Few strongly interacting ultracold fermions in one-dimensional traps of different shapes
FFew strongly interacting ultracold fermions in one-dimensional traps of differentshapes
Daniel Pęcak ∗ and Tomasz Sowiński Institute of Physics, Polish Academy of Sciences,Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
The ground-state properties of a few spin-1/2 fermions with different masses and interacting viashort-range contact forces are studied within an exact diagonalization approach. It is shown that,depending on the shape of the external confinement, different scenarios of the spatial separationbetween components manifested by specific shapes of the density profiles can be obtained in thestrong interaction limit. We find that the ground-state of the system undergoes a specific transitionbetween orderings when the confinement is changed adiabatically from a uniform box to a harmonicoscillator shape. We study the properties of this transition in the framework of the finite-size scalingmethod adopted to few-body systems.
I. INTRODUCTION
With recent experiments on several particles confinedin a one-dimensional optical trap (for fermions as wellas for bosons), quantum engineering has entered a com-pletely new, so far unexplored, area of strongly correlatedquantum systems [1–4]. In these extremely sophisticatedexperiments it is possible to control the total numberof particles, their mutual interactions, and the shape ofexternal potential with very high accuracies [3–7]. Asa consequence, a deep analysis of many properties ofone-dimensional few-body systems is performed experi-mentally. For example, fermionization of distinguishableparticles [8], pairing for attractive forces [9], ground-stateproperties in double-well schemes [10], or the formationof the Fermi Sea [11, 12] have been observed already. Inparallel, on a theoretical level many interesting resultshave been obtained under the assumption that particlesare confined in a harmonic trap [13–31]. They are await-ing experimental confirmations. Some results also forother confinements, like the double-well potential, havebeen discussed recently [32, 33] and the dynamical prop-erties of such systems have been analyzed.Apart from a few exceptions [34–41], it has commonlybeen assumed that particles of different kinds have thesame mass and the main impact on properties of the sys-tem comes from an imbalance of the number of particles.However, recently it was shown that for particles confinedin a harmonic trap, the mass difference between differentfermionic components leads to their spatial separation ifinteractions are strong enough [42]. The mechanism wasshown to be universal with respect to the number of par-ticles and also very robust to external perturbations. Aremaining open question concerns the properties whendifferent shapes of the trap are considered. This ques-tion is interesting also from an experimental point ofview, since shape-manipulation is one of the standardexperimental methods that are well controlled in labo-ratories. Recently, it was even possible to perform the ∗ [email protected] first Bose-Einstein condensation in a purely uniform boxconfinement [7]. Motivated by this background, here weexplore the properties of a spatial separation mechanismfor a two-flavoured mixture of fermions confined in a one-dimensional trap with a tunable shape. We show that,depending on the shape, in the strong interaction limitspatial separation in the many-body ground-state mayoccur for either the lighter or the heavier component.Moreover, the system undergoes a kind of critical transi-tion that is induced by an adiabatic change of the exter-nal potential. This mechanism appears to be very generaland it is present always whenever fermions of differentmass are being considered. We believe that our resultsmay shed some light on the quantum magnetism [43–50]and the role of mass imbalance in spatial separation ofthe density profiles [51, 52].The article is organized as follows. In an introductorySection II we describe the system to be studied and wedefine the tunable shape of the external trap that willbe considered in further analysis. Then, in Section IIIwe briefly summarize the exact diagonalization method– our main tool for studying different properties of few-body problems. The spectral properties of the few-bodyHamiltonian from the point of view of different masscomponents as well as different trap shapes are stud-ied in Section IV. Subsequently, in Section V we focuson properties of the ground-state of the system and wediscuss the spatial separation of density profiles inducedby different masses in a uniform box potential. We alsooutline the similarities and differences in comparison toharmonic confinement. Section VI emphasizes the funda-mental differences regarding single-particle densities be-tween systems with the same and with different massesof the components. In this section, basing on