Direct mapping of the finite temperature phase diagram of strongly correlated quantum models
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Direct mapping of the finite temperature phase diagram of strongly correlatedquantum models
Qi Zhou , Yasuyuki Kato , Naoki Kawashima , and Nandini Trivedi Department of Physics, The Ohio State University, Columbus, OH 43210 Institute for Solid State Physics, University of Tokyo,5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan (Dated: November 10, 2018)Optical lattice experiments, with the unique potential of tuning interactions and density, haveemerged as emulators of nontrivial theoretical models that are directly relevant for strongly corre-lated materials[1, 2, 3, 4, 5, 6, 7, 8, 9]. However, so far the finite temperature phase diagram has notbeen mapped out for any strongly correlated quantum model. We propose a remarkable method forobtaining such a phase diagram for the first time directly from experiments using only the densityprofile in the trap as the input. We illustrate the procedure explicitly for the Bose Hubbard model,a textbook example of a quantum phase transition from a superfluid to a Mott insulator[10]. Using“exact quantum Monte Carlo simulations in a trap with up to 10 bosons, we show that kinks in thelocal compressibility, arising from critical fluctuations, demarcate the boundaries between superfluidand normal phases in the trap. The temperature of the bosons in the optical lattice is determinedfrom the density profile at the edge. Our method can be applied to other phase transitions evenwhen reliable numerical results are not available. The grand challenge of condensed matter physics is tounderstand the emergence of novel phases arising fromthe organization of many degrees of freedom especiallyin regimes where particle interactions dominate over thekinetic energy. Brought to the forefront by the discov-ery of high temperature superconductivity in complexcopper-based oxides, it is absolutely astounding that asimple model, like the Hubbard model is able to captureso many essential aspects of the physics of these mate-rials. However, whether the repulsive fermion Hubbardmodel really contains a d-wave superconducting groundstate is still an open question after decades of study.Optical lattice experiments with cold atoms are emerg-ing as an amazing laboratory for making realizations ofsuch bose and fermi Hubbard and Heisenberg-type mod-els and observing phase transitions without the uncer-tainty posed by the complex materials. In this back-drop quantum Monte Carlo simulations in strongly in-teracting regimes are emerging as an important bridgebetween materials based condensed matter physics andcold atoms, highlighted in Fig. 1. The scale of the nu-merical simulations possible today with up to 10 parti-cles is able to match the experimental cold atom systemsthereby allowing a direct comparison.The most important task for the quantum emulatorof a Hamiltonian H is to determine the correspondingphase diagram in the temperature T , chemical potential µ and g space, where the latter are parameters of H . Sofar such a mapping has not been determined experimen-tally in any model. Even for the simplest Bose Hubbardmodel there have been many challenges arising from (i)lack of a clear diagnostic of how to identify phases. Tra-ditional methods using the sharpness of the interferencepatterns are not reliable to distinguish the normal andsuperfluid phases; (ii) complications due to coexistenceof different phases in the same confining potential; (iii) Novel Materials
Cuprate SuperconductorsQuantum Magnets...
