.163 ASHCROFT, N. W.; MERMIN, N. D. Solid state physics . Fort Worth Saunders:College Publishing, 1976164 BLOCH, F. Über die Quantenmechanik der Elektronen in Kristallgittern. Zeitschriftfür Physik , v. 52, p. 555, 1929. DOI: 10.1007/BF01339455.165 WANNIER, G. H. The structure of electronic excitation levels in insulating crystals. Physical Review , v. 52, n. 3, p. 91, 1937.166 WANNIER, G. H. Dynamics of band electrons in electric and magnetic fields. Reviews of Modern Physics , v. 34, n. 4, p. 645, 1962.167 KITTEL, C. Introduction to solid state physics . 7th ed. Hoboken: John Wiley &Sons, 1976.168 HUBBARD, J. Electron correlations in narrow energy bands Proceedings of theRoyal Society of London A , v. 276, n. 1365, p. 238, 1963.169 GERSCH, H. A.; KNOLLMAN, G. C. Quantum cell model for bosons. PhysicalReview , v. 129, n. 2, p. 959-967, 1963.170 BOER, J. H.; VERWEY, J. W. Semi-conductors with partially and with completelyfilled d -lattice bands. Proceedings of the Physical Society , v. 49, p. 59, 1937. DOI:10.1088/0959-5309/49/4S/307.171 MOTT, N. F.; PEIERLS, R. Discussion of the paper by de Boer and Verwey. Proceedings of the Physical Society , v. 49, n. 4S, p. 72, 1937.172 MOTT, N. F. The basis of the electron theory of metals, with special reference tothe transition metals. Proceedings of the Physical Society A , v. 62, n. 7, p. 416, 1949. eferences 173 ABRAMOWITZ, M.; STEGUN, I. A. Handbook of mathematical functions: withformulas, graphs, and mathematical tables. Mineola: Dover Publications, 1965.174 GRADSHTEYN, I. S.; RYZHIK, I. M. Table of integrals, series, and products . 7thed. Cambridge: Academic Press, 2007.175 ZWERGER, W. Mott–Hubbard transition of cold atoms in optical lattices. Journalof Optics B: quantum and semiclassical optics, v. 5, n. 2, p. S9, 2003.176 ALBUS, A.; ILLUMINATI, F.; EISERT, J. Mixtures of bosonic and fermionic atomsin optical lattices. Physical Review A , v. 68, n. 2, p. 023606, 2003.177 EHRENFEST, P. Phasenumwandlungen im ueblichen und erweiterten Sinn,classifiziert nach dem entsprechenden Singularitaeten des thermodynamischen Potentiales. Verhandlingen der Koninklijke Akademie van Wetenschappen , v. 36, p. 153, 1933.178 JAEGER, G. The Ehrenfest classification of phase transitions: introduction andevolution. Archive for History of Exact Sciences , v. 53, n. 1, p. 51, 1998.179 MIRANSKY, V. A. Dynamical symmetry breaking in quantum field theories .Singapore: World Scientific, 1993.180 KLEINERT, H.; SCHULTE-FROHLINDE, V. Critical properties of Φ -theories Singapore: World Scientific, 2001.181 ZINN-JUSTIN, J. Quantum field theory and critical phenomena . London: OxfordUniversity Press, 1996.182 STANLEY, E. H. Introduction to phase transitions and critical phenomena . London:Oxford University Press, 1971.183 LANDAU, L. D.; LIFSHITZ, E. M. Statistical physics . 3rd ed. New York: PergamonPress, 1980.184 SANT’ANA, F. T.; PELSTER, A.; SANTOS, F. E. A. Finite-temperaturedegenerate perturbation theory for bosons in optical lattices. Physical Review A , v. 100, n.4, p. 043609, 2019.185 KÜBLER, M. et al. Improving mean-field theory for bosons in optical lattices viadegenerate perturbation theory. Physical Review A , v. 99, n. 6, p. 063603, 2019.186 HOFFMANN, A. Bosons in optical lattices . 