Dirty bosons on the Cayley tree: Bose-Einstein condensation versus ergodicity breaking
DDirty bosons on the Cayley tree:Bose-Einstein condensation versus ergodicity breaking
Maxime Dupont,
1, 2
Nicolas Laflorencie, and Gabriel Lemari´e Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Laboratoire de Physique Th´eorique, IRSAMC, Universit´e de Toulouse, CNRS, UPS, 31062 Toulouse, France
Building on large-scale quantum Monte Carlo simulations, we investigate the zero-temperaturephase diagram of hard-core bosons in a random potential on site-centered Cayley trees with branch-ing number K = 2. In order to follow how the Bose-Einstein condensate (BEC) is affected by thedisorder, we focus on both the zero-momentum density, probing the quantum coherence, and theone-body density matrix (1BDM) whose largest eigenvalue monitors the off-diagonal long-range or-der. We further study its associated eigenstate which brings useful information about the real-spaceproperties of this leading eigenmode. Upon increasing randomness, we find that the system un-dergoes a quantum phase transition at finite disorder strength between a long-range ordered BECstate, fully ergodic at large scale, and a new disordered Bose glass regime showing conventionallocalization for the coherence fraction while the 1BDM displays a non-trivial algebraic vanishingBEC density together with a non-ergodic occupation in real-space. These peculiar properties canbe analytically captured by a simple toy-model on the Cayley tree which provides a physical pictureof the Bose glass regime. I. INTRODUCTION
The pioneer work of Anderson on the localization ofnon-interacting electrons in a random potential [1–3]paved the way for the study of disorder-induced phases ofmatter in quantum systems. Beyond a critical amount ofrandomness, a system can undergo drastic changes in itsphysical properties, generically from a delocalized quan-tum state to a localized one, such as a metal-to-insulatortransition for electrons, or a superfluid-to-insulator tran-sition for bosonic degrees of freedom [4].In absence of interaction, the fate of electrons in adisordered environment has been, and is still intensivelystudied. If the transition is now well understood in finitedimension [2, 3], the case of graphs of infinite effective di-mensionality, such as the Cayley tree or random graphshas recently aroused great interest [5–34], due to the anal-ogy between this problem and many-body localization(MBL) which can occur at any arbitrary energy [35–38].At low-energy, the interplay of interaction and disorderin bosonic systems has received a great deal of attentionfollowing experiments on superfluid Helium in porous me-dia [39, 40] and the discovery of a novel localized phase ofmatter at low-temperature, the Bose glass state [41–45].It can be described as an inhomogeneous gapless com-pressible fluid with short-ranged correlations preventingany global phase coherence responsible of delocalizationproperties. Known as the “dirty boson” problem, thelocalized Bose glass phase and its transition from a de-localized superfluid have been theoretically and numer-ically studied from one to three dimensions in variouscontexts [46–78], and also reported in several experimen-tal setups, from disordered superconductors [79–83] totrapped ultracold atoms [84–87], as well as chemicallydoped antiferromagnetic spin compounds [88–97].In this paper, we investigate the low-temperature prop- erties of strongly interacting dirty bosons on the Cay-ley tree. Together with an on-site random potential,the bosons have a nearest-neighbour hopping amplitudeand an infinite repulsive contact interaction (hard-coreconstraint). This system can be efficiently simulated byextensive quantum Monte Carlo (QMC) simulations [98–100], an unbiased (“exact”) numerical method, with morethan a thousand particles on the lattice for the largestsystem sizes accessible.The first interest of the Cayley tree for this prob-lem is the effective infinite dimension ( d = ∞ ) of thegraph, while all quantum Monte Carlo studies have fo-cused on finite dimensional systems d ≤ et al . in their seminal work on thelocalization-delocalization transition for bosons [45], itis unclear what is the correct scenario for the transi-tion in high dimension (typically d > d c beyond which conventional onset of mean-field the-ory usually takes place, and that d c = ∞ . Contrarily,based on the exact treatment of an infinite-range hoppingmodel [45], which is effectively infinite dimensional, no lo-calized phase is found, raising the question on whetheror not boson localization can actually happen in high di-mension. However, long-range hopping might be patho-logical, since the physics in the presence of disorder differssignificantly from that of the short-range problem [41].Some of these questions resonate with the problem onthe Cayley tree addressed in this paper.The second interest lies in the search of non-ergodicphases. At strong disorder, the Bose glass phase shouldhave, as its name suggests, glassy non-ergodic proper-ties, however they have only been little characterized(see, e.g., Refs. [101–104]). The Cayley tree is one ofthe key models of glassy physics where the non-ergodicproperties of classical disordered systems are best under-stood [105, 106]. Recently, the case of quantum disor- a r X i v : . [ c ond - m a t . d i s - nn ] J un dered systems on the Cayley tree has attracted a stronginterest. In particular, the Anderson transition on theCayley tree presents new remarkable non-ergodic prop-erties: The delocalized phase can be multifractal (wherethe states lie in an algebraically small fraction of the sys-tem) in a finite range of disorder [11–18], contrarily tothe finite-dimensional case where multifractality appearsonly at criticality. Moreover the localized and criticalregimes inherit a glassy non-ergodicity where the eigen-states explore only few branches [8, 9, 20, 34].Finally, we aim at comparing exact quantum MonteCarlo results to an approximate cavity mean-field ap-proach, coming from glassy physics [107–111]. Inparticular, Feigel’man, Ioffe and M´ezard [110, 112]have described through this method the disorder-driven superconducting-insulator transition consideringthe boundaryless counterpart of the Cayley tree, theBethe lattice. They have predicted the existence of anon-ergodic delocalized phase. Experimentally, the ob-servation at strong disorder of large spatial fluctuationsof the local order parameter in strongly disordered super-conducting films [81, 83] has been interpreted as the sig-nature of a persistence of glassy, non-ergodic properties inthe superconducting phase. Although the distributionsof the local order parameter observed experimentally dif-fer from the cavity mean-field predictions on the Cayleytree [83], these results have confirmed the importance ofnon-ergodic properties in this problem.The rest of the paper is organized as follows. In Sec. IIwe present the model, the numerical method, and brieflyprovide details about the 1BDM. In Sec. III, first numeri-cal evidences for the disorder-induced BEC depletion arepresented. We then discuss microscopic aspects of theproblem in Sec. IV, where real-space properties of bothoff-diagonal correlations and the leading orbital are an-alyzed. In Sec. V, we look at the critical properties ofthe transition by performing a careful finite-size scaling,yielding estimates of the critical parameters. We thendiscuss the peculiar properties of the localized Bose glassregime, building on both numerical results and an ana-lyticaly solvable toy-model. We finally present our con-clusions and discuss some open questions in Sec. VI. II. MODEL AND METHODSA. Dirty hard-core bosons on Cayley trees
We consider hard-core bosons at half-filling on a site-centered Cayley tree with N lattice sites, described bythe Hamiltonianˆ H = − (cid:88) (cid:104) i,j (cid:105) (cid:16) ˆ b † i ˆ b j + H . c . (cid:17) + (cid:88) Ni =1 µ i ˆ n i , (2.1)where ˆ b † i (ˆ b i ) is the bosonic creation (annihilation) op-erator on lattice site i , and ˆ n i = ˆ b † i ˆ b i the local densityoperator with the constraint (cid:104) ˆ n i (cid:105) ≤ g FIG. 1. Site-centered Cayley tree with branching number K = 2 and G = 4 generations (generations from 0 to G = 4are denoted by g ). The different colors of the vertices corre-spond to a given random configuration of chemical potential µ i in the Hamiltonian Eq. (2.1). ing infinite repulsive interaction. The sum (cid:104) i, j (cid:105) restrictsthe tunneling to nearest-neighbor sites, and the randomchemical potential µ i is drawn from a uniform distribu-tion µ i ∈ [ − µ, + µ ] with µ characterizing the disorderstrength. The model Eq. (2.1) possesses a global contin-uous U(1) symmetry due to the conservation of its totalparticle number, i.e., [ ˆ H , (cid:80) Ni =1 ˆ n i ] = 0.The site-centered Cayley tree is defined by its branch-ing number K > K = 2 case in thefollowing) and its total number of generations G . SeeFig. 1 for an example. The number of sites N scales ex-ponentially with G as N = 1 + ( K + 1)( K G − / ( K − N has the dimension of a volume and G ∼ ln N of a length.Moreover, the number of lattice sites at the boundary isa finite fraction (1 − K − at large size) of the total num-ber of sites, which may lead to macroscopic boundaryeffects to the Cayley tree, as compared its boundarylesscounterpart, the Bethe lattice. B. The one-body density matrix
A central target for this numerical study is the 1BDM,known to be an insightful object for bosonic systems, atthe heart of the Penrose-Onsager criterion [113–115] forBose-Einstein condensation. It has also proven to be suc-cessful in the study of the high-energy many-body local-ization transition [116–121] in one dimension. Moreover,there has been a recent proposal to measure the 1BDMfor hard-core bosons in an optical lattice [122], makingthe quantity experimentally relevant. The 1BDM C , de-fined by C ij = (cid:68) ˆ b † i ˆ b j (cid:69) , (2.2)is square, positive, real, and symmetric. Its diagonalelements correspond to the local densities (cid:104) ˆ n i (cid:105) , such that . .