numeri-cal results, we also postulate that for any confinementone of two types of separation will always occur in thesystem when particles of different flavours have differentmasses. This observation leads us to make a numericalstudy of the transition between different density order-ings in Section VII. In that Section we adopt the wellknown finite-size scaling method to a few-body system.Finally, we conclude in Section VIII. a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t λ = 0 V σ / ( m σ ω L ) λ = 0.5Position x/L V σ / ( m σ ω L ) λ = 0.25 λ = 1Position x/L FIG. 1. The shape of the potentials V σ ( x, λ ) for different val-ues of the parameter λ in natural units of a given flavour.For λ = 0 a uniform box potential is restored. For increas-ing λ , the confinement transforms to the standard harmonicoscillator. II. THE SYSTEM UNDER STUDY
In this paper we consider a two-flavour mixture ofseveral ultra-cold fermionic atoms confined in an effec-tively one-dimensional external potential. Experimen-tally, a one-dimensional geometry is obtained by apply-ing a very strong harmonic confinement in the two re-maining spatial directions[6, 8, 12]. Depending on theexperimental realization, atoms in the two flavours canhave the same or different masses. The latter system isrealized simply by trapping different chemical elements.The most promising fermionic mixture of this type isthe lithium-potassium combination. Obtaining a mix-ture of fermions of the same mass is a more sophisti-cated procedure and can be achieved when two differentnuclear spin projections of the same element are undercontrol. A typical example is the mixture of two differ-ent Li atoms with total atomic spin belonging to thespin- / and spin- / representations, respectively. Re-gardless of the situation, in both scenarios particles ofdifferent flavours can be treated as fundamentally distin-guishable, i.e. each fermion always belongs to one of thetwo flavours and during the whole experiment its naturecannot be changed [6]. This is a kind of superselectionprinciple originating in the observation that interactionsbetween atoms cannot change neither the mass of theatoms nor the spin projection of their nuclei.It is a very good approximation to assume that ultra-cold fermions of different kinds interact only via spher-ically symmetric forces modeled by a zero-range δ -likepotential [53]. In this approximation, fermions belongingto the same flavour do not interact at all due to the an-tisymmetry of the wave function when written in termsof relative positions. In this approximation the Hamilto- nian of the system reads ˆ H = N ↓ (cid:88) i =1 (cid:20) − (cid:126) m ↓ ∂ ∂x i + V ↓ ( x i , λ ) (cid:21) ++ N ↑ (cid:88) i =1 (cid:20) − (cid:126) m ↑ ∂ ∂y i + V ↑ ( y i , λ ) (cid:21) + g N ↓ ,N ↑ (cid:88) i,j =1 δ ( x i − y j ) , (1)where V σ ( x, λ ) is an external potential acting on fermions σ . We model the external potential as follows: V σ ( x, λ ) = (cid:26) λm σ ω x if | x | < L ∞ if | x | > L, (2)where λ is a dimensionless geometric parameter that de-termines the shape of the trap. For clearness, we usedifferent letters for positions of particles belonging to dif-ferent components.The confinement reproduces a uniform box potential oflength L in the limit of λ → and a cropped harmonicoscillator trap of frequency ω in the limit of λ → . Ofcourse, in the latter case hard walls affect and modify thesingle-particle eigenstates of the Hamiltonian. However,for low excited states and for a large enough L , the differ-ence between an exact harmonic oscillator potential andone modeled by V σ ( x, can be neglected. This conclu-sion comes from the observation that the wave functionsof the harmonic oscillator decay exponentially and do notpenetrate the regions in the vicinity of the hard walls ofthe uniform box [54].In Fig. 1 we schematically show the shape of the ex-ternal potential for different values of λ in natural unitsof a given flavour. It is worth noticing that potential(2) seems to be quite natural from an experimental pointof view. It resembles the technique of turning off a har-monic oscillator potential in the presence of an additionaluniform potential with hard walls [7]. Nevertheless, wehave also checked a few other scenarios of crossover froman uniform box to a harmonic trap and found that the re-sults described here do not depend qualitatively on thesedetails.The effective interaction coupling strength g is re-lated to its three-dimensional counterpart and can beobtained by integrating out two remaining degrees ofmotion[55]. From the point of view of our model theimportant information is that the interaction strengthcan be tuned experimentally over the whole range of itspossible values, i.e. from minus to plus infinity [56–58].Note that in contrast to higher dimensions, in a one-dimensional case, the Dirac δ function is a well definedself-adjoint Hermitian operator and can be used