Finite temperature phase transition for disordered weakly interacting bosons in one dimension
FFinite temperature phase transition for disordered weakly interacting bosons in onedimension
I.L. Aleiner , B.L. Altshuler , and G.V. Shlyapnikov , Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA Laboratoire de Physique Th´eorique et Mod`eles Statistique, Universit´e Paris Sud, CNRS, 91405 Orsay, France Van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65/67, 1018 XE Amsterdam, The Netherlands
It is commonly accepted that there are no phasetransitions in one-dimensional (1D) systems ata finite temperature, because long-range correla-tions are destroyed by thermal fluctuations. Herewe demonstrate that the 1D gas of short-range in-teracting bosons in the presence of disorder canundergo a finite temperature phase transition be-tween two distinct states: fluid and insulator.None of these states has long-range spatial corre-lations, but this is a true albeit non-conventionalphase transition because transport properties aresingular at the transition point. In the fluid phasethe mass transport is possible, whereas in the in-sulator phase it is completely blocked even at fi-nite temperatures. We thus reveal how the inter-action between disordered bosons influences theirAnderson localization. This key question, firstraised for electrons in solids, is now crucial forthe studies of atomic bosons where recent exper-iments have demonstrated Anderson localizationin expanding very dilute quasi-1D clouds.
The absence of finite temperature phase transitions inone-dimensional (1D) systems is considered as almost adogma. Its justification is based on another dogma whichstates that any phase transition is related to the appear-ance/disappearance of a long-range order or at least long-range spatial correlations. Thermal fluctuations in 1Dsystems destroy any long-range order, lead to exponen-tial decay of all spatial correlation functions and thusmake phase transitions impossible . Non-interactingquantum particles in a one-dimensional random poten-tial show a similar behavior in the sense that all single-particle eigenfunctions are localized, i.e. decay exponen-tially in space . The same statement holds for two di-mensional systems without spin-orbit interactions . Bycontrast the single-particle states in three dimensions areeither localized or extended as a result of the Andersontransition . In this paper we demonstrate that the 1Dgas of weakly short-range interacting bosons in the pres-ence of disorder exhibits a finite temperature phase tran-sition between two distinct states, fluid and insulator ,and the transition temperature depends on the disorder.None of these states is characterized by a long-range or-der or long-range spatial correlations. Moreover, ther-modynamic functions, such as specific heat, do not havesingularities at the transition point. From this point ofview, the dogma is not violated. Nevertheless, this isa true albeit non-conventional phase transition, becausetransport and energy dissipation properties of the fluid and insulator phases are dramatically different and aresingular at the transition. The difference between thefluid and insulator phases can be qualitatively under-stood by comparing two many-body 1D systems: inter-acting particles without disorder and the 1D Andersoninsulator (disorder without interactions). In the fluidwithout disorder, the dissipation of energy of an arbi-trarily slow external field and mass transport are pos-sible. At the same time, even at finite temperaturesthere is no mass transport in Anderson insulators andthe energy dissipation vanishes for the frequency of theexternal field tending to zero. Here we show that inter-acting 1D bosons in disorder demonstrate one of the twotypes of behavior and describe the phase diagram in thetemperature-disorder plane. We thus provide an answerto the subtle question of how the interaction betweendisordered particles may suppress Anderson localizationand permit them to acquire the fluid behavior. Thiswas the key problem for charge transport in electronicsystems, and it is now emerging in a new light in thestudies of disordered ultracold bosons. These studies aredriven by fundamental interest and by potential applica-tions of atom waveguides on a chip . Recent remarkableexperiments have demonstrated Anderson localizationin expanding extremely dilute quasi-1D Bose gases, andthe investigations of effects of the interparticle interac-tion in relatively dense clouds are underway. I. SINGLE PARTICLE LOCALIZATION IN 1D
Let us first discuss the density of states (DoS) for asingle particle with mass m in 1D and introduce relevantenergy and distance scales. In the absence of disorder theparticle eigenstates are plane waves with energies (cid:15) > ν ( (cid:15) ) = (cid:113) m/ π ¯ h (cid:15) and it diverges at (cid:15) → (cid:15) the DoS is only slightly affected by disorder: ν ( (cid:15) ) ≈ ν ( (cid:15) ) ∼ / √ (cid:15) . At the same time, the disordercuts off the DoS divergence at (cid:15) = 0 and transforms itinto a peak with a finite height ν ∗ and width E ∗ as shownin Fig. 1. Also, the disorder creates states with negativeenergies, thus making the DoS finite at (cid:15) <