01 1 n/N . ∏ n N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 N = 1534 N = 3070
10 100 1000 N . ∏ ∏ ∏ ∏N/ .
001 0 . n/N . N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766
10 100 1000 N . ∏ ∏ ∏ ∏N/ ( c )( a ) μ = 3.0 ∝ N ∼ cst ( b ) ( d ) ∼ cst μ = 3.0 μ = 8.0 μ = 8.0 FIG. 2. Disorder-averaged occupation numbers λ n sorted in descending order, λ ≥ λ ≥ · · · λ N , for two disorder strengths:(a-b) µ = 3 and (c-d) µ = 8. In panels (a) and (c), the average value is plotted versus the normalized index n/N for differentsystem sizes N . The symbols are here to highlight specific eiegenvalues, λ , λ , λ and λ N/ . In panels (b) and (d), the averagevalue is plotted versus the system size N for n = 1 , , n = N/
2. According to the Onsager-Penrose criterion [113–115], Bose-Einstein condensation will occur in a system which displays (at least) one occupation number of the order of N as N → + ∞ , which is what is observed for the largest occupation number λ at µ = 3 in panel (b). The next occupation numbers λ and λ both have a sublinear scaling with the system size ∝ N . , while the middle one, λ N/ , is constant with N . At µ = 8,the first few occupation numbers, including the largest one, have a very weak sublinear scaling ∝ N . , and the middle one isconstant. Not all occupation numbers can scale with N , or the constraint (cid:80) Nn =1 λ n ∼ O ( N ) of Eq. (2.3) would be violated. tr( C ) = (cid:88) Ni =1 (cid:104) ˆ n i (cid:105) = (cid:10) ˆ N b (cid:11) (cid:39) N/ , (2.3)with (cid:104) ˆ N b (cid:105) the total number of bosons in the system.The right-hand side of Eq. (2.3) means that we workin the grand-canonical ensemble where the particle num-ber conservation is not enforced and therefore not re-stricted to half-filling, although half-filling is statisticallyachieved with disorder average [123]. The eigenvectors ofthe 1BDM Eq. (2.2) are the natural orbitals, C | φ n (cid:105) = λ n | φ n (cid:105) , (2.4)and the eigenvalues λ n ≥ (cid:80) n λ n = (cid:104) ˆ N b (cid:105) . Sorting the eigenpairsin descending order, i.e., λ ≥ λ ≥ · · · λ N , (at least)one of the eigenvalues will be of the order of the numberof particles for a Bose-Einstein condensed system. Thiscondition is known as the Onsager-Penrose criterion forBose-Einstein condensation [113–115]. The correspond-ing eigenmode | φ (cid:105) is called the leading orbital and takesthe form, | φ (cid:105) = (cid:88) Ni =1 a i | i (cid:105) , with (cid:88) Ni =1 | a i | = 1 , (2.5)where i designates the lattice site index. The coefficients | a i | account for the distribution of this leading orbitalin real space. C. Numerical investigation
1. Quantum Monte Carlo
So far, the few numerical studies addressing many-body interacting problems on tree-like geometries have resorted to tensor network techniques [124–128], but inthe context of disorder-free models. Here, we instead relyon the quantum Monte Carlo method, using stochasticseries expansion with directed loop updates [98–100] tosimulate the disordered bosonic model Eq. (2.1). For thisproblem, we can in practice access finite-size systems upto G = 10 generations (coresponding to N = 3070 lat-tice sites), with a sufficiently low temperature such thatthe algorithm is probing ground state properties. Ad-ditional informations and data are provided in App. Aregarding the convergence of our results versus the tem-perature. We compute the elements of the 1BDM [129]by performing between 10 and 10 measurements afterthermalization.We note that the presence of “open boundary condi-tions” on the Cayley tree makes inaccessible the compu-tation of the superfluid density [130, 131], a very valu-able quantity in the study of disorder-induced phases forbosonic systems.
2. Disorder average
The disorder average is performed over a large numberof independent disordered samples, between N s = 300and N s = 2000, depending on the system size. The exactnumbers are provided in App. A, where we also discussthe convergence of the main disorder-averaged quanti-ties considered in this paper versus N s . For a physicalquantity O , we note its disorder-averaged value O andits typical value exp(ln O ). III. DISORDER-INDUCED BEC DEPLETIONA. Spectrum of the one-body density matrix
We start with an analysis of the eigenvalues of C . InFig. 2 the disorder-averaged occupation numbers λ n with λ ≥ λ ≥ · · · λ N is shown for various system sizes N for two representative disorder strengths ( µ = 3 and µ = 8) [132]. At weak disorder, the first eigenvalue λ is singular, while the next ones decay smoothly to zeroas the index n increases. More precisely, considering thefirst few occupation numbers ( n = 1 , ,
3) and one in themiddle ( n = N/
2) versus the system size, one observesthat λ ∝ N at large N , signalling Bose-Einstein conden-sation, according to the Onsager-Penrose criterion [113–115]. The next two eigenvalues λ and λ show a sublin-ear scaling ∝ N . with the system size (this exponentdecreases with the disorder strength, data not shown),and the middle one remains constant. For µ = 8, thelargest eigenvalue has a similar behavior to λ and λ ,with a slow sublinear scaling ∝ N . with the systemsize, clearly showing that no Bose-Einstein condensationoccurs for this value of disorder. The middle occupationnumber λ N/ is constant versus N . Note that becauseof Eq. (2.3), not all eigenvalues can scale with N , or theconstraint (cid:80) Nn =1 λ n ∼ O ( N ) would be violated. B. Condensed and coherent densities
From the largest occupation number λ and its rela-tion to Bose-Einstein condensation, one can define thecondensed density of bosons, ρ cond = λ /N. (3.1)Having ρ cond ∼ constant as N → + ∞ is equivalent to theexistence of off-diagonal long-range order in the system,associated with a spontaneous breaking of the continuousU(1) symmetry [115]. Hence, Eq. (3.1) plays the role ofthe order parameter. In homogeneous systems with awell-defined momentum k (this is not the case of theCayley tree), a more common (and convenient from bothcomputational and experimental purposes) definition ofthe order parameter is usually based on the momentumdistribution function, (cid:101) ρ ( k ) = 1 N (cid:88) r ij e − i k · r ij (cid:68) ˆ b † i ˆ b j (cid:69) , (3.2)with r ij the distance between lattice sites i and j . Inthis case, bosons generically condense into a single mo-mentum component, the k = mode, meaning that (cid:101) ρ ( )serves as definition for the order parameter. In systemsthat are not fully translation invariant, the componentwith zero momentum is known as the coherent density, ρ coh = 1 N N (cid:88) i =1 N (cid:88) j =1 (cid:68) ˆ b † i ˆ b j (cid:69) . (3.3) ( a ) ( b )( d )( c ) . . . . Ω c o h ° N , µ ¢ µ = 0 . µ = 0 . µ = 1 . µ = 1 . µ = 2 . µ = 2 . µ = 3 . µ = 3 . ° ° ° ° ° Ω c o h all NN ∏ N ∏ N ∏
10 100 1000 N . . . Ω c o nd ° N , µ ¢ µ = 4 . µ = 4 . µ = 5 . µ = 5 . µ = 6 . µ = 7 . µ = 8 . µ ° ° ° ° ° Ω c o nd FIG. 3. Left: The symbols are the disorder-averaged QMCdata for (a) the coherent density of Eq. (3.3) and (c) the con-densed density of Eq. (3.1), both displayed versus the systemsize for various disorder strengths, as indicated on the plot.The bold lines are fits to the form Eq. (3.4), taking into ac-count all points with N ≥
46. These estimates ρ ∞ ( µ ) areshown in panels (b) and (d), for four different fitting win-dows. One can roughly locate a transition in the vicinity of µ ≈ However, strictly speaking, this quantity does not ac-count for all the condensed bosons of the system Eq. (3.1)since what is refered as the “glassy component” with k (cid:54) = is left out [133–136], and which results in the prop-erty that ρ coh ≤ ρ cond . We note that the subtle links be-tween these different quantities as well as the superfluiddensity (not considered in this paper) were initiated byJosephson [137], and are still under active research, es-pecially within the cold atom community (see Ref. 136for a recent paper with references therein). Finally, al-though momentum is not defined on the Cayley tree, westill consider the coherent density Eq. (3.3) in the follow-ing, which we have found to provide relevant informationon the nature of the system.Fig. 3 shows the size and disorder dependence of bothcoherent (top) and condensate (bottom) densities. Wefind that, independently of the disorder strength µ , theyboth agree with the following form ρ ( N, µ ) = ρ ∞ ( µ ) + a ( µ ) N − ζ ( µ ) , (3.4)with ρ ∞ ( µ ), a ( µ ) and ζ ( µ ) positive disorder-dependentparameters. The precise finite-size correction form willbe discussed and analyzed in more details in Sec. V. Nev-ertheless, from this first simple analysis, one can alreadymake important observations. First, both coherent andcondensate densities display similar behaviors, and wealways observe ρ coh ≤ ρ cond . The extrapolated valueof the coherent and condensed densities in the thermo-dynamic limit ρ ∞ ( µ ) clearly show a transition in theregime µ c ≈
5. This indicates that hard-core bosons onthe Cayley tree display a long-range ordered phases at µ . . . . . r N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 N = 1534 N = 3070 ( a ) . . . r P ( r ) N = 46 N = 94 N = 190 N = 382 N = 766 P ( r ) = 2(1+ r ) ( b ) FIG. 4. (a) Disorder-average adjacent gap ratio r Eq. (3.5)plotted against the disorder strength µ for different systemsizes N . The transition from the delocalized BEC phase tothe disordered regime is visible around µ c ≈ −
6, with clearchange from r → r → − (cid:39) .