withoutany regularization[59].For a given shape of the confinement λ , we numeri-cally find single-particle states φ ( λ ) nσ ( x ) and correspondingenergies E ( λ ) nσ with a direct diagonalization of the single-particle Hamiltonian H ( λ ) σ = − (cid:126) m σ d d x + V σ ( x, λ ) (3)The diagonalization is performed in the position domainon a dense grid with spacing δx . In this representationany single-particle Hamiltonian has a simple tridiago-nal form. Therefore, a diagonalization is straightforwardwith standard numerical recipes [60]. It is quite obvi-ous that along with decreasing δx , eigenstates and theireigenenergies converge to exact values. Here, to makenumerical analysis possible, we assume that convergenceis achieved when the relative numerical error of a num-ber n cutoff of the lowest states is smaller than . Thestates φ ( λ ) nσ ( x ) serve as the basis for further many-bodyanalysis.In the limiting case of the harmonic oscillator( λ → ), the single-particle eigenfunctions of bothflavours are related by the following scaling: φ (1) n ↑ ( x ) =( m ↑ /m ↓ ) / φ (1) n ↓ ( (cid:112) m ↑ /m ↓ x ) . This means that the wavefunctions of the heavier particles are more localized inthe center of the trap. In this case, the eigenenergies donot depend on the mass of the particle and they dependlinearly on the main quantum number n = 1 , , , ... : E (1) n = (cid:126) ω (cid:18) n − (cid:19) . (4)Note, that for consistence of the whole analysis, we enu-merate the single-particle states in such a way that theground state is denoted by n = 1 and not by n = 0 asusually used in the literature for the harmonic oscilla-tor problem. It is also worth noticing that for a highenough excitation n , corrections from the hard-wall con-straints become relevant. As explained before, to avoidthis problem in our numerical approach, we set the size ofthe hard-wall box large enough to assure that the single-particle states that are appreciably occupied are not dis-turbed. We have checked that for our choice of L , theresults of a pure harmonic oscillator confinement are re-stored for λ = 1 . Therefore, in the following we will treat λ = 1 as a pure harmonic oscillator confinement.In the opposing case of a uniform box potential ( λ =0 ), the shapes of the wave functions do not depend onthe mass and they have the well known form φ (0) nσ ( x ) = (cid:114) L sin (cid:20) nπ ( x + L )2 L (cid:21) . (5)However, in this case, the single-particle eigenenergiesdepend on mass and the quantum number n = 1 , , , ...E (0) nσ = (cid:126) π n m σ L ∝ n . (6)In what follows we will express all quantities in har-monic oscillator units of the spin- ↓ particles, i.e. alllengths are measured in units of (cid:112) (cid:126) / ( m ↓ ω ) , energies in (cid:126) ω , momenta in units of (cid:112) (cid:126) m ↓ ω , etc. We also introducethe dimensionless mass ratio parameter µ = m ↑ /m ↓ .This is substantially greater than unity for the lithium-potassium mixture, µ = 40 / . In these units, the single-particle Hamiltonians (3) have the form H ( λ ) ↓ = −
12 d d x + 12 λx , H ( λ ) ↑ = − µ d d x + µ λx . (7)To make the later analysis clear, we fix the size of thesystem in such a way that the single-particle spectra ofthe extreme Hamiltonians (i.e. those for a box trap andfor a harmonic oscillator potential) have energy gaps ofthe same order of magnitude i.e. E (0)2 ↓ − E (0)1 ↓ ≈ E (1)2 − E (1)1 , which corresponds to the following condition ≈ (cid:126) π m ↓ ωL . (8)This condition determines an appropriate size of the sys-tem for numerics, L ≈ . (cid:112) (cid:126) / ( m ↓ ω ) . To make surethat the walls do not noticeably affect the single-particledensities in the case of the cropped harmonic poten-tial, we set the position of the walls to a larger value, L = 7 (cid:112) (cid:126) / ( m ↓ ω ) . With this condition, the energy gapsare still of the same order of magnitude.For our numerical purposes it is convenient to rewritethe Hamiltonian (1) in a dimensionless form in the secondquantization formalism as follows: ˆ H = (cid:88) σ (cid:90) L − L d x ˆΨ † σ ( x ) H ( λ ) σ ˆΨ σ ( x )++ g (cid:90) L − L d x ˆΨ †↑ ( x ) ˆΨ †↓ ( x ) ˆΨ ↓ ( x ) ˆΨ ↑ ( x ) , (9)where the dimensionless interaction strength is g = g (cid:112) m ↓ / ( ω (cid:126) ) . All integrations are performed over thewhole space where the particles could be present, i.e. inthe region between the walls ( − L, L ) . The field operator ˆΨ σ ( x ) annihilates fermions of spin σ at a position x . Thequantum fields obey canonical anti-commutation rela-tions for same spin particles (cid:110) ˆΨ σ ( x ) , ˆΨ † σ ( x (cid:48) ) (cid:111) = δ ( x − x (cid:48) ) and (cid:110) ˆΨ σ ( x ) , ˆΨ σ ( x (cid:48) ) (cid:111) = 0 . In contrast, due to the fun-damental distinguishability of opposite spin fermions ex-plained before, the final result and the values of calcu-lated observables do not depend on the choice of thecommutation relations for