0. For largenegative (cid:15) the DoS is exponentially small, and this partof the curve ν ( (cid:15) ) is known as Lifshitz tail .Both ν ∗ and E ∗ are determined by the statistics ofthe random potential U ( x ). For simplicity we assumea short-range Gaussian potential with the amplitude a r X i v : . [ c ond - m a t . d i s - nn ] O c t U and correlation length σ such that condition U (cid:28) ¯ h /mσ holds . Then the only relevant energy andlength scales are : E ∗ ∼ ( U σ m/ ¯ h ) / ; (1) ζ ∗ = ¯ h/ (cid:112) mE ∗ = (¯ h /U σm ) / . (2)They determine the width of the DoS peak and themaximum DoS value ν ∗ ∼ /E ∗ ζ ∗ . In order to obtainthese scales consider a weakly bound state of a parti-cle in the potential U ( x ), with an extension of the wavefunction, ζ (cid:29) σ . The particle energy can be written as E ∼ (¯ h / mζ − U (cid:112) σ/ζ ), where for the Gaussian dis-order the potential energy term is obtained multiplyingthe contribution of each potential well, U σ/ζ , by (cid:112) ζ/σ , which is the square root of the number of wells on thelength scale ζ . The energy E reaches a minimum value E ∗ at ζ ∼ ζ ∗ , with E ∗ and ζ ∗ given by Eqs. (1) and (2).We thus see that the single-particle spectrum can bedivided into three parts: high-energy states with (cid:15) (cid:29) E ∗ and ν ( (cid:15) ) ∼ (cid:113) m/ ¯ h (cid:15) , low-energy states located in theregion of the DoS peak, and Lifshitz tail at negative en-ergies. As we already noted, all single-particle eigenfunc-tions in 1D are localized with an energy-dependent lo-calization length ζ ( (cid:15) ). For high-energy states, (cid:15) (cid:29) E ∗ ,we have ζ ( (cid:15) ) ∼ (cid:15)ζ ∗ /E ∗ , whereas in the Lifshitz tail ζ ( (cid:15) ) ∼ ζ ∗ (cid:112) E ∗ / | (cid:15) | . For the low-energy states in theDoS peak, | (cid:15) | < ∼ E ∗ , the localization length is ζ ( (cid:15) ) ∼ ζ ∗ . U ( x ) U σ ψ ( x ) ψ ( x ) xx (cid:15)ζ ( (cid:15) ) ν ( (cid:15) ) ζ ζE ∗ ζ ∗ E ∗ ζ ∗ FIG. 1: Properties of single particle localization. In (a) the brown curve shows the disorder potential U ( x ), with blue solid linesand dashed red lines indicating the location of tail and high-energy states, respectively. The shape of the wavefunctions of thesestates is shown by blue (tail) and red (high-energy) curves ψ ( x ). In (b) the density of states ν and localization length ζ versusenergy (cid:15) are shown in blue for the DoS peak and tail states, and in red for high-energy states. For studying the many-bodylocalization transition, ν ( (cid:15) ) and ζ ( (cid:15) ) in the DoS peak (low-energy states) can be approximated by the green dotted lines. II. MANY-BODYLOCALIZATION-DELOCALIZATIONTRANSITION
Repulsive short-range interaction between bosons in1D gives rise to two other energy scales in addition to E ∗ :the temperature of quantum degeneracy T d = ¯ h n /m (we use the units with Boltzmann constant k B = 1), andthe interaction energy per particle, ng , where g is thecoupling constant for the interaction, and n is the meanboson density. We focus on the weakly interacting regimewhere the dimensionless coupling strength is small: γ ≡ ng/T d (cid:28) t = Tng = 1 γ TT d . (4)Another dimensionless parameter characterizes thestrength of the disorder: κ ≡ E ∗ /ng (5)so that large values of κ correspond to strong disorder.At this point we should make two important state-ments. If the disorder is extremely strong ( κ → ∞ ),the bosons occupy only states in the Lifshitz tail. Asthe DoS in the tail is exponentially small, the bosonsare distributed among the ”lakes” located exponentiallyfar from each other. The bosons can not hop betweenthe lakes, and the system is in an insulating state. Thereduction of κ at T = 0 eventually (for κ = κ c (cid:39)
1) trans-forms the insulator into an algebraic superfluid (spatialphase correlations do decay, but only algebraically). ThisKosterlitz-Thouless type transition was first analyzed inRef. and more recently discussed in relation to disor-dered Josephson chains and to cold atomic gases .However, well before the transition from insulating tosuperfluid state most of the particles find themselves inlow-energy states where the DoS is much higher than in the Lifshitz tail. Thus, we may neglect the tail in ourdiscussion of the fluid-insulator transition and consideronly low-energy and high-energy states (green and red inFig. 1b).Second, although the interaction between bosonsrenormalizes (screens) the disordered potential U ( x ), thisdoes not change the picture of single-particle eigenstatesin our discussion. The reason is that relevant particleenergies are of the order of E ∗ or larger. For κ > ∼ ng , and as we will seethere is no need to consider κ (cid:28)