386 at large disorder strength. Thereis a strong size dependence, except for the largest disorderstrengths, where all system sizes converge onto the Poissonvalue. (b) Adjacent gap ratio distribution P ( r ) in the dis-ordered phase, at disorder strength µ = 8 for system sizes N ≥
46. The poisson distribution P ( r ) = 2 / (1 + r ) , usuallyexpected for a localized system, is in very good agreementwith the numerical data. small disorder, while beyond a critical disorder strength µ c the system is driven to a disordered phase where Bose-Einstein condensation has disappeared. C. Gap ratio from the largest occupation numbers
Level statistics of the eigenvalues of disordered Hamil-tonians is well-knwon to be a powerful way to de-tect localization-delocalization transitions at high en-ergy [138–140]. Here, despite the fact that we work atzero temperature, one can study level statistics of the1BDM of Eq. (2.2). More precisely, looking at the statis-tics of the three largest occupation numbers λ , λ and λ provides insightful information. We define the adjacentgap ratio, r = min (cid:0) δ , δ (cid:1) max (cid:0) δ , δ (cid:1) , (3.5)with δ n = λ n − λ n +1 the local gap between two consec-utive occupation numbers. In a BEC phase, λ ∝ N as N → + ∞ , while the next occupation numbers havea sublinear scaling with N , as discussed in Fig. 2. Inthis case, the denominator of Eq. (3.5) will always scalefaster with N than the numerator, resulting in r → r → − (cid:39) .
386 if the λ n follow a Poisson distribution [140]. In Fig. 4 (a), thesetwo limiting behaviors are clearly observed at small and strong disorder, respectively. In agreement with our pre-vious analysis for the order parameters, here again onecan roughly locate a transition around µ c ≈
5. However,the strong size dependence of the gap ratio makes dif-ficult a precise determination. Note that similar driftsof the gap ratio with the system size are also observedin the context of the many-body localization transitionat high energy [140–142] and the Anderson transition onrandom graphs [9, 19, 34].Here the absence of finite-size crossing signals thatthere is presumably no intermediate statistics at the tran-sition, in contrast with the Anderson localization case onregular lattices [138, 143, 144]. Nevertheless, Fig. 4 (a)confirms the existence of a spectral transition for thelargest occupation numbers, from a BEC regime with r = 0, to a disordered phase with Poisson statistics. Thisis also clear form the distribution P ( r ) shown in Fig. 4 (b)for strong disorder ( µ = 8), where a very good agreementis found with a Poisson distribution P ( r ) = 2 / (1 + r ) .Despite the fact that spectral properties of the leadingeigenvalues of the 1BDM unambiguously shows a Poissonbehavior, it does not necessarily mean that the associatedeigenmodes are strictly localized. Indeed, a multifractalbehavior is also possible, as recently found for the MBLphase of the random-field Heisenberg chain at high en-ergy [145] (see also [9, 34] for the Anderson transition onrandom graphs). In the following section, we will addressthis question in a quantitive way by directly studying thelocalization and ergodicity properties of the leading or-bital in real-space. IV. REAL-SPACE AND ERGODICITYPROPERTIESA. Local density of bosons
We start this analysis by looking at the local densi-ties (cid:104) ˆ n i (cid:105) , which correspond to the diagonal entries of the1BDM, see Eq. (2.3). We show in Fig. 5 its probabil-ity distribution for two representative disorder strengths, µ = 2 (Bose-Einstein condensed phase) and µ = 8 (dis-ordered regime). The sites i are sorted according to thegeneration g to which they belong. Because of the re-duced connectivity of the boundary sites ( K + 1 in thebulk, and only K − g = G oneobserves a strong deviation from the occupations (cid:104) ˆ n i (cid:105) inthe bulk. This is true for both phases: The probabilitydistributions have a larger weights around the extremevalues 0 and 1, meaning that boundary sites are morelocalized, as naturally expected from the reduced con-nectivity. However, this is not a mere finite-size effectsince about half of the sites belong to the boundary onthe Cayley tree with branching number K = 2. Moregenerally, the double-peak U-shape structure observed inFig. 5 (b) in the disordered phase is also observed in thecontext of many-body localization at high energy, and isa fingerprint of ergodicity breaking [146–150]. . . . . . . ≠ ˆ n i g Æ . . P ≥ ≠ ˆ n i g Æ ¥ g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 . . . . . . ≠ ˆ n i g Æ g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 . . . . . . ≠ ˆ n i G Æ . . P ≥ ≠ ˆ n i G Æ ¥ N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 N = 1534 N = 3070 . . . . . . ≠ ˆ n i G Æ N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 N = 382, μ = 2.00 ( a ) ( b ) BOUNDARY N = 382, μ = 8.00 BOUNDARY μ = 2.00 ( c ) μ = 8.00 ( d ) FIG. 5. Probability distribution of the local density (cid:104) ˆ n i (cid:105) ver-sus the generation g to which a site i belongs: g = 0 is thecenter site and g = 7 the boundary, see Fig. 1. A systemsize with G = 7 generations ( N = 382) is considered for (a) µ = 2 and (b) µ = 8. Only the densities at the boundary ofthe Cayley tree ( g ≡ G ) display a strong deviation from thosein the bulk. They are more easily localized by having largerprobabilities around the extreme values 0 and 1. Panels (c)and (d) show the absence of finite-size effect for the boundarysites i ∈ G . In Fig. 6, we provide a real-space picture for these occu-pations, focusing on two representative finite-size ( G = 7, N = 382) samples for both regimes: The BEC phase at µ = 2 (left column), and the disordered state at µ = 8(right column). The top row displays the deviations fromcomplete localization δ i = min( (cid:104) ˆ n i (cid:105) , −(cid:104) ˆ n i (cid:105) ), from whichwe clearly observe that spatial inhomogeneities developwith increasing randomness. In particular, at strong dis-order an apparent non-ergodic behavior settles in, withonly a finite number of branches in the Cayley tree whichhost particle fluctuations, while a large fraction of thegraph displays almost frozen sites with δ i (cid:28) B. Off-diagonal correlations
The second row of Fig. 6 shows a snapshot of the off-diagonal correlation function measured from the root ofthe tree C i ≡ (cid:68) ˆ b † ˆ b i (cid:69) , (4.1)here again for two representative samples from bothphases at µ = 2 and µ = 8. The spatial structure ob-served for the density (top row of Fig. 6) is also clearlyvisible in the correlators, as shown by panels (c) and (d).Note the logarithmic scale. μ = 2.00( a ) ( b )min ( ⟨ ̂ n i ⟩, 1 − ⟨ ̂ n i ⟩ ) ( c ) ( d )Leading orbital | a i | ( e ) ( f )Correlation ⟨ ̂ b †0 ̂ b i ⟩ μ = 8.00 FIG. 6. Real space representation of various physical quanti-ties for a given random sample of size N = 382 lattice sites, atsmall ( µ = 2, left column) and strong ( µ = 8, right column)disorder strengths. The scales on the panels are independent.(a-b) Deviation from perfect (non)occupation of the latticesites measured by δ i = min( (cid:104) ˆ n i (cid:105) , − (cid:104) ˆ n i (cid:105) ). The radius of thecircles is proportional to [ δ i − min( δ i )] / [max( δ i ) − min( δ i )].(c-d) Two-point correlation C i = (cid:104) ˆ b † ˆ b i (cid:105) from the centersite, in log-scale, with the radius of the circles proportionalto [ln C i − min(ln C i )] / [max(ln C i ) − min(ln C i )]. (e-f)Leading orbital | φ (cid:105) = (cid:80) Ni =1 a i | i (cid:105) of Eq. (2.5). The radiusof the circles is in log-scale and proportional to [ln | a i | − min(ln | a i | )] / [max(ln | a i | ) − min(ln | a i | )]. This figure is dis-cussed throughout Sec. IV.