opposite spin operators [61].However, as commonly used for distinguishable parti-cles, we assume commutation of the field operators inthis case, (cid:104) ˆΨ ↑ ( x ) , ˆΨ †↓ ( x (cid:48) ) (cid:105) = (cid:104) ˆΨ ↑ ( x ) , ˆΨ ↓ ( x (cid:48) ) (cid:105) = 0 . Notethat in the Hamiltonian (9) there are no terms thatchange the number of particles of a given flavour. Asa consequence, the total number of fermions of a givenflavour, ˆ N σ = (cid:82) L − L ˆΨ † σ ( x ) ˆΨ σ ( x )d x , commutes with themany-body Hamiltonian (9). This property of the modelcorresponds to realistic experimental situations where thenumber of particles can be controlled with an extremeprecision[6, 12]. From the numerical point of view, it en-ables one to perform a complete analysis of the Hamilto-nian independently in each of the subspaces correspond-ing to a given number of particles. III. EXACT DIAGONALIZATION APPROACH
The ground-state properties of the system are stud-ied straightforwardly within an exact diagonalizationapproach for the many-body Hamiltonian. Recently,the method has been successfully used for equal massfermions confined in a harmonic trap [13, 62, 63] as well asfor fermions of different masses [42]. First we decomposethe field operators ˆΨ σ ( x ) into the basis of the eigenfunc-tions of the corresponding single-particle Hamiltonians(3), ˆΨ σ ( x ) = (cid:88) n φ ( λ ) nσ ( x )ˆ a nσ , (10)where an operator ˆ a nσ annihilates a fermion of the σ -type in level n , i.e. a fermion in a single-particle statedescribed by the wave function φ ( λ ) nσ ( x ) . Note that forsimplicity we omit the superscript λ in the definition ofan annihilation operator since it should not lead to anyconfusion.The expansion (10) is exact provided the sum runs overall n . In practice, to perform numerical calculations wecut the summation at a value n cutoff chosen in such a waythat the final results do not change significantly when n cutoff is increased. Of course, for stronger interactions g ,more single-particle levels should be taken into account toachieve the convergence. For example, for g = 4 , N ↓ = 2 ,and N ↑ = 3 , we use 12 single-particle eigenstates for eachcomponent, i.e. the dimension of the many-body Hilbertspace is 14520.With the expansion (10) the Hamiltonian (9) can berewritten in the form ˆ H = (cid:88) σ (cid:88) n E ( λ ) nσ ˆ a † nσ ˆ a nσ + (cid:88) ijkl U ( λ ) ijkl ˆ a † i ↑ ˆ a † j ↓ ˆ a k ↓ ˆ a l ↑ , (11)where E ( λ ) nσ is a single-particle energy. The interactionenergy has the form U ( λ ) ijkl = g (cid:90) d x ¯ φ ( λ ) i ↑ ( x ) ¯ φ ( λ ) j ↓ ( x ) φ ( λ ) k ↓ ( x ) φ ( λ ) l ↑ ( x ) . (12)The Hamiltonian (11) is represented using all its matrixelements between states belonging to the Fock space of allthe possible many-body configurations of N ↑ and N ↓ par-ticles occupying the first n cutoff single-particle orbitals.Finally, an exact diagonalization of the matrix obtained isperformed using the Implicitly Restarted Arnoldi method[64] available in the ARPACK Fortran library. This al-lows us to find the many-body ground-state of the system | G (cid:105) , several excited states | G i (cid:105) and their eigenenergies E i .In this way complete information about the structure ofthe many-body ground-state (and excited states if neces-sary) can be obtained. In what follows, we concentrate onthe simplest quantity that can be measured experimen-tally in a straightforward way, namely the single-particledensity profile (normalized to the number of particles ina given flavour) ρ σ ( x ) = (cid:104) G | ˆΨ † σ ( x ) ˆΨ σ ( x ) | G (cid:105) . (13) IV. MANY-BODY SPECTRAL PROPERTIES
First let us study how the spectral properties of themany-body Hamiltonian are affected by the shape of theexternal potential λ and mass ratio µ . The results fora harmonic oscillator (shown in the upper panels of Fig.2) were recently discussed with all details in [42]. There,it was shown that along with an increasing mass ratio µ the quasi-degeneracy of the many-body spectrum issplit in the limit of strong interactions. This is caused bythe lifting of some global symmetries of the Hamiltonianthat are present only in equal mass systems. As a con-sequence, separation between spin components appearsin the ground-state of the system for strong enough in-teractions, i.e. the heavier particles always concentratein the middle of the trap and the cloud of light parti-cles is divided into two parts and pushed out from thecenter. It was noticed that the separation of the den-sity profile induced by a mass imbalance always displaysthe same features regardless of the number of particles inboth flavours.The situation is very similar for the case of a uniformbox potential ( λ = 0 ). The many-body spectrum be-comes more complicated for strong interactions wheneverdifferent masses of constituents are introduced (bottomright panel of Fig. 2). The main qualitative differenceappears in the limit of vanishing