1. In this respect, themain effect of the interaction on the ground state of thesystem is not screening the random potential but rathercontrolling the occupation of single-particle states. c)a)b) µ U t N t ’ gζ ( E ) TE ∝ exp( −√ κ ) U t ’ gζ ( T ) ’ ζ ∗ √ κ | i i| k i | j i | l i FIG. 2: Scattering processes leading to the many-body localization-delocalization transition for a classical Bose gas (
T > T d )in (a), for a degenerate thermal Bose gas ( T d √ γ < T < T d ) in (b), and for the low-temperature regime ( T < T d √ γ ) in (c). At a finite temperature T it is crucial to take into ac-count two-body processes that change occupation num-bers and can dramatically affect the properties of thesystem. For example, the system of interacting localizedfermions (electrons) can have a finite DC conductivityeven in the absence of coupling with any outside bath ,whereas without interactions the conductivity is exactlyzero at any T . In the presence of the interactions the con-ductivity remains zero unless the temperature exceeds acritical value T c . This transition can be thought of as Anderson localization of many-body wavefunctions.The critical temperature T c depends on the interactionstrength. This is the many-body analog of the mobilityedge that separates bands of localized and extendedstates in the single-particle Anderson transition.The Anderson transition is based on the fact that twoquantum states belonging to different lattice sites hy-bridize provided that the hopping matrix element be- tween these states exceeds the difference in onsite en-ergies. As soon as the density of the hybridized pairsexceeds a critical value the eigenstates turn out to beextended . The many-body localization-delocalizationtransition can be qualitatively understood by extendingthis physical picture to states of more than one particle.Consider an occupied localized single-particle state | i (cid:105) with energy (cid:15) i . The interaction of a particle occupyingthis state with a particle in the state | k (cid:105) can transferthe | i (cid:105) -state particle to the state | j (cid:105) , transferring simul-taneously the | k (cid:105) -state particle to another state | l (cid:105) (seeFig. 2). Let U ik,jl be the matrix element of this pro-cess. Due to the exponential decay of the localized wavefunctions one may assume that U ik,jl = 0 unless all fourstates are located near each other. Moreover, it turnsout that the matrix element rapidly decreases withan increase in the energy transfer | (cid:15) i − (cid:15) k | . We thus mayconfine ourselves to the case where the states | i (cid:105) and | k (cid:105) are nearest neighbors in the energy space. For simplic-ity we replace U ik,jl by a certain typical value U t pro-vided that the states | i (cid:105) , | j (cid:105) , | k (cid:105) , | l (cid:105) are localized nearbyand pairwise are nearest neighbors in energy.In a random system the energies of the final and initialstates can not be matched exactly. As long as the energymismatch ∆ ik,jl = | (cid:15) i + (cid:15) k − (cid:15) j − (cid:15) l | exceeds U ik,jl the effectof the interactions on the quantum state of 4 particles isnegligible. One may say that single-particle excitationsdo not decay . Suppose that the interaction is weakand a typical mismatch ∆ t exceeds the matrix element U t . Does it mean that single-particle excitations havean infinite lifetime? The answer depends on the num-ber of channels, N , for the decay of a given excitation(more precisely, N is the number of possible processes | i (cid:105) , | k (cid:105) → | j (cid:105) , | l (cid:105) that involve a given state | i (cid:105) ). Indeed,with probability of order unity, these processes shouldhave a channel with the mismatch that is smaller by a fac-tor of N than the typical value ∆ t . Therefore, U t shouldbe compared with ∆ t /N . Note that N plays a role ofthe number of nearest neighbors in the single-particle lo-calization problem. However, in the many-body case N is determined by the density of thermal excitations andis temperature dependent. Since characteristic single-particle energies and localization lengths are determinedby the temperature, both ∆ t and U t are also temperaturedependent. As a result, there is a critical temperature T c following from the equation∆ t ( T c ) = U t ( T c ) N ( T c ) . (6)At T > T c many-body states are extended, i.e. theyare linear combinations of one-, two-particle, etc. excita-tions, and the number of terms is infinite. This leads tothe fluid behavior. For T < T c the many-body localiza-tion takes place and one expects the insulating behavior.Note that the arguments given above and Eq. (6) aregeneral and independent of quantum statistics of the par-ticles. At the same time, both U t ( T ) and N ( T ) dodepend on the statistics. For disordered bosons thesequantities are determined by the density of single parti-cle states ν ( (cid:15) ) and by the occupation number N ( (cid:15) ). Thelatter is controlled by the chemical potential µ which isrelated to the mean density n and temperature T by thenormalization condition n = (cid:82) d(cid:15)N ( (cid:15) ) ν ( (cid:15) ), with N ( (cid:15) ) de-termined by the Bose-Einstein distribution. More pre-cisely, as long as the interaction is weak ( γ (cid:28)