1. Average and typical correlations
Disorder averaging has also been performed for thetwo-point correlation, as displayed in Fig. 7 as a functionof the distance. While the BEC phase is characterizedby a slow decay at large distance towards a constant,signalling off-diagonal long-range order, the disorderedregime shows short-ranged correlations with an exponen-tial decay of the form C i ∝ exp (cid:0) − g/ξ (cid:1) , (4.2)where g measures the distance between the root and site i . This exponential decay is clearly visible for µ = 8 inFig. 7 (b) where both average and typical correlators are ( a ) ( b ) ∝ e xp ( − g / ξ a vg /t yp ) μ = 2.0 μ = 8.0 . . . . . g . . . D ˆ b † ˆ b g E N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 N = 1534 N = 3070 g ° ° ° ° ° ° N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 a v e r a g e t yp i ca l AverageTypical
FIG. 7. Disorder-averaged and typical two-point correlationsbetween the site at the center (generation g = 0) and thesites at the generation g for different system sizes N at tworepresentative values of the disorder strength: (a) µ = 2 and(b) µ = 8. Exponential fits to the form Eq. (4.2) at µ = 8yield ξ avg ≈ .
15 and ξ typ ≈ . plotted. Here two remarks are in order: (i) Finite-sizeeffects are essentially absent in the disordered phase, incontrast with the BEC regime shown in panel (a), and(ii) while at weak disorder average and typical values arevery similar (except at the boundary), in the disorderedphase they decay with two different characteristic lengths ξ avg / typ . Such a difference between average and typicalcorrelations is a qualitative sign of non-ergodicity (seee.g. [22, 34, 151, 152]).
2. Distributions
In order to better explore microscopic properties andthe spatial features of the off-diagonal correlations, weshow different types of distributions in Fig. 8, again forweak ( µ = 2) and strong ( µ = 8) disorder. This quantityis indeed central in studies of non-ergodicity on this typeof graphs [21, 34, 110, 152, 154]. We have consideredthe distribution of the correlator C i over all sites i fordifferent system sizes [panels (a) and (b) of Fig. 8], C g over all sites at generation g for a fixed large system size[panels (c) and (d)] and C G/ for different values of thetotal number of generations G [panels (e) and (f)].In the Bose-Einstein condensed phase at weak disorder µ = 2, the different correlators allow to clearly identifythe localizing effect of the boundary (also seen in Fig. 5).Similarly to Fig. 7 for the disorder averaged correlator C g which decreases much faster close to the boundarythan in the bulk of the tree, one observes in panel (c) asharp broadening of the distribution of C g close to theboundary. A similar localizing effect of the boundaryarises also in the Anderson localization problem on theCayley tree [13]. On the contrary, in panel (e), the dis-tribution of C G/ in the bulk of the tree reaches a sta-tionary distribution, characteristic of long-range order.In panel (a), the correlator C i over all sites i is clearlydominated by the boundary sites, which represent half of ° ° ° D ˆ b † ˆ b i E ° ° ° ° P ≥ l n D ˆ b † ˆ b i E ¥ N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 N = 1534 N = 3070 ° ° ° D ˆ b † ˆ b i E ° ° ° ° N = 10 N = 22 N = 46 N = 94 N = 190 N = 382 N = 766 ° ° ° D ˆ b † ˆ b g E ° ° P ≥ l n D ˆ b † ˆ b g E ¥ g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10 ° ° ° D ˆ b † ˆ b g E ° ° ° ° g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 ° ° ° D ˆ b † ˆ b G/ E ° ° ° P ≥ l n D ˆ b † ˆ b G / E ¥ G = 2 G = 4 G = 6 G = 8 G = 10 ° ° ° D ˆ b † ˆ b G/ E ° ° ° ° G = 2 G = 4 G = 6 G = 8 ( b ) ( d ) N = 766 ( a ) ( c ) μ = 2.00 μ = 8.00 N = 3070 ( e ) ( f ) FIG. 8. Distribution of different types of correlators in theBose-Einstein condensed phase (left panels, µ = 2) and inthe disordered phase (right panels, µ = 8): The correlator C i over all sites i for different system sizes (upper panels (a)and (b)), C g over all sites at generation g for a fixed largesystem size (panels (c) and (d)) and C G/ for different valuesof the total number of generations G (panels (e) and (f)). Inthe BEC phase (left panels), a clear localizing effect of theboundary is observed in panel (c) with a sharp broadeningof the distribution for g close to G = 10. In the bulk ofthe tree shown in panel (e), the distribution is stationary atsufficiently large system size, indicating long-range order. Inthe disordered phase, the distribution follows a traveling waveregime, i.e. drifts towards lower values of ln C at constantspeed 1 /ξ typ with a fixed shape and a right tail close to power-law P ( C ) ∼ C − (1+ B ) with B ≈ . the total number of sites.In the disordered phase at µ = 8, one clearly ob-serves in the panels (b) and (d) a traveling wave regimewhere the distribution of the correlator drifts towardslower value of C g by always keeping the same shapeat a constant speed 1 /ξ typ , where ln C g = − g/ξ typ .Moreover, the distribution P (ln C ) develops at large g or N a right tail close to exponential decrease P (ln C ) ∼ exp( − B ln C ) which translates into a power-law tail for P ( C ) ∼ C − (1+ B ) with an exponent B ≈ .