interactions – in con-trast to the case of the harmonic oscillator, the spectrumof the uniform box with noninteracting particles changeswith µ . This is a direct consequence of the form of thesingle-particle energies (4) and (6).At this point it is also worth noting that for an equalmass system, and for any confinement, there exist many-body eigenstates that are absolutely insensitive to theinteraction strength (seen as horizontal lines in the leftpanels of Fig. 2). These states, commonly named afterGirardeau [66], are straightforwardly constructed using asingle Slater determinant of N ↑ + N ↓ single-particle or-bitals. Such wave functions are antisymmetric under theexchange of the positions of any two fermions, regard-less of their spin. Thus they are the eigenstates of theinteraction part of the Hamiltonian. This constructionof completely antisymmetric states can only be adoptedfor equal mass systems since only then the single-particleorbitals are the same for both flavours. This is the rea-son why the Girardeau states are not present in the right µ = 1 E ne r g y Interaction g E ne r g y µ = 40/6 C r opped H a r m on i c T r ap ( λ = ) Interaction g U n i f o r m B o x ( λ = ) FIG. 2. Spectra of the system consisting of N ↓ = 3 and N ↑ = 1 fermions as a function of the dimensionless interactionstrength g . The top row corresponds to a harmonic oscillatorpotential and the bottom row to a box trap potential. Quasi-degenerate energy bands seen in the left column split up whenmass imbalance µ (cid:54) = 1 is introduced (right column). Theenergy is given in the natural units of harmonic oscillator, (cid:126) ω . panels of Fig. 2. V. SEPARATION OF FLAVOURS IN THEUNIFORM BOX
As mentioned previously, in harmonic confinement, themass difference between fermions of different flavoursleads to the separation of density profiles of oppositespecies for strong enough repulsions. In the case of theuniform box potential a separation of the density also oc-curs in the system. However, in this case, the separationis present always in the heavier component (see Fig. 3).The direct reason why a mass difference acts differentlyfor different confinements can be explained intuitivelyvia energetic arguments. As mentioned previously, inthe uniform box case, the single-particle wave functionsare exactly the same for both components and they arecompletely independent of mass difference. Therefore,the part of the energy cost for exciting a particle to ahigher state that comes from the interaction, is indepen-dent of the flavour. The only difference in energies comesfrom the single-particle part of the Hamiltonian. The en-ergy cost for exciting heavier particles is smaller (see eq.(6)) and therefore the separation in heavier componentis favoured. This argumentation is completely oppositeto that in the case of harmonic confinement (see [42] for details) and therefore the separation is governed by anopposite rule.These intuitive pictures and arguments are confirmedby our numerical calculations. In Fig. 3, the single-particle density is plotted for a strongly interacting sys-tem of two species characterized by a mass ratio of µ = 40 / . We have checked that the separation occursin the strong interaction limit for any number of parti-cles up to seven. From the same calculations we haveseen that for mass ratios µ closer to , a much strongerinteraction is needed to create the separation in densityprofiles. This observation is also in accordance with ourintuitive picture, i.e. for almost equal masses neithercomponent is favored and much stronger interactions areneeded to break the symmetry and support separation.For completeness, in Fig. 4, we show a comparison ofseparations for the two confinements considered, i.e. theuniform box (left panels) and the harmonic trap (rightpanels). Matching plots are obtained for the same num-ber of particles in both components and the same inter-action strengths. From this comparison it is obvious thatthe separation mechanism induced by a mass imbalanceacts completely differently in the two cases. VI. COMPARISON TO THE EQUAL MASSSYSTEM
Before we analyze the transition in the ground-statebetween the two orderings described above, let us com-pare the situation to the case when both flavours have thesame mass. It is known that in this case the separationcan be induced only by a difference in the number of par-ticles, N ↑ − N ↓ . This arises directly from the general sym-metry under global exchange of both families of particles.As consequence, whenever N ↑ = N ↓ , both flavours havethe same single-particle density profile and no separationof the density profile can be observed. The situation ismodified when the system is slightly imbalanced in thenumber of particles. As an example we concentrate onthe system with N ↑ = 3 and N ↓ = 2 particles. As seen inthe left panels of Fig. 5, characteristic alternating oscilla-tions in the densities of the ground-state