1) the oc-cupation is N ( (cid:15) ) = { exp[( (cid:15) HF − µ ) /T ] − } − , where (cid:15) HF differs from the single-particle energy (cid:15) by the Hartree-Fock corrections. We will see that these corrections be-come important only at sufficiently low T and are neg-ligible at high temperatures where in the vicinity of thetransition relevant particle energies greatly exceed ng . III. PHASE DIAGRAM
We now use Eq. (6) and analyze the phase diagram ofweakly interacting disordered 1D bosons. It is convenient to represent the phase diagram in terms of the dimension-less temperature t = T /ng and find the dependence of t c on the strength of the disorder κ . The relation t = t c ( κ )determines the boundary between the fluid and insulatorphases in the ( t, κ ) plane. Alternatively, one can speakof a temperature-dependent critical disorder κ c ( t ).At high temperatures T > T d , or t > γ − , the Bosegas is not degenerate and a characteristic energy of par-ticles is of the order of T (cid:29) ng . As will be seen, near thetransition temperature we have E ∗ < T so that mostof the particles are in the high-energy states and theoccupation of all the states is small and described buBoltzmann distribution. In this case the typical matrixelement U t ( T ) in Eq. (6) does not depend on the occu-pation and can be estimated as U t ( T ) ∼ g/ζ ( T ). Thetypical mismatch is the meen nearest neighbor energyspacing: ∆ t ∼ [ ν ( T ) ζ ( T )] − . The quantity N ( T ) isgiven by the number of particles localized within a dis-tance ∼ ζ ( T ) from a given state | i (cid:105) , i.e. N ( T ) ≈ ζ ( T ) n .Then, Eq. (6) is reduced to ngν ( T c ) ζ ( T c ) ∼
1. Using thehigh-energy localization length ζ ( T ) = ζ ∗ T /E ∗ and thedensity of states ν ( T ) = 1 / (cid:112) E ∗ ζ ∗ T , with the help ofEqs. (4) and (5) we obtain the high temperature relationfor the critical disorder : κ c ( t ) ∼ t / ; t > γ − . (7)Close to the transition we have E ∗ /T = κ/t ∼ t − / (cid:28) γ / (cid:28)
1, which justifies our initial assumption that mostof the particles are in the high-energy states.As the temperature is reduced below T d the Bose gasbecomes degenerate. In the absence of disorder the chem-ical potential is µ = − T /T d as long as T > T d √ γ ,or t > / √ γ . Characteristic energies of particles are ∼ | µ | < T . However, as will be seen below, they stillexceed both ng and E ∗ . Therefore, most of the 1D dis-ordered interacting bosons in the temperature interval γ − / < t < γ − occupy high-energy states and onecan use the ideal Bose gas distribution N ( (cid:15) ), i.e. neglectHartree-Fock corrections to single-particle energies. Themajor part of particles has energies (cid:15) < ∼ | µ | and is charac-terized by a multiple occupation N ( (cid:15) ) ∼ T /(cid:15) >
1. Thismanifests itself in the dependence of interaction matrixelements on N ( (cid:15) ).In this regime there are two energy scales ( T and µ (cid:28) T ) characterizing the distribution of particles. What arethe particle energies that determine the many-body de-localization? Let us apply Eq. (6) to particles with ener-gies (cid:15) ∼ E in the energy interval of width ∼ E . A typicalvalue of the energy spacing is ∆ t = 1 /ν ( E ) ζ ( E ), andthe typical matrix element of the two-body interaction isenhanced due to a multiple occupation of single-particlestates: U t = [ g/ζ ( E )]( T /E ). The number N of occupiedlevels at distances smaller than ζ ( E ) from a given stateis N ( E ) ∼ Eν ( E ) ζ ( E ) and Eq. (6) takes the form N U t / ∆ t ∼ gT c ν ( E ) ζ ( E ) . Then, using the high-energy density of states ν ( E ) =1 / (cid:112) E ∗ ζ E and localization length ζ ( E ) = ζ ∗ E/E ∗ wefind a remarkable result: the criterion of delocalizationdoes not involve the single-particle energy scale! Thetransition temperature follows from the relation gT c ∼ E ∗ ζ ∗ = (cid:113) ¯ h E ∗ /m . (The fact that this is valid for allenergy scales suggests that the expression for the criticaltemperature/disorder can contain a prefactor logarithmicin γ , which we neglect). In terms of the parameters t and κ the relation for the critical disorder becomes κ c = t / γ / ; γ − / < t < γ − . (8)Equation (8) shows that κ c > ∼ γ − / < t < γ − . Hence, close to the transitionwe have E ∗ > ∼ ng and characteristic particle energies are ∼ | µ | > ∼ E ∗ > ∼ ng . This justifies our assumption that themajor part of particles occupies high-energy states, andthe interparticle interaction affects neither the occupa-tion numbers N ( (cid:15) ) nor the chemical potential µ .Consider now T = 0. For κ (cid:29) i is formed by N i bosons in the single-particle eigenstate | i (cid:105) which ischaracterized by energy (cid:15) i and localization length ζ i ≈ ζ ∗ . The energy cost E i of bringing an extra particle tothis lake is enhanced by the repulsive interaction betweenthe bosons, E i ≈ (cid:15) i + gN i /ζ ∗ , and it should be equalto the global chemical potential µ measured from thelowest low-energy state (cid:15) = 0. The bosons thus occupyonly states below the chemical potential, (cid:15) i < µ , withthe occupation numbers N i ≈ ( µ − (cid:15) i ) ζ ∗ /g , and as longas µ < E ∗ only low-energy states are occupied. Thedensity of these states is ν ∗ = 1 /E ∗ ζ ∗ , and thus the meandensity of bosons is related to the chemical potential by n = µ / gE ∗ . The chemical potential can be expressedin terms of the parameter κ as µ = E ∗ / √ κ , i.e. thechemical potential is indeed smaller than E ∗ providedthat κ >