Optical Lattices
Alkali atoms
Models
Hubbard-Type ModelsHeisenberg-Type Models... (Li, Na,K, Rb,Cs)
FIG. 1: Interaction between novel materials, models and op-tical lattice emulators. The models capture the fundamentalphysics of complex materials with novel phenomena such ashigh temperature superconductivity and quantum magnetismand form a bridge with experiments on ultracold atoms inoptical lattices that emulate these models in a clean environ-ment. lack of thermometry of the Bose gas in the optical lattice;(iv) and until very recently, the lack of QMC simulationson experimentally relevant sizes. The work reported hereovercomes all of the above obstacles.The Hamiltonian for the single band Bose HubbardModel (BHM) is given by: H BHM = − tz X h i,j i ( b † i b j + h.c )+ U X i n i ( n i − − µ X i n i (1)where b i ( b † i ) is the boson destruction (creation) oper-ator at a site i , n i = b † i b i is the density operator, z = 6is the coordination number in 3D, and µ controls thedensity of bosons. The relative strength, g = t/U ofthe tunnelling t of bosons between nearest neighbor sitesvs. the repulsive interaction U between bosons, tunesthe system through a superfluid to Mott transition at t/U SuperfluidMottVacuum µ/U
Mott !" ! "&’()%*+,-).. ∆ = Tµ c t/U / µ/U µ a b ! ! ! ! c μ/U t/U=0.15t/U=0.05 C r i t i c a l t e m pe r a t u r e FIG. 2: Phase diagram of BHM. (a) µ − t/U plane at T = 0 (schematic); (b) µ − t/U plane at T = 0 (schematic). The superfluidphase is characterised by a non-zero order parameter h b † i 6 = 0 and a non-zero superfluid density ρ s = 0, both of which vanishin the normal phase. The Mott insulator at T = 0, characterized by integer filling and a finite gap to excitations ∆ , crossesover to the normal state for T ≈ ∆ . At finite temperatures, the critical chemical potential µ c ( T, t/U ) demarcates the S-Nphase boundary. In trapped atomic gases, the local chemical potential decreases from the centre of the trap to the edge (brownvertical line). (c) QMC simulations and finite size scaling to calculate the phase diagram in the µ − T plane for a fixed t/U . T = 0 (Fig. 2(a)). At finite T the system shows a phasetransition from a superfluid (S) to a normal(N) phase(Fig. 2(b)). In optical lattice experiments, bosons areconfined in an additional harmonic trap of frequency ω modelled by H = H BHM − mω P i r i n i where m is themass of the bosons. We simulate the Hamiltonian H withthe trap using quantum Monte Carlo techniques that in-clude the effect of strong interactions “exactly” withinstatistical errors. Recent modifications of the directed-loop algorithm for the world-line quantum Monte Carlomethod[11] have allowed us to significantly improve theefficiency near a critical point[12] for large 3D systemswith up to 10 bosons in a 64 lattice for the first time.Thus the phase diagram in the µ − T plane is calculatedby finite size scaling for fixed t/U .(Fig.2(c)).Experimentally, the challenge of obtaining the phasediagram at finite T lies in identifying measurable prop-erties that can diagnose different phases. Tradition-ally the momentum distribution n ( ~k ) imaged in the in-terference patterns of the expanding cold atom cloudshas been used to identify the phases. In a long-time ballistic expansion the interference pattern ˜ n ( ~r ) =( m/ ¯ hτ ) | W (cid:16) ~k = m~r ¯ hτ (cid:17) | n (cid:16) ~k = m~r ¯ hτ (cid:17) essentially providesan image of n ( ~k ) = P i,j h b † i b j i e i~k · ( ~r i − ~r j ) before expan-sion by convoluting with W ( ~k ) , the Fourier transformof the Wannier function within a single site. Here τ isthe expansion time. The final image detects the col-umn integrated momentum distribution N ⊥ ( k x , k y ) = R dk z | W ( ~k ) | n ( ~k ). We have previously shown for a ho-mogeneous BHM that sharp peaks in the interferencepattern are not reliable for identifying a superfluid[13,14, 15]. What is the effect of a confining potential on n ( ~k )?To answer this question we calculate both n ( ~k ) andthe density profile ρ ( r ) in a harmonic trap at different T and tuning t/U (Fig. 3). For comparison, within local density approximation (LDA), we also calculate ρ h ( µ ( r )),the density ρ h ( µ ) for a homogeneous system using QMC,where µ ( r ) = µ − mω r /