2011. (Diploma thesis) - FreieUniversität Berlin, Berlin, 2011.187 HUBAC, I.; WILSON, S. Brillouin-Wigner methods for many-body systems .Switzerland: Springer Nature, 2010.188 THOMAS, L. H. The calculation of atomic fields. Mathematical Proceedings of theCambridge Philosophical Society , v. 23, n. 5, p. 542, 1927.189 FERMI, E. Un metodo statistico per la determinazione di alcune prioprietàdell’atomo. Rendiconti Academia Dei Lincei , v. 6, p. 602, 1927.190 FÖLLING, S. et al. Formation of spatial shell structure in the superfluid to Mottinsulator transition. Physical Review Letters , v. 97, n. 6, p. 060403, 2006. References 191 GERBIER, F. Boson Mott insulators at finite temperatures. Physical Review Letters ,v. 99, n. 12, p. 120405, 2007.192 GRIFFITHS, D. J. Introduction to quantum mechanics . 2nd ed. Upper Saddle River:Pearson Prentice Hall, 2004.193 MESSIAH, A. Quantum mechanics . New York: Dover Publications, 2014.194 ARFKEN, G. B.; WEBER, H. J.; HARRIS, F. E. Mathematical methods forphysicists . 7th ed. Waltham: Academic Press, 2012.195 DECAMP, J. Symmetries and correlations in strongly interacting one-dimensionalquantum gases . 2018. 150p Ph. D. Thesis (Physics) - Institut de Physique de Nice,Université Côte d’Azur, 2018.196 BLEINSTEIN, N.; HANDELSMAN, R. A. Asymptotic expansions of integrals . NewYork: Dover Publications, 2010.197 BRACEWELL, R. N. The Fourier transform & its applications . 3rd ed. New York:McGraw-Hill„ 1999.198 NARASCHEWSKI, N.; GLAUBER, R. J. Spatial coherence and density correlationsof trapped Bose gases. Physical Review A , v. 59, n. 6, p. 4595-4607, 1999.199 PÂT , U O. I.; KLÜMPER, A. Universal Tan relations for quantum gases in onedimension. Physical Review A , v. 96, n. 6, p. 063612, 2017.200 FEYNMAN, R. P. Forces in molecules, Physical Review , v. 56, n. 4, p. 340-343, 1939.201 PAREDES, B. et al. Tonks-Girardeau gas of ultracold atoms in an optical lattice. Nature , v. 429, n. 6989, p. 277, 2004.202 KINOSHITA, T.; WENGER, T.; WEISS, D. S. Observation of a One-DimensionalTonks-Girardeau Gas. Science , v. 305, n. 5687, p. 1125, 2004203 GIRARDEAU, M. D.; WRIGHT, E. M.; TRISCARI, J. M. Ground-state propertiesof a one-dimensional system of hard-core bosons in a harmonic trap. Physical Review A , v.63, n. 3, p. 033601, 2001.204 FORRESTER, P. J. et al. Finite one-dimensional impenetrable Bose systems:occupation numbers. Physical Review A , v. 67, n. 4, p. 043607, 2003.205 KOLOMEISKY, E. B. et al. Low-dimensional Bose liquids: beyond theGross-Pitaevskii approximation. Physical Review Letters , v. 85, n. 6, p. 1146-1149, 2000.206 VIGNOLO, P.; MINGUZZI, A.; TOSI, M. P. Exact particle and kinetic-energydensities for one-dimensional confined gases of noninteracting fermions. Physical ReviewLetters , v. 85, n. 14, p. 2850-2853, 2000.207 REICHL, L. E. A modern course in statistical physics . 