5. This largeright tail is responsible of the different decay of the aver-aged and typical correlator, see Fig. 7 (b). Such a behav-ior is characteristic of a non-ergodic phase and is oftenrelated to the characteristic directed polymer physics onthe Cayley tree [8, 106, 110, 153]. In this context, an ex-ponent
B <
C. Ergodicity properties of the leading orbital
The leading orbital | φ (cid:105) = (cid:80) Ni =1 a i | i (cid:105) , associated tothe largest eigenvalue of the 1BDM, is the most delocal-ized one, corresponding to the condensed mode in theBEC regime. In the last row of Fig. 6, we representthe weights | a i | in real space for the same samples as inthe above rows of the same figure with µ = 2 (left) and µ = 8 (right). It is quite remarkable that the same spa-tial structure observed for the correlators in the middlepanels also emerges for this leading orbital.In order to be more quantitative, we study the partic-ipation entropy S q [155], derived from the q th momentsof the eigenmode | φ (cid:105) . This quantity informs us on its(de)localization properties in real space. It is defined by, S q = 11 − q ln (cid:32)(cid:88) Ni =1 | a i | q (cid:33) . (4.3)In the thermodynamic limit, one gets S q = ln N for aperfectly delocalized mode whereas S q = constant if themode is localized. In an intermediate situation, S q ∝ ln( N D q ) ∝ D q ln N with 0 < D q < q -dependent) (multi)fractal dimension. In this case, themode is delocalized (the participation entropy still growswith N ) but non-ergodic (the scaling is slower than inthe perfectly delocalized case, meaning that it does notoccupy uniformly the whole space). The extreme cases D q = 1 and D q = 0 correspond to perfect delocalizationand localization, respectively.In the following, we mainly focus on the q = 2 case,which recovers the usual inverse participation ratio (IPR)with S = − ln(IPR) [156]. We show in Fig. 9 (a) thedisorder-averaged participation entropy S of the lead-ing mode versus the system size N for various disorderstrengths µ . As expected, we observe a logarithmic in-crease ln N , with a prefactor which seems to graduallychange with increasing randomness. While the multi-fractal dimension is defined in the thermodynamic limit,it is instructive to consider its finite-size version at fixed N to then try to extract its N → + ∞ value, D q (cid:0) N (cid:1) = d S q (cid:0) N (cid:1) d ln N with D q ≡ D q (cid:0) N → ∞ (cid:1) . (4.4)This local slope is displayed in Fig. 9 (b) as a functionof system size N . In absence of disorder, the leading S q = µ = 0 . µ = 0 . µ = 1 . µ = 1 . µ = 2 . µ = 2 . µ = 3 . µ = 3 . µ = 4 . µ = 4 . µ = 5 . µ = 5 . µ = 6 . µ = 7 . µ = 8 .
10 100 1000 N . . . . . D ( N )
10 100 1000 N . . . D q ( N ) . . . . . q . . . D q ( a )( b ) ( c ) μ = 3.00 μ = 6.00 q = 2 q = 1 q = 1/2 μ = 6.00, N = 1532 μ = 3.00, N = 3070 ( d ) FIG. 9. (a) Disorer-averaged participation entropy S q =2 ofthe leading orbital versus the system size N for various dis-order strengths µ , see definition of Eq. (4.3). (b) Local slopeof the disorer-averaged participation entropy Eq. (4.4) versusthe system size is displayed, and should saturate to D ( µ )in the limit N → + ∞ . At small disorder, we observe a non-monotonous behavior characterized by a minimum before get-ting D (cid:39) D seems to saturate to a finite value smaller thanone, signaling nonergodicity. (c) Local slope of the disorer-averaged participation entropy of index q ( q = 0 . q = 1 and q = 2) versus the system size N for µ = 3 and µ = 6. No q dependence is observed at small disorder as N → + ∞ , while D q ( N ) saturates to slightly different values depending on q ,suggesting multifractality. (d) Same data as panel (b) for afixed system size N at µ = 3 and µ = 6, versus the index q .Multifractality is confirmed at strong disorder, with D q being q -dependent. orbital is perfectly delocalized with D ( N ) = 1 for allsystem sizes. When introducing disorder in the system,the local slope becomes non-monotonous by developinga minimum at N = N ∗ ( µ ) before increasing towards D ( N ) → N , S q = ln N + b q , with b q <
0. Thisnegative constant correction can be physically related toa finite nonergodicity volume Λ S q = exp( − b q ), as arguedlater. The position N ∗ ( µ ) of the minimum increases withthe disorder, resulting in the nonergodicity volume alsoincreasing with µ . A detailed scaling analysis will beperformed below, in Sec. V.At stronger disorder, in the regime where the BEC or-der parameter was found to vanish, we clearly observe adifferent behavior for the prefactor D , with an appar- ( c )( b )( a ) μ c = 5.5 ° ° N/ § Ω cond ( µ )110100 Ω c o nd ( N , µ ) . Ω c o nd ( N , µ c ) F ( x ) µ = 0 . µ = 0 . µ = 1 . µ = 1 . µ = 2 . µ = 2 . µ = 3 . µ = 3 . µ = 4 . µ = 4 . µ = 5 . µ = 5 . µ = 5 . µ = 5 . µ = 5 . ° ° N/ § Ω coh ( µ )110100 Ω c o h ( N , µ ) . Ω c o h ( N , µ c ) ° ° N/ § S ( µ )012345 S ( N , µ ) ° S ( N , µ c ) μ c = 5.5 μ c = 5.5 ∝ x ζ c ond c ∝ x ζ c oh c ∝ ( − D c ) l n ( x ) FIG. 10. Finite-size scaling analysis of the disorder-averaged (a) condensed density ρ cond , (b) coherent density ρ coh , and (c)participation entropy of the leading orbital S in the regime µ ≤ µ c . The best scaling of the data is obtained for a volumic scalingEq. (5.1) at µ c ≈ .
5. The green curve shows the scaling function F ( N/ Λ) of Eq. (5.2) fitted to the data, with a quantity-dependent and disorder-dependent scaling parameter Λ( µ ). The divergence of the non-ergodicity volume Λ at criticality isshown in Fig. 11. The dashed lines correspond to the behavior of the scaling function for the three quantities for N (cid:29) Λ,according to Eq. (5.6) and Eq. (5.7). ent saturation at a value D <
1, thus signalling thatthe leading orbital associated to the most delocalizedmode is no longer ergodic on the Cayley tree, but rather(multi)fractal. Panel (c) of Fig. 9 shows such a differ-ence between the two regimes, for three values of theR´enyi parameter q = 0 . , ,
2. For µ = 3 (BEC phase)full ergodicty of the leading orbital is recovered for largeenough system size with D q →
1. Instead, in the dis-ordered regime at µ = 6, D q clearly saturates to a non-ergodic value, with an additional signs of multifracatalityas a non-trivial q -dependence is found. This is better vis-ible in Fig. 9 where one also sees strong multifractality atsmall q (as also observed for the Anderson transition ininfinite dimension [13, 34], or for gapped ground-states ofspin chains [157]), followed by an almost q -independentregime. V. QUANTUM CRITICAL PROPERTIESA. Scaling analysis across the transition
We established in the previous sections that a transi-tion takes place in the system around µ c ≈ − µ ≤ µ c . In the disordered phaseat µ > µ c , the numerical simulations are limited in sizeand strength of the disorder so that we could not performa conclusive finite-size scaling analysis.The finite-size scaling analysis of localization transi-tions in graphs of effective infinite dimensionality suchas the Cayley tree is particularly subtle. This was il-lustrated recently in the Anderson transition on random graphs [20, 21, 34], in the MBL transition [145, 150] andin certain classes of random matrices [33, 158, 159]. Thedifficulty comes from the fact that the volume of the sys-tem N (the number of sites) varies exponentially withthe linear size of the system, i.e., the number of genera-tions G in the Cayley tree. This implies that a volumicscaling law F ( N/ Λ) depending on the ratio of volume N by a characteristic volume Λ (e.g., a correlation volume)is distinct from a linear scaling F ( G/ξ ) depending on theratio of the size G over the characteristic length ξ . Thesedifferent types of scaling have important implications forthe nature of the transition and of the different phases.In particular, a linear scaling can imply (depending onthe critical behavior) a non-ergodic delocalized phase, seeRefs. [20, 34, 145, 150].We carried out a detailed scaling analysis of the be-havior of S , ρ cond and ρ coh according to the size of thesystem, and tested these various scaling assumptions (lin-ear and volumic). The results show a quantitative agree-ment of the data with a volumic scaling assumption, at-testing to the ergodic character of the delocalized phase,with compatible values of µ c and critical exponent for S , ρ cond and ρ coh . We detail this analysis in this sec-tion. The approach