1. In this regime only a small fraction ( ∼ µ/E ∗ )of low-energy states is occupied and neighboring lakes areseparated by a distance l ( κ ) ∼ ζ ∗ √ κ , while their size is ∼ ζ ∗ . Using equations (1)-(5) one can show that a typicallake ( (cid:15) i ∼ µ ) contains N i ∼ nζ ∗ √ κ = 1 / √ γ bosons.The fact that l ( κ ) (cid:29) ζ ∗ implies that for κ (cid:29) κ . As soon as κ isreduced to the value of the order of unity, the distancebetween neighboring lakes becomes of the order of theirsize ζ ∗ and the interlake coupling drives the system to thefluid state. So, the insulator-fluid Berezinskii-Kosterlitz-Thouless transition at T = 0 occurs for κ c ∼ t ∼ γ − / , the critical disorder is also κ c ∼
1. Therefore, one expects that in the entire tem-perature range t < γ − / it remains κ c ∼ κ >
1, i.e. E ∗ > ng ? Let thedisorder be as strong and reduce the temperature be-low √ E ∗ T d ( t < (cid:112) κ/γ ). Under these conditions we have | µ | < E ∗ and only a few bosons are hosted by high-energystates. For high-energy bosons ( (cid:15) > | µ | ) the condition (6) is not satisfied because their density is too small for themany-body delocalization. It turns out that the mainbody of the bosons, which occupy low-energy states, alsoforms an insulator. Indeed, for low-energy states we have (cid:15) < E ∗ and the number of channels N ∼ (cid:15)ν ( (cid:15) ) ζ ( (cid:15) )is smaller than unity, since ν ( (cid:15) ) ζ ( (cid:15) ) ∼ E − ∗ . In otherwords, most of the particles occupy single-particle stateswhich are separated from each other by distances exceed-ing the localization length ζ ∗ . This causes exponentialreduction of U t and, according to Eq. (6), prevents delo-calization. Finite temperature in this situation does notlead to any increase in the phase volume for availabletransitions. By contrast temperature fluctuations of thenumber of particles in each lake lead to a growth of theenergy mismatch by an amount ∼ (cid:112) T g/ζ ∗ , which fur-ther suppresses the probablity to hybridize several states.The physical situation is somewhat similar to the one de-scribed in Ref. , where it was demonstrated numericallythat in a finite-width band with less than one state perlocalization length the insulator remains stable with re-spect to the interactions at arbitrarily high temperatures.If the disorder is weak, κ <
1, the chemical potentialis determined by the interaction and always exceeds E ∗ .As a result, we have N ( T ) >
1. The condition (6) thenindicates that the insulator is unstable, and one dealswith the fluid state.The arguments presented above indicate that the phasetransition line has to be almost horisontal in the region t < / √ γ . We can also realize that at t → κ c ( t ) should terminate at the quantum phase transitionpoint κ = 1. Indeed, assuming κ c ( t = 0) < T = 0 has delocalized excitations (phonons) at lowenergies. On the other hand, the assumption of κ c > Insulator Fluid κ = E ∗ / ( gn )1 t = T/ ( gn )1 1 / √ γ /γ κ c = t / κ c = t / γ / κ c ( t ) FIG. 3: Phase diagram for weakly interacting disorderedbosons. The green line shows the zero temperature alge-braic superfluid. The red part of the curve κ c ( t ) follows fromEqs. (7) and (8), and the blue part is an estimate for thelow-temperature regime. This completes our description of the finite temper-ature fluid-insulator phase transition for 1D interactingbosons. The phase diagram is presented in Fig. 3. De-tailed calculations and a more accurate description of thetemperature range t < γ − / (blue part of the curve κ c ( t )in Fig. 3) will be presented elsewhere. IV. DYNAMICS OF EXPANSION
The observation of the fluid-insulator phase transi-tion described above is feasible in experiments with coldbosonic atoms in the 1D geometry. For electrons insolids one measures e.g. the DC conductivity. In quan-tum (neutral-atom) gases the analysis of transport prop-erties is based of the dynamics induced by significantexternal perturbations which may drive the system farfrom the initial state. We believe that the localization-delocalization transition can be identified in the dynam-ics of expansion of disordered bosonic clouds releasedfrom the superimposed trapping potential.In an array of harmonically trapped quasi-1D tubeswith about a hundred of atoms per tube, the density n can be made ∼ cm − so that the length L of each tubeis ∼ µ m and the temperature of quantum degeneracy T d is of the order of tens of nanokelvins. Tuning the inter-action strength by Feshbach resonances or by variationsof the tight transverse confinement one can achieve theinteraction energy ng of the order of nanokelvins or evensmaller and make γ ∼ − − − . For the correlationlength of the disorder σ (cid:39) . µ m as in the experiment and typical values of U , the localization length ζ ∗ isin the range of 1 −