2. Fig.3 shows the excellentagreement of ρ ( r ) and ρ h ( µ ( r )), not unexpected for ashallow harmonic trap. We next turn to a key diagnos-tic, the local superfluid density distribution in the trap ρ hs ( µ ( r )), obtained from a knowledge of µ ( r ) and the su-perfluid density ρ hs ( µ ) in the homogeneous system. Themost significant observation from these data is that evenin the presence of a trap when all the atoms are in thenormal state, the interference pattern continues to showsharp peaks. Strong repulsive interactions that expandthe bosonic cloud and suppress T c in the optical latticecontribute to a sharpening up of n ( ~k ) even in the normalstate. With the lowering of T , as more regions within thetrap become superfluid, there is a distinct change in theshape of the sharp peak (Fig. 3(d)). The emergence ofa singular feature of width ∼ /L , limited by the cloudsize, indicates phase coherence throughout the sample.It is evident from Fig. 3 that n ( ~k ) which integratesover the entire trap is unable to provide local informa-tion about the distribution of the phases. Also the finiteexpansion time, as well as resolution problems compli-cates the analysis of the interference pattern ˜ n ( ~r ) and itsrelation to n ( ~k )[16]. We therefore explore direct methodsof identifying phase boundaries in the trap.Our proposal relies on extracting the local compress-ibility in the trap defined by κ diff = − mω r dρdr from ahigh-resolution scan of ρ ( r ) in a trap. κ diff agrees verywell with the compressibility κ h ( µ ( r )) = dρ h dµ for a bulksystem (Fig.4(a)). A novel feature of the local compress-ibility is the existence of kinks at specific locations inthe trap. The origin for the kink feature becomes evi-dent if we consider in addition the behavior of the localsuperfluid density in the trap ρ hs ( µ ( r )) (Fig.5). We seeclearly that the kinks in the compressibility occur at theS-N phase boundary. At fixed T , the superfluid order ! ! " Π ! Π ! !" ! ! ! ! " Π ! Π ! !" ! ! ! ! " Π ! Π ! !" ! ! ! ! " Π ! Π ! !" ! %& ! ! %&’()’ * ! " ! "! ! "! ! "! n ( k ) n ( k ) k k t/U = 0 . t/U = 0 . T/t = 0 . T/t = 0 . t/U = 0 . T/t = 0 . T/t = 0 . t/U = 0 . ! "! $! "!!%$!%&!%’!%( ! "! ! "! $! "!!%$!%&!%’!%( ! "! r/d n ( r ) r/d n ( r ) r/d n ( r ) r/d n ( r ) a bc d n ( k ) n ( k ) FIG. 3: Interference pattern in a trap with N ≈ bosons:The insets show the local density profile ρ ( r )(red boxes) ob-tained using QMC as a function of the radial coordinate inthe trap. Also shown is the excellent agreement with ρ h ( µ ( r ))calculated within local density approximation (LDA) (purplediamonds). The superfluid density ρ hs ( µ ( r )) is non-zero in asignificant portion of the trap only at sufficiently low temper-atures in panel (d). Note that sharp peaks are seen in n ( ~k )in panels (a), (b) and (c) even when all the atoms are in thenormal state. The peak in panel (d) is much sharper. Thedifference between the normal state and the superfluid stateis evident from the system-size dependence of the width ofthe peak. becomes weaker when approaching the phase boundary.The increase of fluctuations is reflected in the growth of κ h ( r ). Starting from deep in the Mott-like region, wherethe particle number per site is essentially unity, κ h ( r )increases when approaching the phase boundary with asuperfluid resulting in a singular feature in κ h ( r ). Thephase transition in the BHM is in the XY universalityclass with κ h ( r ) ∼ | µ − µ c | − α and α = − ρ s vanishes inthe trap, the kinks disappear in the absence of a phaseboundary and the compressibility changes smoothly.The importance of Fig. 4 and 5 is that purely froma high resolution scan of the local density it is possibleto obtain the local compressibility and from the exis-tence of kinks deduce that there must be superfluid re-gions in the trap separated from normal regions, eventhough both of them have non-integer filling and the den-sity changes smoothly across the phase boundary. Wewould like to point out that there are significant differ-ences between our work and previous studies that haveprimarily focused on the compressibility of Mott and su-perfluid phases at T = 0[17, 18, 19]. As seen from Fig.5(a), the compressibility of the Mott state is exactly zeroand becomes finite in the superfluid. At finite T , noth-ing singular happens as we move from Mott-like to morecompressible regions in the trap. It is only at the S-N boundary (Fig. 5(b,c,d)) even though formed betweentwo compressible phases, that the system shows enhanced !" !"