we have used is very similar to whathas been done in the context of Anderson localization onrandom graphs and the MBL transition [20, 34, 145, 150].We assume some value of µ c which belongs to the set of µ ’s that we have simulated. We then consider the scalingobservable O ≡ ρ ( N, µ ) /ρ ( N, µ c ) for ρ cond and ρ coh and O ≡ S (cid:0) N, µ (cid:1) − S (cid:0) N, µ c (cid:1) for S (the substraction in-stead of the division by the critical behavior comes fromthe fact that S is an entropy) and test the validity ofthe volumic or linear scaling assumptions: O = F vol O (cid:16) N/ Λ (cid:17) or O = F lin O (cid:16) G/ξ (cid:17) . (5.1)To do this, we perform a Taylor expansion of the scaling0functions around µ ≡ µ c [160, 161]: F (cid:16) Θ N /ν (cid:17) = n (cid:88) j =0 a j (cid:16) Θ N /ν (cid:17) j , (5.2)with N = N the volume or N = G the depth of the treeand Θ = (cid:0) µ − µ c (cid:1) + m (cid:88) j =2 b j (cid:0) µ − µ c (cid:1) j , (5.3)The orders of expansion have been set to n = 5 and m = 3. Therefore, N dof = n + m +1 parameters are to befitted (including the critical exponent ν ). The goodnessof fit (calculated from the chi-squared statistic dividedby the number of degrees of freedom) should be of orderone for an acceptable fit.A systematic test of different choices of µ c and volumicand linear scaling hypotheses is represented in Fig. 16 ofthe appendix. It gives a clear indication that the datain the condensed phase µ < µ c are compatible with avolumic scaling with µ c ≈ . µ = 5 . ≈ µ c is well described by ρ (cid:0) N, µ c (cid:1) ∼ N − ζ c , (5.4)and S (cid:0) N, µ c (cid:1) ∼ D c2 ln N + b c2 , (5.5)with ζ condc ≈ . ζ cohc ≈ . D c2 ≈ . µ c within the range µ c = 5 and µ c = 6. The volumic scaling behavior, represented inFig. 10, together with the ergodic behavior at small µ , ρ cond ≈ ρ ∞ cond and ρ coh ≈ ρ ∞ coh and S = ln N + b pre-dicts an ergodic condensed phase for N (cid:29) Λ. Indeed,the scaling function F behave as F ρ (cid:0) x (cid:1) ∼ x ζ c for x (cid:29) , (5.6)for the condensed and coherent densities, and as F S (cid:0) x (cid:1) ∼ (cid:0) − D c2 (cid:1) ln( x ) for x (cid:29) , (5.7)for the participation entropy of the leading orbital, asshown by the dashed lines in Fig. 10. B. Critical exponents
1. Correlation volumes
The divergence of the correlation volumes Λ is shownin Fig. 11. It is difficult to conclude whether the scal-ing volume diverges exponentially or algebraically at thetransition. On the one hand, our approach presupposesan algebraic divergence, see Eq. (5.2), but we considered non-linear corrections so that it can describe also an ex-ponential divergence. On the other hand, volumes varyexponentially with lengths on the Cayley tree, and if thedivergence of Λ is to be associated with an algebraic di-vergence of a characteristic length, then Λ must divergeexponentially. In the panel (a) of Fig. 11, algebraic fitsof the three Λs as a function of ( µ c − µ ) give exponents ν ≈ . ρ cond and S and ν ≈ . ρ coh . Thesevalues are quite large and may suggest an exponential di-vergence, which is shown in panel (b) of Fig. 11. There,ln Λ versus ( µ c − µ ) are fitted by a power-law with a com-mon exponent ν (cid:48) ≈ .
25 for all three observables ρ cond , ρ coh and S . This scaling analysis confirms the ergodicnature of the condensed phase at small µ < µ c when thesystem volume N (cid:29) Λ, with Λ a characteristic volumediverging at a critical value of the disorder µ c ≈ .
2. Order parameter
The order parameter usually vanishes at criticality as ρ ∼ (cid:12)(cid:12) µ − µ c (cid:12)(cid:12) β , (5.8)which defines the critical exponent β . Assuming that thecharacteristic volume diverges asΛ ∼ (cid:12)(cid:12) µ − µ c (cid:12)(cid:12) − ν , (5.9)we expect ρ ∼ Λ − β/ν . (5.10)Therefore, at criticality, for finite-size N ≤ Λ, we have ρ ∼ N − β/ν , (5.11) . µ c ° µ § . µ c ° µ l n § ( a ) ( b ) l n Λ ∼ ( μ c − μ ) − ν ′ μ c = 5.5 ρ cond μ c = 5.5 Λ ∼ ( μ c − μ ) − ν ρ coh S ν ≈ 2.6(7) ν ≈ 3.3(10) ν ≈ 2.6(3) ρ cond ρ coh S ν ′ ≈ 0.22(7) ν ′ ≈ 0.24(13) ν ′ ≈ 0.25(8) FIG. 11. Divergence of the characteristic scaling volumes Λof the three disorder-averaged observables ρ cond , ρ coh and S obtained from Fig. 10. (a) Test of an algebraic divergence ofΛ: Power-law fit Λ ∼ ( µ c − µ ) − ν close to µ c . The extractedcritical exponents are given with the corresponding error bar,estimated from the change of ν with the choice of µ c between µ c = 5 and µ c = 6. (b) Similar to the other panel with atest of an exponential divergence of Λ, where ln Λ is fitted toln Λ ∼ ( µ c − µ ) − ν (cid:48) .
10 100 1000 N . . . . Ω c o h ° N , µ ¢ , Ω c o nd ° N , µ ¢ µ = 5 . µ = 5 . µ = 6 . µ = 6 . µ = 7 . µ = 7 . µ = 8 . µ . . . . . . ≥ ° µ ¢ All NN ∏ N ∏ N ∏ ( a ) ( b ) ρ coh ρ cond ρ coh ρ cond ∼ N − ζ ( μ ) FIG. 12. (a) Finite-size decay of the disorder-averaged co-herent (circles) and condensate (diamonds) densities in thedisordered regime µ > .
5. Lines are power-law fits. (b)Decay exponents ζ coh / cond , estimated from fits to the formEq. (5.15), are plotted against disorder strength µ . Resultsfor different fitting windows show that ζ coh → ζ cond ( µ ) < thus leading to the simple identification ζ = 2 β/ν. (5.12)If instead of a power-law, the correlation volume divergesexponentially, see Fig. 11 (b), followingln Λ ∼ (cid:12)(cid:12) µ − µ c (cid:12)(cid:12) − ν (cid:48) , (5.13)that would imply an exponential vanishing of the orderparameter ln ρ ∼ (cid:12)(cid:12) µ − µ c (cid:12)(cid:12) β (cid:48) , (5.14)in order to verify the observed critical algebraic decayEq. (5.11).From our numerics, a direct estimate of the order pa-rameter exponent is very difficult, as seen in Fig. 3 (rightpanels). However, it is also clear that the vanishing of theorder parameter is very fast, suggesting a large value of β , in agreement with the large value of ν (Fig. 11). QMCdata would also be compatible with an exponential decayEq. (5.14). C. Strong disorder regime
1. QMC results
In the ordered phase both coherent and BEC densitiestake finite values of comparable magnitude. However, inthe disordered regime, for µ > µ c the two order parame-ters vanish in the thermodynamic limit as power-laws ρ coh / cond ∝ N − ζ coh / cond , (5.15) with different decay exponents ζ coh (cid:54) = ζ cond . This is clearfrom Fig. 12 where the decay of ρ coh ( N ) is compatiblewith a conventional 1 /N behavior while the condensatedensity shows a slower decay with ζ cond <
2. Toy-model for the Bose glass phase
We want to build the simplest realistic toy-modelwhich captures all relevant features of the disorderedregime. Building on our QMC results, in particu-lar the real-space properties shown in Fig. 6, and theRefs. [20, 34], we propose a simple two-parameter ansatzwhich describes both the pair-wise correlations C ij andthe coefficients a i of the leading orbital | φ (cid:105) = (cid:80) Ni =1 a i | i (cid:105) associated to the largest eigenvalue λ . We first modelthe inhomogeneity, clearly visible in the right panels ofFig. 6, by dividing the Cayley tree in two subsets, assketched in Fig. 13: An ergodic region E where all theweights are finite and of comparable magnitude, and alocalized subset L where instead C ij and the coefficients a i are very small and decay exponentially.This behavior is modeled by the following ans¨atze, | a i | ∼ (cid:26) exp( − g i /ξ ) if i ∈ L constant if i ∈ E , (5.16)where g i is the distance across the tree generations fromthe localization center, and C ij ∼ (cid:26) exp( − g ij /ξ ) if i, j ∈ L constant if i, j ∈ E , (5.17)where g ij is the distance between two sites. If i and j belong to different subsets, we will also assume an expo-nential decay. In addition to the length scale ξ , whichcontrols the localized part (and which should depend onthe disorder strength), we introduce a second disorder-dependent parameter α in order to describe the size ofthe ergodic support N E ∝ N α , (5.18)with 0 ≤ α < L has a dominant scaling N L ∼ N (withsubleading corrections). a. Coherent density— In order to estimate the co-herent density Eq. (3.3), we have to perform a summationover all possible pairs of correlators. Using the ansatzEq. (5.17) we arrive at ρ coh ≈ N + c e G (ln K − /ξ ) N + c N − α ) , (5.19)where c and c are constants. The first term accounts forparticle density (half-filling) contribution, the second onecomes from the localized support, and the third one fromthe ergodic part. Using the fact that G = ln N/ ln K , weget a decay exponent governing ρ coh ∼ N − ζ coh ζ coh = min (cid:20) , ξ ln K , − α ) (cid:21) . (5.20)2 ℰ (ergodic)ℒ (localized)