10 microns and the energy E ∗ > ∼ ng .The dimensionless strength of the disorder, κ (5), rangesfrom about unity to large values and one can to studyall temperature regimes of the phase diagram in Fig. 3.Note, however, that the conditions described here arequite different from those in the experiment , where theinteraction energy was greatly exceeding E ∗ and the sys-tem was deeply in the fluid state.Switching off abruptly the 1D trap but still keepingthe disorder (and the transverse tight confinement) likein the experiments , is expected to cause the expansionof the cloud. Not very far from the fluid-insulator transi-tion, the localization length ζ of single-particle states inthe initial cloud is much smaller than its size L . Underthis condition, the size rapidly increases by an amountof ∼ ζ (cid:28) L . If the entire initial cloud is in the insulatorphase, i.e. locally κ > κ c ( t ), the expansion then stops(see Fig. 4a). If the central part of the cloud is in thefluid phase (while the outskirts are insulators), two-bodyscattering processes induce further expansion of the cen-tral part. In this stage the expansion is a slow diffusiveprocess governed by the diffusion equation: ∂n∂τ = ∂∂x D ( n, t ) ∂n∂x , (9)where τ denotes the time and t is the dimension- less temeperature (4). The diffusion coefficient D ( n, t )strongly depends on the local density n ( x ) and vanishesat the border of the insulating phase, i.e. for the den-sity n c ( t ) = E ∗ / ( gκ c ( t )). Hence, the local decrease ofthe density stops when it becomes equal to n c ( t ). Thus,the density profile of the expanded cloud represents aplateau with n = n c ( t ) in the central part and the initialinsulating wings with n ( x ) < n c (see Fig. 4b).Moderately far from the transition, i.e. at ( n − n c ) /n c ∼
1, the diffusion coefficient can be estimated asfollows. The occupation N i of the state | i (cid:105) changes by ∼ h/U t ∼ ¯ hζ/gN i . The distance of a typical hop is ∼ ζ .Therefore, on a time scale τ ∼ N i ¯ h/U t ∼ ¯ hζ/g each bo-son will move by a distance ∼ ζ , and D ∼ ζ /τ ∼ ζg/ ¯ h .For high-energy states the localization length at typi-cal energies (cid:15) is ζ ( (cid:15) ) = ζ ∗ (cid:15)/E ∗ , and the diffusion coef-ficient is given by D ∼ (cid:15)ζ ∗ g/ ¯ hE ∗ . Locally, the decreaseof the density ceases when it reaches the critical value n c ( t ). Thus, for the initial central density comparablewith n c ( t ), the evolution of the cloud to the final shapeof the plateau and wings (see Fig. 4) requires a charac-teristic time τ ∗ ∼ L /D . Since L ∼ ( (cid:15)/mω ) / , where ω is the initial trap frequency, we obtain τ ∗ ∼ ω ¯ hE ∗ mωζ ∗ g ∼ ω (cid:16) ng ¯ hω (cid:17) κ / γ / . (10)However, in the temperature interval T d √ γ < T < T d most of the particles, which have energies (cid:15) ∼ | µ | = T /T d , and particles with (cid:15) ∼ T will expand with dif-ferent velocities. So, the redistribution of particles in thecourse of the expansion may become important and itcan slightly modify the above estimate for the time τ ∗ .The time τ ∗ by far exceeds the time ω − of ballisticexpansion of the clean thermal cloud. For example, nearthe lower bound of Eq. (7), i.e. for T ∼ T d and κ ∼ γ − / ,we have τ ∗ ∼ T d / ¯ hω . For typical values T d ∼
50 nK and ω ∼
10 Hz this estimate yields τ ∗ ∼ T ∼ T d √ γ and κ ∼
1, thetime τ ∗ reduces by a factor of √ γ , i.e. it is of the orderof 0 . T < T d √ γ ( t < γ − / ). The analysis of the diffusion coefficient forthis case is beyond the scope of the present paper. V. PHASE DAGRAM IN HIGHERDIMENSIONS
We conclude our discussion of the properties of disor-dered interacting bosons with a brief sketch of the phasediagram in higher dimensions. As is well known, in theabsence of disorder bosons form a superfluid (algebraicsuperfluid in 2D) below a critical temperature ¯ T c . At b)a) x xn ( x ) n ( x ) n c FIG. 4: Manifestation of the many-body localization-delocalization transition in the expansion of a quasi-1D cloud.In (a) the the entire initial cloud is an insulator, whereas in(b) initially the central part is in the fluid and the outskirtsin the insulating phase. Initial and final shapes of the cloudare shown in black and red, respectively. high temperatures,