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10 20 40 600.20.40.60.81.0 a b r/d r/d n ( r ) r/d n i n t ( r ) FIG. 4: Kinks in the local compressibility and mapping thephase diagram: (a) The local density ρ ( r )(blue dots) agreeswith ρ h ( r )(purple diamonds) calculated for a homogeneoussystem. We also show that the compressibility κ diff ( r ) (redboxes) directly obtained from the density profile agrees with κ h ( µ ( r )) (pink triangles) calculated within LDA. The signalto noise in the derivative method to extract κ diff ( r ) can beimproved by taking an angular average of ρ ( r ). Experimen-tally, ρ ( r ) is obtained by an inverse Abel transformation ofthe column density[20] or by identifying the planar-integrateddensity with the pressure. Recently, the column density hasbeen measured in high resolution experiments[21]. Alterna-tively, ρ ( x, y, z ) at fixed z can be obtained directly[22] usinga sliding technique to get the density. (b) Determination oftemperature by fitting (shown by lines) of the tail of the den-sity profile (QMC results shown by symbols) to the ideal gasbehavior in the dilute regime e βµ ≪ | µ ( r )+2 t | ≫ Un sointeractions can be ignored. Inset shows the fitting to the an-gular integrated densities at the corresponding temperaturesthat have larger signals and therefore better accuracy. T /t are 0 .
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44 from top to bottom. critical fluctuations detected in the local compressibility.Note that in Fig. 5(b, d) there are kinks even in theabsence of any Mott regions.Usually strong interaction effects in an optical lat-tice hamper the determination of the temperature T and the chemical potential µ at the center in the op-tical lattice. We propose two methods: (i) By fitting ρ ( r ) with ρ ( µ ( r ) , T ) constrained by the known numberof bosons in the trap according to N = R d rρ ( r ) = R µ −∞ dµ √ µ − µ )( mω ) / ρ ( µ ). (ii) By fitting the tail of the den-sity profile in the trap to the expected ideal gas behav-ior ρ ideal = R d k (2 π ) (cid:16) e β ( ǫ k − µ + mω r / − (cid:17) − as demon-strated in Fig. 4(b). This is a direct method with-out any input from simulations. Thus from the location r c of kinks in κ diff ( r ), experimentalists can determine µ c = µ − mω r c / µ − T plane for a fixed t/U and compare with theoreticalresults in Fig. 2(c).Mapping the phase diagrams of strongly correlatedquantum models is the central goal of condensed matterphysics. In that context our proposed method of prob-ing fluctuations of the density, specifically singularites orcusps, arising from critical fluctuations, provides a gen-eral method for identifying the phase boundaries. Thecold atom emulator of a mathematical model in a confin-ing trap, essentially generates a chemical potential scan r/d µ/U T/t = 0 . T/t = 0 . t/U = 0 . t/U = 0 . r/d µ/U t/U = 0 . t/U = 0 . T/t = 0 . T/t = 0 . a bc d FIG. 5: Origin of kinks: The local density ρ ( µ ( r ))(purple),compressibility κ h ( µ ( r ))(red) and superfluid density ρ hs ( µ ( r ))(blue) as functions of the radial coordinate inthe trap. Panel (a) shows when ρ s = 0 in the trap, κ ( r )showsa smooth variation. As T is reduced or t/U is increased(b,c,d) and a finite ρ s develops in some portion of the trap, κ ( r ) shows sharp kinks. The location of these kinks coincideswith the S-N boundary in the trap. Notice the location ofthe kinks shifts as the S-N boundary changes. of the phase diagram from a single measurement. Ourstudies open up new directions for mapping the finitetemperature phase diagrams of strongly correlated quan-tum models, usually not available by other means, byprobing local properties of trapped atoms in optical lat-tices rather than relying on the momentum distribution.of the phase diagram from a single measurement. Ourstudies open up new directions for mapping the finitetemperature phase diagrams of strongly correlated quan-tum models, usually not available by other means, byprobing local properties of trapped atoms in optical lat-tices rather than relying on the momentum distribution.