FIG. 13. Sketch for the toy-model inspired from the QMCresults shown in Fig. 6. The Cayley tree is divided in twosubsets: the ergodic part E where the weights (of both thecorrelations and the leading orbital) are finite and of compa-rable magnitude, and the localized subset L where instead C ij and the coefficients a i are very small, exponentially localized,see Eqs. (5.25) and (5.17). Our QMC results, see Fig. 12, strongly suggest that ζ coh = 1, which constraints ξ ≤ / ln K and α ≤ / b. BEC density— Then, one can also get an esti-mate for the largest eigenvalue λ of the 1BDM C | φ (cid:105) = λ | φ (cid:105) . (5.21)We have to solve a C + a C + · · · + a N − C N − = λ a , (5.22)which, using Eqs. (5.25) and (5.17), the fact that ξ < / ln K , and after a proper normalization of the leadingorbital (see below), yields for the dominant term λ ∝ N α/ . (5.23)This gives a BEC density ρ cond = λ /N ∝ N − α/ , andtherefore a decay exponent ζ cond = 1 − α c. Participation entropies— The third quantitywhich can be estimated from our toy-model is the par-ticipation entropy of the leading orbital, as previouslydefined in Eq. (4.3). Using the fact that ξ < / ln K ,the normalization of the ansatz wave-function Eq. (5.25)yields | a i | ∝ N − α/ × (cid:26) exp( − g i /ξ ) if i ∈ L constant if i ∈ E , (5.25)The q -R´enyi entropies will depend on a threshold index q ∗ = ξ ln K − α ) < , (5.26) such that S q ≈ (cid:40) − q ( α + ξ ln K ) − q ln N if q < q ∗ α ln N if q > q ∗ , (5.27) d. Consequences and comparison with QMC— From QMC simulations, see Fig. 12, we expect for thedisordered regime that ζ coh = 1 and ζ cond <
1, a behaviorperfectly reproduced by our toy-model, see Eqs. (5.20)and (5.24), provided that the characteristic length scale ξ < / ln K , and the parameter α < /
2. Moreover,the leading orbital | φ (cid:105) was found to be delocalized andnon-ergodic (see Fig. 9) with a multifractal behaviorat small q , followed by a simpler fractal behavior atlarger q with a constant D q <
1. Here, our toy-modelis also able to capture such a behavior, see Eq. (5.27),with a threshold R´enyi parameter q ∗ <
1, in agreementwith QMC results. The physical interpretation of such(multi) fractal properties is fairly simple: large values of q probe the larger components of | φ (cid:105) , thus essentiallyexploring the ergodic subset E (see Fig. 13) of size N α .Conversely, at small q , the localized component cannotbe ignored, and will also contributes to the multifractaldimension. In Tab. I we give a summary of thesefindings, and a comparison betwen the toy-model andQMC results.It is also worth mentioning that a simple geometricinterpretation of the exponent α can be done, in termsof an effective branching number K eff . Indeed, insteadof having all branches of the tree equally contributing,we allow a reduced branching parameter K eff = pK with1 /K < p ≤ p can be interpretedas the probability to follow a branch. Then we have thesimple relation α = 1 + ln p/ ln K .Finally, this toy-model provides for a nice physical pic-ture for the anomalous power-law scaling λ ∼ N − ζ cond (with ζ cond <
1) which is a direct consequence of the frac-tal behavior of the associated orbital. At large enough q , where all fractal dimensions are the same D q ≡ D , wehave the very simple result1 − ζ cond = D/ . (5.28)A direct comparison of the toy-model results with QMCsimulations gives a surprisingly accurate agreement withEq. (5.28). For instance, at the strongest disorderedstrength that we have considered ( µ = 8), the QMCestimate for 1 − ζ cond = 0 . D / . µ c = 5 . − ζ condc =0 . D c2 / . D → ζ cond →
1. However, it is not3
Coherent density ρ coh ∝ N − ζ coh Condensed density ρ cond ∝ N − ζ cond Participation entropies S q ≈ D q ln N QMC ζ coh ≈ ζ cond < < D q < ζ coh = 1if ξ < K and α < / ζ cond = 1 − α if ξ < K D q ≈ − q (cid:16) α + ξ ln K (cid:17) − q if q < q ∗ α if q > q ∗ , TABLE I. Summary of some properties of the disordered phase of dirty bosons on a Cayley tree with branching number K .QMC estimates are shown, together with analytical results obtained from a two-parameter toy-model (see Sec. V C 2). Thedecay exponents ζ coh / cond of both coherent and condensate densities are displayed, together with the (multi)fractal dimension D q governing the ergodicity properties of the leading orbital of the 1BDM. The threshold R´enyi index q ∗ is given in Eq. (5.26). clear whether or not a second transition towards such afully localized phase will take place at a larger but finitedisorder strength, or perhaps more likely only in the limitof infinite randomness. VI. DISCUSSIONS AND CONCLUSIONA. Summary of our results
In this paper, building on large-scale quantum MonteCarlo simulations, we have investigated the zero-temperature phase diagram of hard-core bosons in a ran-dom potential on site-centered Cayley trees with branch-ing number K = 2. We find that the system undergoes adisorder-induced quantum phase transition at finite dis-order strength µ c ≈ . ρ coh , the condensed density ρ cond as well as theleading orbital and its participation entropy. We haveperformed a careful scaling analysis on the last threequantities as the transition is approached from the weakdisorder side µ < µ c . In the strong disorder side µ > µ c ,we have described the physics using a toy-model whichaccounts for the observed behaviors.All the observations and analyzes agree on the samephysical image of the transition. At low disorder, there isa characteristic volume Λ beyond which the system showsoff-diagonal long-range order: The long-distance correla-tor is constant ; its distribution is stationary ; the co-herent density is finite ρ coh > ρ cond > D = 1.In the strong disorder regime, we observe clear signaturesof a non-ergodic Bose glass phase: the typical and aver-age correlators decrease exponentially with different lo- calization lengths, their distribution has a traveling waveregime and a large power-law tail P ( C ) ∼ C − (1+ B ) withan exponent B <
1, a signature of replica symmetrybreaking, a crucial glassy property. The coherent den-sity decreases as ρ coh ∼ /N while the condensed densitydecreases with a non-trivial power-law ρ cond ∼ N − ζ cond .Moreover, the leading orbital of the one-body densitymatrix is multifractal ( D q < N α sites) along which the correlator shows long-range order and the leading orbital is delocalized, whilethe remaining sites show a strong exponential localiza-tion. In our picture, the Bose glass phase is similar to thenon-ergodic delocalized phase of the Anderson transitionon the Cayley tree, where similar multifractal propertieshave been predicted in a broad range of disorder [11–18]. The two characteristic scales ξ typ and α character-izing the Bose glass phase are also reminiscent of thetwo localization lengths that govern the localized phaseof the Anderson transition on random graphs [34]. Inparticular, different observables are governed by differ-ent characteristic scales. While ρ coh is controlled by thebulk localization properties, i.e., ξ typ , ρ cond and the lead-ing orbital are dominated by the rare delocalized prunedtree, thus by the scale α .The comparison of our results with the predictions ofthe cavity mean field [110] clearly indicates a new con-densed ergodic phase at low disorder which is absent inthe cavity approach. It remains to be studied if the Boseglass phase that we observe, which is a non-ergodic de-localized phase, corresponds to that predicted by cavityapproach and if a second transition to a completely lo-calized phase occurs at stronger disorder. Another pos-sibility is that the cavity mean field describes differentphysics, because of the approximation made when deal-ing with the Ising model, which is clearly different fromour U(1) symmetric bosonic system.Finally, the non-trivial scaling laws that we found sug-gest that there is no finite upper critical dimension d c beyond which conventional onset of mean-field theory4would take place. B. Open questions