T > ¯ T c , the clean system is a nor-mal fluid. The superfluid survives a sufficiently weakdisorder, but the superfluid transition temperature ¯ T c de-creases with increasing the strength of the disorder andvanishes at a critical strength (see the black curve andbrown point in Fig. 5). What is the state of the dis-ordered system at T > ¯ T c , i.e. above the black curvein Fig. 5 ? It follows from our previous discussion thatbosons can form either the normal fluid or the insulat-ing state. The suggested phase diagram is presented inFig. 5.This sketch can be justified in the following way. First,it is safe to assume that at sufficiently strong disorderbosons are in the insulating state. The second obser-vation is that at the critical disorder (brown point inFig. 5) and arbitrarily small but finite T one should ex-pect the normal fluid rather than the insulator. Indeed,the zero-temperature insulator can be thought of as asystem of superfluid lakes which are separated from eachother and have uncorrelated phases. A typical size ofsuch a lake increases with decreasing the strength of thedisorder and diverges at the critical strength. It meansthat although excitations in the insulating phase at T = 0are always localized, their localization length can be ar-bitrarily large. Therefore, at any finite temperature andinteraction strength there is a vicinity of the critical dis-order, where the insulator is unstable with respect to themany-body delocalization. As a result, the insulator -normal fluid phase boundary can not follow the lowerblue curve on Fig. 5.On the other hand, the normal fluid can not be sta-ble at T = 0 . Indeed, the wavefunctions of low-energysingle-particle states have to be localized, otherwise par-ticles can not avoid Bose-Einstein condensation and thesystem would become superfluid. At extremely low butfinite temperatures, the density of thermal excitations isvanishingly small and the interaction between them is un-able to delocalize many-body wavefunctions. This rulesout the phase boundary following the upper blue curvein Fig. 5. Thus, the only possible option is represented Fluid
Disorder Temperature
InsulatorSuperfluid
FIG. 5: Phase diagram for two-dimensional weakly inter-acting bosons. The black curve shows the thermodynamicBerezinskii-Kosterlits-Thouless transition. The red curve isthe many-body localization-delocalization transition. Theblue dashed curves indicate phase boundaries which can beruled out on rather general grounds (see text). by the red solid curve .The arguments given above apply to both 2D and 3Dcases. However there is a big difference, since in 2D allsingle-particle states are known to be localized, and thestrong-disorder phase is a true insulator. This means thatthe diffusion constant is zero even at finite temperatures.At the same time, in three dimensions high-energy statescan be extended. As soon as the extended states appearthey can host thermal excitations and thus allow expo-nentially small but finite diffusion. From this point ofview, there is no qualitative difference between the insu-lator and normal fluid, and the red curve in 3D representsa crossover rather than a true phase transition. VI. CONCLUDING REMARKS
A remarkable possibility to compare disordered inter-acting bosons with non-interacting ones is offered by a Li atomic gas, where the coupling strength g can bevaried by a Feshbach resonance from practically zero tolarge ppositive values . In the 1D case, achieving thestrongly interacting regime where γ > ∼
1, is expected topresent new transparent physics. For γ → ∞ as in therecent cesium experiment , the bosons become impen-etrable and show a clear analogy with non-interactingfermions. All single-particle states are then localized irre-spective of the strength of the disorder. At intermediatevalues of γ one expects a peculiar interplay between theinterparticle interaction and temperature. For T (cid:29) ng the situation should be the same as desribed in SectionIII for the high-temperature regime, T (cid:29) T d , and in thissense equation (7) is universal. In the other extreme, T < ∼ ng , an increase in the interaction strength first leadsto the localization-delocalization transition at small γ ,but then causes a reentrance to the insulating phase at acritical disorder-dependent value of γ . This behavior isexpected because at T = 0 even an infinitesimally smalldisorder leads to the appearance of an insulating (Boseglass) phase if γ is sufficiently large . The comparison ofthe insulating phases emerging at small and large γ witheach other is supposed to shed new light on the structureof Bose glasses. Acknowledgements
We are grateful to Alain Aspect and Jean Dalibardfor interesting discussions, and to Matthew Foster and Leonid Glazman for comments on the manuscript. Weacknowledge support from US DOE contract No. DE-AC02-06CH11357, from the IFRAF Institute of Ile deFrance, and from ANR (Grant ANR-08-BLAN-0165).G.S. was also supported by the Dutch Foundation FOM.Part of the work was performed during the workshop”From Femtoscience to Nanoscience: Nuclei, QuantumDots, and Nanostructures” in the Institute of NuclearTheory at the University of Washington. van Hove, L. Sur l’integrale de configuration pour les sys-temes de particules a une dimension. Physica , 137-143(1950). Landau, L.D. & Lifshits, E.M.