4th ed. New York:Wiley-VCH, 2016.208 LENARD, A. One-dimensional impenetrable bosons in thermal equilibrium. Journalof Mathematical Physics , v. 7, n. 7, p. 1268, 1966. eferences 209 VIGNOLO, P.; MINGUZZI, A. Universal contact for a Tonks-Girardeau gas at finitetemperature. Physical Review Letters , v. 110, n. 2, p. 020403, 2013.210 SANT’ANA, F. T. et al. Scaling properties of Tan’s contact: embedding pairs andcorrelation effect in the Tonks-Girardeau limit. Physical Review A , v. 100, n. 6, p. 063608,2019.211 ZÜRN, G. et al. Fermionization of two distinguishable fermions. Physical ReviewLetters , v. 108, n. 7, p. 075303, 2012.212 WENZ, A. N. et al. From few to many: observing the formation of a Fermi sea oneatom at a time. Science , v. 342, n. 6157, p. 457, 2013..213 BLATT, J. M.; WEISSKOPF, V. F. Theoretical nuclear physics . New York:Springer-Verlag, 1979.214 HUANG, K.; YANG, C. N. Quantum-mechanical many-body problem withhard-sphere interaction. Physical Review , v. 105, n. 3, p. 767-775, 1957.215 OLSHANII, M. Atomic scattering in the presence of an external confinement and agas of impenetrable bosons. Physical Review Letters , v. 81, n. 5, p. 938-941, 1998.216 YANG, C. N. Concept of off-diagonal long-range order and the quantum phases ofliquid He and of superconductors. Reviews of Modern Physics , v. 34, n. 4, p. 694-704,1962. ppendix APPENDIX A – ATOMIC COLLISIONS IN COLD GASES In this appendix we derive the general theory regarding two-body collisions thathappens to give rise to interacting properties of cold atoms in three and one dimensions. A.1 The three-dimensional case Firstly, we want to find a solution for the respective Schrödinger equation " − (cid:126) µ ∇ + V ( r ) − E ψ ( r ) = 0 (A.1)in terms of the relative distance r = r − r and the reduced mass µ = m/ 2. Now, let usexpand the wave function making use of the spherical harmonics: ψ ( r ) = ∞ X ‘ =0 φ ‘ ( r ) Y ‘, ( θ ) . (A.2)Substituting the solution (A.2) into (A.1) we have that d dr u ‘ ( r ) − ‘ ( ‘ + 1) r u ‘ ( r ) + 2 µ (cid:126) [ E − V ( r )] u ‘ ( r ) = 0 , (A.3)where u ‘ ( r ) ≡ rφ ‘ ( r ). In the asymptotic region r (cid:29) r , where r is the range of theinteratomic potential V ( r ), the solution of (A.1) can be regarded as the contributions ofan incident plane wave in the x direction and a spherical scattered wave ψ ( r ) ’ r →∞ e ikx + f ( θ ) e ikr r , (A.4)where f ( θ ) is the scattering amplitude as a function of the angle between ~r and the x -axis θ , which relates the differential cross section dσ to the solid angle element d Ω in elasticscattering processes via dσ = | f ( θ ) | d Ω . (A.5)Therefore, for r (cid:29) r , the term ∝ /r in (A.3) can be neglected, yielding the solution u ‘ ( r ) = A ‘ sin kr − π‘ δ ‘ ! , (A.6)where A ‘ is the normalization constant and δ ‘ is simply a phase factor. By performing theexpansion e ikx = 12 ikr ∞ X ‘ =0 (2 ‘ + 1) P ‘ (cos θ ) h e ikr − e − i ( kr − π‘ ) i (A.7)and choosing A ‘ = (2 ‘ + 1) i ‘ e iδ ‘ , we have that, from (A.2) = (A.4), f ( θ ) = 12 ik ∞ X ‘ =0 (2 ‘ + 1) P ‘ (cos θ ) (cid:16) e iδ ‘ − (cid:17) . (A.8) Chapter A Atomic collisions in cold gases In the ultracold regime, we can consider the atomic energies low enough so that wemust only consider the ‘ = 0 term in the expansion, also known as s -wave approximation. Within such a consideration, the scattering amplitude reduces to f ( θ ) ≈ e iδ − ik = 1 k cot( δ ) − ik . (A.9)So, we now define the so-called s-wave scattering length as a ≡ − lim k → f ( θ ) = − tan( δ ) k . (A.10)It is worth noting that the scattering