This work is the first of its kind, studying by an un-biased numerical method the dirty boson problem on aneffectively infinite-dimensional lattice. While we addressseveral fundamental points such as the existence of aquantum phase transition at finite disorder strength andthe nature of some of its critical properties, several ques-tions remain open.A first one concerns the critical properties of the tran-sition when approached from the localized phase µ > µ c .For instance, different scaling laws on both sides of thetransition were found for the Anderson localization tran-sition on random graphs (which are also effectively infi-nite dimensional) [20]: a volumic scaling on the delocal-ized side and a scaling with the linear size of the systemon the localized side. Would the same phenomenologyapply for boson localization? However, accessing strongdisorder with quantum Monte Carlo is computationallychallenging and expensive, which is why we limited ourscaling analysis to the ordered phase.Another interesting question concerns the possible uni-versal properties of the delocalization-localization transi-ton in infinite dimension. The Cayley tree that we stud-ied in this paper is one example of such effectively infinitedimensional lattice, but other graphs meet the require-ments, such as random graphs or small-world networks.In particular, by studying the dirty boson problem onone of these lattices would allow to quantify the effectof geometrical loops, absent on the Cayley tree, and toquantify the effect of the extensive number of boundarylattice sites, specific to the Cayley tree.A peculiar property of the superfluid–Bose glass tran-sition is the predicted [45] hyperscaling relation z = d ,between the dimension of the system d , and the dy-namical exponent z , numerically verified in for d ≤ z is not read-ily available from the quantities we considered in thiswork, as it is usually inferred from the scaling of the su-perfluid density or the imaginary-time off-diagonal corre-lation function. Yet, accessing it would be interesting inorder to complete the critical properties description of thetransition in infinite dimension, and check on the validityof the hyperscaling relation z = d in infinite dimension,in the absence of a finite upper critical dimension.Regarding the critical exponent ζ , we have identifiedthe hyperscaling relation ζ = 2 β/ν , which in principleis valid for both coherent and BEC densities. Howeverthe fact that at criticality we observed ζ ccoh (cid:54) = ζ ccond mayimply two different order parameter exponents β coh (cid:54) = β cond .Finally, our results could suggest an avalanche sce-nario for the delocalization transition when µ → µ c isapproached from the Bose glass regime, a process shownrigorously for the Anderson transition on the Cayley tree [163, 164] and crucially important in the many-body lo-calization transition [165]. In our case, it may occur whenthe exponential bulk localization does not compensatethe exponential increase of the number of sites with thedistance, i.e. when ξ typ > ξ ctyp with ξ ctyp a critical valuewhich depends only on the branching number. ACKNOWLEDGMENTS
M.D. was supported by the U.S. Department of En-ergy, Office of Science, Office of Basic Energy Sciences,Materials Sciences and Engineering Division under Con-tract No. DE-AC02-05-CH11231 through the ScientificDiscovery through Advanced Computing (SciDAC) pro-gram (KC23DAC Topological and Correlated Matter viaTensor Networks and Quantum Monte Carlo). N.L.and G.L are supported by the French National ResearchAgency (ANR) under projects THERMOLOC ANR-16-CE30-0023-02, MANYLOK ANR-18-CE30-0017 andGLADYS ANR-19-CE30-0013, and the EUR grantNanoX No. ANR-17-EURE-0009 in the framework ofthe “Programme des Investissements d’Avenir”. This re-search used the Lawrencium computational cluster re-source provided by the IT Division at the LawrenceBerkeley National Laboratory (supported by the Direc-tor, Office of Science, Office of Basic Energy Sciences,of the U.S. Department of Energy under Contract No.DE-AC02-05CH11231). This research also used resourcesof the National Energy Research Scientific ComputingCenter (NERSC), a U.S. Department of Energy Office ofScience User Facility operated under Contract No. DE-AC02-05CH11231. Additionally, the numerical calcula-tions benefited from the high-performance computing re-sources provided by CALMIP (Grants No. 2018- P0677and No. 2019-P0677) and GENCI (Grant No. 2018-A0030500225).
Appendix A: Additional information on thequantum Monte Carlo simulations
The quantum Monte Carlo data displayed in the maintext is computed at the inverse temperature β = 1 /T reported in Tab. II. The number of disorder samples N s for each system size is also reported.
1. Convergence with temperature
The stochastic series expansion quantum Monte Carlois a finite temperature method. Therefore, it is importantto perform calculations at sufficiently low temperaturesto capture ground state properties. For three represen-tative system sizes N = 46, N = 382 and N = 3070 atdisorder strength µ = 4, we show in Fig. 14 the conver-gence of the disorder-averaged condensed density ρ andthe disorder-averaged participation entropy S versus the5 G N β N s > > > > > > > > > β = 1 /T used in the quantumMonte Carlo algorithm depending on the system size (num-ber of generations G , total number of lattice sites N ). Thenumber of independent disordered samples N s computed toperform the disorder average is also reported. . . . Ω c o nd N = 46 N = 382 N = 3070 . . . Ω c o h Ø = 1 /T S ( a )( b ) μ = 4.00 ( c ) FIG. 14. Convergence of the disorder-averaged condenseddensity ρ cond , the disorder-averaged coherent density ρ coh ,and the disorder-averaged participation entropy S versus theinverse temperature β = 1 /T . Three system sizes are dis-played N = 46 ( N s = 2000), N = 382 ( N s = 2000) and N = 3070 ( N s = 360) at disorder strength µ = 4. Note thelogarithmic scale fo the x -axis. The data displayed in themain text are those at the largest inverse temperature, seeTab. II. inverse temperature β = 1 /T . The data displayed in themain text are those at the largest inverse temperature, asindicated in Tab. II. We have found that these tempera-tures are low enough to reliably probe the ground statein the quantum Monte Carlo simulations. . . . . Ω c o nd . . . . Ω c o h N s S ( a )( b ) μ = 4.00 N = 46 N = 46 N = 382 N = 382 N = 3070 N = 3070 μ = 4.00 N = 46 N = 382 N = 3070 ( c ) μ = 4.00 FIG. 15. Convergence of the disorder-averaged condenseddensity ρ cond , the disorder-averaged coherent density ρ coh ,and the disorder-averaged participation entropy S versus thenumber of independent samples N s considered to perform theaverage. Three system sizes are displayed N = 46, N = 382and N = 3070 at disorder strength µ = 4. See Tab. II.
2. Convergence with number of samples
The convergence with the number of disorder samplesis checked by performing averages including an increas-ing number of samples N s . We show in Fig. 15 thatconvergence is quickly achieved (with a few tens of sam-ples) by considering three representative system sizes aredisplayed N = 46, N = 382 and N = 3070 at disor-der strength µ = 4. Even at stronger disorder, a fewhundred of samples is sufficient to obtain reliable aver-age estimates. We consider in general > Appendix B: Finite-size scaling analysis
To determine the value of the critical disorder strength µ c and the critical properties of the transition for µ < µ c ,we perform a finite-size scaling analysis, as detailed inSec. V. For the condensed density, the coherent densityand the participation entropy of the leading orbital, botha linear and volumic scaling hypothesis are tested. Theirquality is measured by the chi-squared statistic χ perdegree of freedom N dof of fitting the numerical data tothe corresponding scaling function, which also dependson the choice of critical disorder strength µ c considered.Hence, by plotting χ /N dof versus µ c , we are able to es-timate the best scaling hypothesis and locate the transi-6 ¬ . N d o f ¬ . N d o f µ c ¬ . N d o f ( a )( b ) ρ cond ℱ ( N /Λ ) ℱ ( ln N / ξ ) ( c ) ρ coh S μ c = 5.5(5) FIG. 16. Chi-squared statistic χ per degree of freedom N dof for the best volumic and linear fits of the data ( µ < µ c )obtained following a scaling of the form of Eq. (5.2) versusthe critical disorder strength µ c for (a) the condensed density,(b) the coherent density and (c) the participation entropy ofthe leading eigenmode. The volumic scaling is systematicallybetter, with a minimum around µ c ≈ . ± .
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