are built in thelimit of very strong repulsions and both components takeon an antiferromagnetic ordering. It is seen that an alter-nating ordering is present in the system independently ofthe shape of the external potential. This generalizes theresult obtained recently for harmonic confinement for fi-nite interactions[65], and extends the results obtained forinfinite interactions [67, 68].The situation is very different whenever differentmasses of the components are introduced (see the rightpanels in Fig. 5). Under harmonic confinement, theheavier particles concentrate in the middle and the lighterones are pushed out from the center. For the case of theuniform box, heavier particles are located in the vicin-ity of the walls and lighter ones are in the middle. Themiddle plot shows a generic situation for an intermediate Uniform Box 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =1N ↓ =2 D en s i t y ↑ =2N ↓ =1 D en s i t y ↑ =1N ↓ =3 D en s i t y ↑ =1N ↓ =6Position D en s i t y ↑ =2N ↓ =2 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =3N ↓ =1 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =1N ↓ =4 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =3N ↓ =4Position 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =2N ↓ =3 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =3N ↓ =2 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =4N ↓ =1 0 0.5 1 1.5 -4 -2 0 2 4N ↑ =6N ↓ =1Position FIG. 3. Single-particle densities ρ ↑ ( x ) (thick blue line, heavier flavour) and ρ ↓ ( x ) (thin red line, lighter flavour) calculated inthe ground-state of the system for different numbers of fermions with µ = 40 / and strong interaction g = 4 confined in abox trap. The black vertical lines correspond to the walls of the box trap. In contrast to the separation induced by the massdifference for a harmonic potential [42], in this case the separation always occurs in the heavier fraction, independently of theway the fermions are distributed between flavours. In particular, the separation is present also for an equal number of fermions N ↑ = N ↓ . The positions and the densities are measured in units of (cid:112) (cid:126) / ( m ↓ ω ) and (cid:112) m ↓ ω/ (cid:126) , respectively. confinement shape. It suggests that in this case, sep-aration is not present. One should remember however,that the plot is obtained for strong but finite interactions.Our numerical calculations, performed for many differentarrangements (different confinements and different num-bers of particles) show that for an arbitrary confinementin the range ≤ λ ≤ there exists some critical inter-action strength above which one of the two separationtypes occurs in the system. One can imagine that forinfinite interactions any few-fermion system with imbal-anced mass reveals spatial separation in single-particledistributions. The only question is if the separation isbuilt in the heavier or the lighter component. The answeris directly related to the shape of the confinement. Fromthe above analysis it follows that the system undergoessome kind of transition between different separations inthe limit of infinite interactions, which is driven by anadiabatic change of the potential. As explained below,the properties of this transition can be understood withmethods well known from the theory of quantum phasetransitions. VII. THE TRANSITION DRIVEN BY THESHAPE OF THE TRAP
As explained above, for the two extreme cases of auniform box and a harmonic trap, the density separa-tion induced by the mass imbalance is of an oppositekind. Depending on the spectrum of the single-particleHamiltonians, heavier or lighter particles are pushed outfrom the center for sufficiently large repulsions betweenparticles. In the framework of our model it is possibleto study the transition between these two orderings in-duced by an adiabatic change of the shape. To make thisanalysis not only qualitative but also quantitative oneshould introduce some quantity which indicates the kindof ordering. The choice is obviously not unique, howeverit is quite natural to concentrate on a magnetization-likedistribution defined as follows: M ( x ) = ρ ↑ ( x ) − ρ ↓ ( x ) . (14)It is quite natural that this distribution has an oppositebehavior whenever heavier or lighter particles are pushedout from the center of the trap. Since the distribution isnormalized to the difference of the total number of par-ticles, N ↑ − N ↓ , and also because it is symmetric under Uniform Box Cropped Harmonic Trap 0 0.5 1 1.5 -4 -2 0 2 4 N ↑ =1N ↓ =3 D en s i t y N ↑ =2N ↓ =2 D en s i t y N ↑ =3N ↓ =1 D en s i t y N ↑ =1N ↓ =4 D en s i t y N ↑ =2N ↓ =3 D en s i t y N ↑ =3N ↓ =2 D en s i t y N ↑ =4N ↓ =1 Position D en s i t y N ↑ =1N ↓ =3 N ↑ =2N ↓ =2 N ↑ =3N ↓ =1 N ↑ =1N ↓ =4 N ↑ =2N ↓ =3 N ↑ =3N ↓ =2 N ↑ =4N ↓ =1 Position