length a is deeply related to the initial condition ofthe radial solution (A.6) as 1 u ( r ) du ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =0 = k cot( δ ) = − a . (A.11)Moreover, the definition of the scattering length implies that the total scattering crosssection yields σ = 4 πa , which turns out to be the same as an impenetrable sphere ofradius a . It was shown in HUANG; YANG that, instead of performing all considerationsthat were done until now, the problem can be equivalently formulated by introducing thepseudopotential ˆ V ps ( r ) = 4 π (cid:126) am δ ( r ) ∂∂r r (A.12)in the respective Schrödinger equation " − (cid:126) µ ∇ + ˆ V ps ( r ) − E ψ ( r ) = 0 . (A.13)Finally, we conclude that, via the introduction of the pseudopotential, theinteraction parameter in the three-dimensional case is defined as g ≡ π (cid:126) a m , (A.14)with a being the three-dimensional s -wave scattering length. A.2 The one-dimensional case The one-dimensional scattering amplitude within the pseudopotential approxima-tion was performed by M. Olshanii in 1998. Thus, the considerations under such atreatment are: a) the atomic motion is allowed to happen freely along the x -axis; b) theharmonic potential possesses a frequency ω ⊥ , acting along the yz -plane; c) the interactionbetween particles is of the form of the pseudopotential introduced in Eq. (A.12). Therefore,the respective Schrödinder equation reads " − (cid:126) µ ∇ x + ˆ H ⊥ + ˆ V ps − E ψ ( r ) = 0 , (A.15) .2 The one-dimensional case where ˆ H ⊥ = − (cid:126) µ (cid:16) ∇ y + ∇ z (cid:17) + µω ⊥ y + z ) . (A.16)Assuming that the incident wave corresponds to a particle in the ground state of ˆ H ⊥ , ψ = e ikx φ ( y, z ), the asymptotic solution of (A.15) is given by ψ ( r ) ∼ x →∞ (cid:16) e ikx + f even ( k )e ik | x | + f odd ( k )sgn( x )e ik | x | (cid:17) φ ( y, z ) , (A.17)where f even/odd ( k ) is the even/odd partial wave scattering amplitude. For low velocities kr (cid:28) a ⊥ (cid:28) | a | , and regarding the one-dimensionalpseudopotential V ps ( x ) = − (cid:126) ma δ ( x ) , (A.18)the odd scattering amplitude vanishes, f odd = 0, while the even one yields I f even ( k ) ≈ − (1 + ika ) − , (A.19)where a = − a ⊥ a (cid:18) − C a a ⊥ (cid:19) (A.20)is the one-dimensional s -wave scattering length, a ⊥ ≡ q (cid:126) /µω ⊥ , and C ≡ lim u →∞ Z u du √ u − u X u =1 √ u ! ≈ . . (A.21)In conclusion, the analogous one-dimensional pseudopotential (A.18) retains the properscattering behavior, so that we can define the corresponding interaction strength in 1D as g ≡ − (cid:126) ma . (A.22) I For the detailed analytical calculation, see Ref. 215.43 APPENDIX B – BOSE-EINSTEIN CONDENSATION IN LOW DIMENSIONS In this appendix we present a concise description of the underlying physics involvedin low-dimensional BECs. Let us begin with the D -dimensional ideal Bose gas. Thechemical potential µ is determined by satisfying the number of particles formula X k β ( E k − µ ) − N. (B.1)Now, suppose the chemical potential vanishes, µ → 0, at a critical temperature T c ≡ ( k B β c ) − . Then, as the density of states (DOS) becomes a continuous one, ρ ( E ), we performthe substitution P k → ( L/ π ) D R d D k and then the number of particles