Statistical Physics (Perga-mon Press, London, 1958). Gertsenshtein, M.E. & Vasil’ev, V.B. Waveguides withrandom inhomogeneities and brownian motion In theLobachevsky plane.
Theor. Probab. Appl. , 391-398(1959). Abrahams, E., Anderson, P.W., Licciardello, D.C. & Ra-makrshnan, T.V. Scaling theory of Localization - Absenceof quantum diffusion in 2 dimensions.
Phys. Rev. Lett. ,673-676 (1979). Anderson, P.W. Absence of diffusion in certain randomlattices.
Phys. Rev. , 1492-1505 (1958). Fortagh, J. & Zimmermann C. Magnetic microtraps forultracold atoms.
Rev. Mod. Phys. Billy, J. et al. Direct observation of Anderson localizationof matter waves in a controlled disorder.
Nature Roati, G. et al. Anderson localization of a non-interactingBose-Einstein condensate.
Nature Lifshitz, I.M. Energy spectrum structure and quantumstates of disordered condensed systems.
Sov. Phys. Usp. , 549-573 (1965). Halperin, B.I. & Lax, M. Impurity-band tails in high den-sity limit I. Minimum counting method.
Phys. Rev. ,722-740 (1966). Zittartz, J. & Langer, J.S. Theory of bound states in arandom potential.
Phys. Rev. , 741-747 (1966). In the opposite limit, the random potential gets screenedby the interparticle interaction. The phase diagram inFig. 3 is then modified quantitatively rather than quali-tatively: one may replace U ( x ) by an effective screenedpotential U eff ( x ). For weaky interacting bosons the fluid-insulator transition is expected in the region where theamplitude and correlation length of U eff ( x ) satisfy the in-equality U eff < ∼ ¯ h /mσ eff . This issue will be analyzedelsewhere. Giamarchi, T. & Schulz, H. Anderson localization and in-teractions in one-dimensional metals.
Phys. Rev. B ,325-340 (1988). Altman, E., Kafri, Y., Polkovnikov, A. & Refael, G. Insu-lating phases and superfluid-insulator transition of disor-dered boson chains.
Phys. Rev. Lett . , 170402 (2008). Falco, G.M., Nattermann, T. & Pokrovsky, V.L. Weaklyinteracting Bose gas in a random environment.
Phys. Rev.B , , 104515 (2009). Lugan, P., et al. Ultracold Bose gases in 1D disorder: FromLifshitz glass to Bose-Einstein condensate.
Phys. Rev. Lett. Basko, D.M., Aleiner, I.L. & Altshuler, B.L. Metal-insulator transition in a weakly interacting many-electronsystem with localized single-particle states. Annals ofPhysics , 1126-1205 (2006). Mott, N.F. & Twose, W.D. The theory of impurity con-duction.
Advances in Physics , 107-163 (1961). Mott, N.F. Conduction in non-crystalline systems: 4. An-derson localization in a disordered lattice.
Phil. Mag. ,7-29 (1970). Mirlin, A.D. Statistics of energy levels and eigenfunctionsin disordered systems.
Phys. Rep. , 259-382 (2000). Aleiner, I.L., Brouwer, P.W. & Glazman, L.I. Quantumeffects in Coulomb blockade.
Phys. Rep. , 309-440(2002). Altshuler, B.L., Gefen, Y., Kamenev, A. & Levitov, L.S.Quasiparticle lifetime in a finite system: A nonperturba-tive approach.
Phys. Rev. Lett. , 2803-2806 (1997).te Oganesyan, V. & Huse, D.A. Localization of interactingfermions at high temperature.
Phys. Rev. B , 155111(2007). A similar shape of the expanded cloud was suggested for 3Ddisordered bosons at T →
0: Shklovskii, B.I. “Superfluid-insulator transition in ”Dirty” ultracold Fermi gas”.
Semi-conductors , 909-913 (2008). However, near the fluid-insulator transition for a disordered interacting system inthe limit of T →
0, the entropy production (and heating)during the expansion becomes important, and it stops dueto the many-body localization-delocalization transition de-scribed in our paper. Fisher, M.P., Weichman, P.B., Grinstein, G., & FisherD.S. Boson localization and superfluid-insulator transition.
Phys.Rev. B , 546-570 (1989). Our conclusion for the 2D case contradicts the recent work:Mueller, M. Purely electronic transport and localizationin the Bose glass, arXiv:0909.2260. This paper aussumesthe existence of delocalized states for 2D non-interactingparticles in the absence of spin-orbit scattering. We do notsee any basis for this assumption. Pollack, S.E. et al. Extreme tunability of interactions ina Li Bose-Einstein condensate. Phys. Rev. Lett. ,090402 (2009). Haler, E. et al. Realizatin of an excited strongly correlatedquantum gas phase. Science325