FIG. 4. Comparison of different separation scenarios forfermions with different mass ( µ = 40 / ) driven by differentshapes of the confinement, in the limit of strong interaction g = 4 : in the uniform box (left panels) and cropped har-monic oscillator (right panels). The thick blue and thin redlines represent the single-particle density profiles for heavyand light components, respectively. Note that, independentlyon the number of particles in a given flavour, the separationis always present in the heavier (for the uniform box) or thelighter (for the harmonic oscillator) component. The posi-tions and the densities are measured in units of (cid:112) (cid:126) / ( m ↓ ω ) and (cid:112) m ↓ ω/ (cid:126) , respectively. D en s i t y µ = 1 D en s i t y Position D en s i t y λ = µ = 40/6 λ = . λ = Position
FIG. 5. Single-particle densities ρ ↑ ( x ) (thick blue line, heavierflavour) and ρ ↓ ( x ) (thin red line, lighter flavour) calculatedin the ground-state of the system of N ↑ = 3 and N ↓ = 2 inthe limit of strong interaction, g = 4 . The positions and thedensities are measured in units of (cid:112) (cid:126) / ( m ↓ ω ) and (cid:112) m ↓ ω/ (cid:126) ,respectively. spatial reflections with respect to x = 0 , therefore thefirst distinction between the two orderings being consid-ered is manifested by the value of the second moment ofthe distribution σ = (cid:90) L − L d x x M ( x ) . (15)In Fig. 6 we show the dependence of σ on the trap pa-rameter λ for different numbers of particles and differentinteractions. It is seen in the two extreme confinements,that σ saturates to the two completely distinct valuescorresponding to the two different orderings. It meansthat σ plays the role of an order parameter and can beused as an indicator of a given ordering. As long a givenordering is present in the system, the parameter σ is al-most constant. Near the transition point, however, (apoint that is different for different numbers of particles)its value rapidly changes. Moreover, for stronger inter-actions, the transition is more sharp. Therefore, one cananticipate that for infinitely strong repulsions a charac-teristic ’step-like’ function is obtained. All the abovepoints mean that the transition between orderings ap-pearing for strong interactions has many of the propertiesof a phase transition [69, 70] and it can be analyzed withthe methods known from the theory of quantum phasetransitions [71, 72]. Here, the roles of the order parame-ter and the parameter of control are played by the secondmoment of the magnetization-like distribution σ , and theshape of the trap λ , respectively. From this point of view,the thermodynamic limit is mimicked by the limit of in- -10-7-4-1 0.3 0.5 0.7 0.9 1.1 λ σ N ↑ =1N ↓ =3 -12-6 0 6 0.3 0.5 0.7 0.9 1.1 λ σ N ↑ =2N ↓ =3 -8-4 0 4 8 12 0.3 0.42 0.54 0.66 λ σ N ↑ =2N ↓ =2 -6 0 6 12 0.3 0.5 0.7 0.9 λ σ N ↑ =3N ↓ =2 λ σ N ↑ =3N ↓ =1 -15-12-9-6-3 0.5 0.7 0.9 1.1 λ σ N ↑ =1N ↓ =4 FIG. 6. The second moment σ of the magnetization distri-bution (15) as a function of the shape of the confinementfor different interaction strengths (from g = 4 to g = 5 ).Each plot corresponds to given numbers of particles in bothflavours. Note that in extreme confinements, σ saturates toa well defined value, while it changes rapidly in the vicinityof the transition point. The second moment σ is given in thenatural units of a harmonic oscillator, (cid:126) / ( m ↓ ω ) . finitely strong repulsions between particles.To characterize the transition between different order-ings one should study not only the behavior of the orderparameter but also its derivatives. Naturally, the mostimportant of these is the lowest derivative that is diver-gent at the transition point. Our numerical results sug-gest that, in the case studied, the first derivative of σ has this property in the limit of infinite interactions. Inanalogy to the physics of phase transitions this quantityhas all the properties of the susceptibility since it mea-sures changes of magnetization under small changes ofthe parameter of control χ ( λ ) = d σ ( λ )d λ . (16)We numerically calculate the susceptibility χ for differ-ent numbers of particles and for different interactions g (examples for N ↓ + N ↑ = 4 are shown in the left pan-els of Fig. 7). The susceptibility calculated in this wayhas a natural behavior well known from the theory ofphase transitions. Its maximum grows with interactionsalong with a small shift of its position. One can antic-ipate that for infinitely large interactions the suscepti-bility will be divergent at the position of the transitionpoint. This behavior is a direct consequence of the sharp-ening of the σ function. The analogy with the theoryof quantum phase transitions is seen to be even closer when we adopt the well known finite-size scaling methodto determine the position of the transition point in thelimit of infinite interactions. First we assume that theorder parameter defined by σ has some natural scaling inthe vicinity of the transition point λ c , i.e. it is a homo-geneous function of its