formula reads Z ∞ dE ρ ( E )e β c E − N. (B.2)However, as the DOS depends on the dimensionality as ρ ( E ) ∝ E D/ − , the integral in(B.2) diverges for D ≤ 2. This implies that µ = 0 in D ≤ T > 0. In the 3D scenario, ρ ( E ) ∝ √ E , thence ρ ( E ) E → → 0, meaning that a macro-scopic occupation of the lowest-energy state is favorable. Differently, in 1D we have that ρ ( E ) ∝ E − / , implying that ρ ( E ) E → → ∞ , disfavoring the occupancy of the lowest-energyconfiguration. In 2D, the DOS is independent of E and the analysis for the occurrence ofBEC is a bit more subtle: the main point is that in 2D the off-diagonal long-range orderbreaks down, but a quasi long-range order associated with a topological quantum phasetransition emerges. I When the system is spatially confined, the picture changes and BEC can emerge inlow dimensions d ≤ This happens because there is a change in the DOS-dependencyon the energy due to the dependency on the size occupied by such a nonuniform gas, e.g. , consider the power law potential V ( r ) ∝ r α ; because of L D ∝ E D/α , the density ofstates changes according to ρ ( E ) ∝ L D E D/ − = E D/α + D/ − . Thus, the condition for theintegral (B.2) to converge becomes D > αα + 2 , (B.3)which results in D > / V ( r ) ∝ r and D > V ( r ) ∝ r . B.1 Hohenberg-Mermin-Wagner theorem In this section we are concerned with the derivation of the Hohenberg-Mermin-Wagner theorem , that states the absence of the U (1)-symmetry breaking at T > D ≤ I For a detailed explanation, see UEDA. Chapter B Bose-Einstein condensation in low dimensions and is associated with the proof of Bogoliubov’s inequality: h{ ˆ A, ˆ A † }ih [ ˆ B † , [ ˆ H, ˆ B ]] i ≥ k B T |h [ ˆ A, ˆ B ] i| . (B.4)Here ˆ A and ˆ B are arbitrary operators, ˆ H is the Hamiltonian of the system, [ · · · ] and {· · · } are, respectively, the commutator and the anticommutator, while h· · ·i stands forthe thermal average of an arbitrary operator ˆ O , h ˆ Oi ≡ X n P n h n | ˆ O| n i , P n ≡ e − βE n Tr h e − β ˆ H i , (B.5)where ˆ H | n i = E n | n i . Let us define( ˆ A, ˆ B ) ≡ X m = n h n | ˆ A † | m ih m | ˆ B | n i P m − P n E n − E m . (B.6)Now, making use of the hyperbolic inequality x − tanh( x ) ≤ 1, we have thattanh [ β ( E m − E n ) / β ( E m − E n ) / p m − p n ) β ( p m − p n )( E n − E m ) ≤ ⇒ < p m − p n E n − E m ≤ β p m − p n ) . (B.7)Thence, it follows that( ˆ A, ˆ A ) = X m = n |h m | ˆ A | n i| p m − p n E n − E m ≤ β X m,n |h m | ˆ A | n i| ( p m − p n ) = β h{ ˆ A, ˆ A † }i . (B.8)In addition, let ˆ C ≡ [ ˆ B † , ˆ H ], so we work out( ˆ C, ˆ C ) = X m = n h n | ˆ C † | m ih m | [ ˆ B † , ˆ H ] | n i p m − p n E n − E m = X m,n h n | ˆ C † | m ih m | ˆ B † | n i ( p m − p n )= h [ ˆ B † , ˆ C † ] i = h [ ˆ B † , [ ˆ H, ˆ B ]] i , (B.9)and ( ˆ A, ˆ C ) = X m = n h n | ˆ A † | m ih m | [ ˆ B † , ˆ H ] | n i p m − p n E n − E m = X m,n h n | ˆ A † | m ih m | ˆ B † | n i ( p m − p n )= h [ ˆ B † , ˆ A † ] i . (B.10)Now, let us introduce the Schwarz inequality ( ˆ A, ˆ A )( ˆ C, ˆ C ) ≥ | ( ˆ A, ˆ C ) | . (B.11)By the direct substitution of (B.8), (B.9), and (B.10) into (B.11), we prove the Bogoliubovinequality (B.4). .1 Hohenberg-Mermin-Wagner theorem Now, let the operators be ˆ A = ˆ a † p and ˆ B = ˆ ρ p ≡ P k ˆ a † k ˆ a k + p , then h{ ˆ A, ˆ A † }i = 2 n p + 1 , n p ≡ h ˆ a † p ˆ a p i , (B.12a)[ ˆ B † , [ ˆ H, ˆ B ]] = [ ˆ ρ † p , [ ˆ H, ˆ ρ † p ]] = N p m , II (B.12b)[ ˆ A, ˆ B ] = − ˆ a † . (B.12c)By using these results into Eq. (B.4), the Bogoliubov inequality results in n p ≥ mk B Tp |h ˆ a i| N − . (B.14)Therefore, it follows from (B.14) the absence of the U (1)-symmetry breaking in D ≤ T > P p n p (unless h ˆ a i = 0). In the zero-temperature regime, h ˆ a i 6 = 0 can happen in 2D, but it cannot in 1D. In order to showthis, let us make use of the following inequality: h{ ˆ A † , ˆ A }ih{ ˆ B † , ˆ B }i ≥ |h [ ˆ A † , ˆ B ] i| . (B.15)Similarly to the previous association, let the operators be ˆ A = ˆ a † p and ˆ B = ˆ ρ p . Hence, h{ ˆ A, ˆ A † }i = 2 n p + 1 , (B.16a) h{ ˆ B † , ˆ B }i = 2 h ˆ ρ p ˆ ρ † p i , (B.16b)[ ˆ A, ˆ B ] = − ˆ a † . (B.16c)Consequently, the inequality (B.15) results in n p ≥ |h ˆ a i| h ˆ ρ p ˆ ρ † p i − . (B.17)In the low-momentum limit p → 0, we have that h ˆ ρ p ˆ ρ † p i ≤ N p mv s , (B.18)where v s is the sound velocity. II Consider the Hamiltonianˆ H = X p E p ˆ a † p ˆ a p + 12 X k , p , q V k ˆ a † p + k ˆ a † q − k ˆ a q ˆ a p = X p E p ˆ a † p ˆ a p + 12 X k V k ˆ ρ † k ˆ ρ k − N X p V p . By performing some algebraic manipulations and considering E k = k / m and ˆ N ≡ P k ˆ a † k ˆ a k ,we have that [ ˆ H, ˆ ρ p ] = X k ( E k − E k + p )ˆ a † k ˆ a k + p , [ ˆ ρ † p , [ ˆ H, ˆ ρ p ]] = X k ( E k + p + E k − p − E k )ˆ a † k ˆ a k = ˆ N p m . Chapter B Bose-Einstein condensation in low dimensions Proof. Starting with [ ˆ ρ † p , [ ˆ H, ˆ ρ p ]] = ˆ N p m , (B.19)multiplying both sides by e β ˆ H / Tr h e β ˆ H i and taking the trace, we obtain Z ∞ (cid:126) ωS ( p , ω ) dω = E p N, (B.20)where S ( p , ω ) ≡ h e β ˆ H i X m,n e βE m (cid:16) |h m | ˆ ρ † p | n i| + |h m | ˆ ρ p | n i| (cid:17) δ ( (cid:126) ω − (cid:126) ω nm ) , (B.21)is called dynamic structure factor and ω nm ≡ ω n − ω m . From the Schwarz inequality (B.11),one obtains S ( p ) = Z + ∞−∞ S ( p , ω ) dω ≤ sZ + ∞−∞ (cid:126) ωS ( p , ω ) dω Z + ∞−∞ ( (cid:126) ω ) − S ( p , ω ) dω, (B.22)with S ( p ) being called static structure factor . From (B.20), the inequality reduces to S ( p ) ≤ s E p N Z + ∞−∞ ( (cid:126) ω ) − S ( p , ω ) dω. (B.23)In the low-momentum limit, it can be shown that lim p → Z + ∞−∞ ( (cid:126) ω ) − S ( p , ω ) dω = N (cid:126) mv s , (B.24)which results in S ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p → ≤ N pmv s . (B.25)Consequently, we have that n p ≥ mv s | ˆ a | N p − . (B.26)The aftermath of Eq. (B.26) is the following: because P p n p diverges in 1D, we concludethat the U (1)-symmetry breaking does not occur, even at T = 0, unless h ˆ a i vanishes. Onthe other hand, for D = 2, the sum P p n p is a convergent one, thence it is possible theappearance of the U (1)-symmetry breaking.To conclude this appendix, we showed that the Hohenberg-Mermin-Wagner theoremforbids the breaking of the U (1) symmetry in low dimensions. However, BEC can existeven without the presence of the U (1)-symmetry breaking. APPENDIX C – BOSE-EINSTEIN CONDENSATION AND SUPERFLUIDITY The occurrence of BEC is associated with an important