relevant parameters: interactionstrength g and the normalized shape of the trap definedas τ = ( λ − λ c ) /λ c . Consequently, the same property isshared by all its derivatives. Regarding the susceptibility,this means that there exists one universal function ˜ χ ( ξ ) that determines the shapes of all susceptibilities for dif-ferent confinements and interaction strengths. To makethe analogy to the theory of quantum phase transitions asclose as possible we assume the following scaling ansatz[69, 70]: χ ( τ, g ) = g γ/ν ˜ χ ( g /ν τ ) , (17)where ν and γ are appropriate critical exponents of themodel. If the assumption of the scaling property of thesusceptibility is correct, then there exists an appropriatechoice of critical exponents for which all numerical datapoints form the one universal curve determined by ˜ χ . Toshow that indeed our system has this scaling propertywe performed appropriate numerical calculations basedon the data-collapse method (for details see for example[69–71]). As the result of this numerical approach, weobtain the plots shown in the middle panels of Fig. 7.It is clearly visible that after appropriate scaling, allthe curves for a given system collapse to one universalcurve for a large range of normalized potential shapes τ .The position of the transition point λ c , as well as criti-cal exponents, are presented in the legend of their corre-sponding plots. Note that, depending on the number ofparticles, different values of the critical parameters areobtained. Finally, to make the presentation complete, inthe right panels of Fig. 7 we show the second momentof the distribution σ when the same scaling transforma-tion is performed. It is seen that also in this case alldata points collapse to one universal curve. Together, allthese results suggest that the transition between differ-ent orderings driven by an adiabatic change of the shapeof the trap, in the limit of very strong interactions, hasmany properties similar to those known from the the-ory of quantum phase transitions. This means that inthe limit of infinite interactions, for a given shape of thetrap, the system has a well established ordering. In thevicinity of the transition point, the system undergoes arapid transition – single-particle densities change to forma new ordering. VIII. CONCLUSIONS
To conclude, in this article we have discussed the prop-erties of several fermions confined in a one-dimensionaltrap in the very strong interaction limit. We show thatthe mass difference between components, independently χ χ λ χ g - γ / ν χ λ c = 0.0097 γ = 0.53 ν = 0.39 4.04.24.44.64.85.0 g - γ / ν χ λ c = 0.0054 γ = 0.23 ν = 0.05 4.04.24.44.64.85.0 g γ τ g - γ / ν χ λ c = 0.0029 γ = 0.33 ν = 0.17 4.04.24.44.64.85.0 -10-7-4-1 -8 -6 -4 -2 0 σ N ↓ = , N ↑ = γ = 0.53 λ c = 0.0097 4.04.24.44.64.85.0 -8-4 0 4 8 12 -120 -60 0 60 120 σ N ↓ = , N ↑ = γ = 0.23 λ c = 0.0054 4.04.24.44.64.85.0 g γ τ σ N ↓ = , N ↑ = γ = 0.29 λ c = 0.0033 4.04.24.44.64.85.0 FIG. 7. Scaling properties of a few-body system. Left panels: Susceptibility χ as a function of the shape of the confinement λ for different values of interactions and different number of particles. A characteristic peak of the susceptibility, whose heightincreases with g , is clearly visible. The vertical red line corresponds to the critical value λ c obtained after extrapolation of theresults to infinite repulsion. Middle panels: Rescaled susceptibility as a function of a rescaled confinement shape parameterobtained after adopting the data-collapse method. Note that all data points collapse to one well defined curve. Right panels:The second moment of the distribution σ when the same scaling procedure is performed for the trap shape parameter. Thesusceptibility χ is given in the natural units of a harmonic oscillator, (cid:112) (cid:126) / ( m ↓ ω ) . of the confinement’s shape, always leads to a spatial sep-aration between flavours. However, the nature of the sep-aration depends on the shape, i.e. for a given shape thedensity profile of either lighter or heavier particles is splitinto two parts and pushed out from the center of the trap.This observation subsequently led us to the concept of atransition between orderings driven by the shape of thetrap. We show that this transition has many properties incommon with standard quantum phase transitions, andcan be similarly analyzed within the finite-size scalingframework. In this way we find critical shape values fordifferent numbers of particles for which the system un-dergoes transitions and we have estimated the relevantcritical exponents for these transitions. It is worth notic- ing that in the case of one-dimensional systems, typicallysmooth crossovers rather than rapid transitions betweendifferent phases are suspected. From this point of viewthe transition predicted here is quite a rare and interest-ing phenomenon. ACKNOWLEDGMENTS
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