quantity that, dependingon the textbook, is known as one-body density matrix or one-body correlator , % (1) ( r , r ) ≡ Tr h ˆ % ˆΨ † ( r ) ˆΨ( r ) i = h ˆΨ † ( r ) ˆΨ( r ) i , (C.1)where ˆ % is the density operator, while ˆΨ † ( r ) and ˆΨ( r ) are the field operators that createand annihilate a particle at the spatial position r , respectively. This physical quantity hasa clear interpretation: it is the probability amplitude of the annihilation of a particle atposition r followed by the creation of a particle at position r . As the system undergoesBEC, the waves of the individual particles overlap over each other, enhancing theirindistinguishability and, consequently, producing the effect of increasing the correlationbetween long-distance particles. Mathematically, this fact corresponds to lim | r − r |→∞ % (1) ( r , r ) = 0 , (C.2)and is known as off-diagonal long-range order (ODLRO).As we shall discuss, the concepts of BEC and superfluidity are strongly related,although there is a subtle relation between them. For example, the special case of anideal gas that undergoes BEC presents no superfluidity and a 2D superfluid presents noBEC. Despite of these two referred cases, there are many systems in which BEC andsuperfluidity coexist.Let us consider a time-dependent system and its associated one-body density matrix % (1) ( r , r ; t ) = h ˆ ψ † ( r , t ) ˆ ψ ( r , t ) i = X ν n ν ( t ) ψ ∗ ν ( r , t ) ψ ν ( r , t ) , (C.3)with n ν ( t ) satisfying X ν Z ∞ n ν ( t ) dt = N. (C.4)Considering that ν = 0 represents the BEC mode, it follows thatlim | r − r |→∞ % (1) ( r , r ; t ) = n ψ ∗ ( r , t ) ψ ( r , t ) . (C.5)We can interpret Ψ( r , t ) ≡ √ n ψ ( r , t ) (C.6)as being the condensate wave function as well as % ( r , t ) ≡ Ψ ∗ ( r , t )Ψ( r , t ) (C.7) Chapter C Bose-Einstein condensation and superfluidity as being the particle density in the BEC state. From the continuity equation ∂∂t % ( r , t ) + ∇ · j ( r , t ) = 0 , (C.8)the current of particles yields j ( r , t ) = − i (cid:126) m [Ψ ∗ ( r , t ) ∇ Ψ( r , t ) − Ψ( r , t ) ∇ Ψ ∗ ( r , t )] . (C.9)Decomposing the BEC wave function into an amplitude times a phase,Ψ( r , t ) = A ( r , t )e iφ ( r ,t ) , (C.10)the density and the current possess the form % ( r , t ) = A ( r , t ) , (C.11a) j ( r , t ) = (cid:126) m A ( r , t ) ∇ φ ( r , t ) . (C.11b)Now, from the definition of the superfluid velocity, v s ( r , t ) ≡ j ( r , t ) % ( r , t ) = (cid:126) m ∇ φ ( r , t ) , (C.12)it is possible to realize the connection between BEC and superfluidity: the phase of thecondensate wave function φ plays the role of the velocity potential in the superfluid. nnex ANNEX A – PUBLICATIONS M. Kübler, F. T. Sant’Ana, F. E. A. dos Santos, and A. Pelster, Improving mean-field theory for bosons in optical lattices via degenerate perturbation theory, Phys. Rev. A , 063603 (2019).Felipe Taha Sant’Ana, Axel Pelster, and Francisco Ednilson Alves dos Santos,Finite-temperature degenerate perturbation theory for bosons in optical lattices, Phys.Rev. A , 043609 (2019).F. T. Sant’Ana, F. Hébert, V. Rousseau, M. Albert, and P. Vignolo, Scalingproperties of Tan’s contact: Embedding pairs and correlation effect in the Tonks-Girardeaulimit, Phys. Rev. A100