Absence of two-body delocalization transitions in the two-dimensional Anderson-Hubbard model
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Absence of two-body delocalization transitions in the two-dimensionalAnderson-Hubbard model
Filippo Stellin ∗ and Giuliano Orso † Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013, Paris, France (Dated: July 22, 2020)We investigate Anderson localization of two particles moving in a two-dimensional (2D) disorderedlattice and coupled by contact interactions. Based on transmission-amplitude calculations for rela-tively large strip-shaped grids, we find that all pair states are localized in lattices of infinite size. Inparticular, we show that previous claims of an interaction-induced mobility edge are biased by severefinite-size effects. The localization length of a pair with zero total energy exhibits a nonmonotonicbehavior as a function of the interaction strength, characterized by an exponential enhancement inthe weakly interacting regime. Our findings also suggest that the many-body mobility edge of the2D Anderson-Hubbard model disappears in the zero-density limit, irrespective of the (bosonic orfermionic) quantum statistics of the particles.
I. INTRODUCTION
It is well known that in certain disordered media wavepropagation can be completely halted due to the back-scattering of the randomly distributed impurities. Thisphenomenon, known as Anderson localization , has beenreported for different kinds of waves, such as light wavesin diffusive media or in disordered photonic crystals ,ultrasound , microwaves and atomic matter waves .Its occurrence is ruled by the spatial dimension of thesystem and by the symmetries of the model, which deter-mine its universality class . When both spin-rotationaland time-reversal symmetries are preserved, notably inthe absence of magnetic fields and spin-orbit couplings,all wave-functions are exponentially localized in one andtwo dimensions. In three and higher dimensions theenergy spectrum contains both localized and extendedstates, separated by a critical point, dubbed the mobil-ity edge, where the system undergoes a metal-insulatortransition . Anderson transitions have recently been de-tected using noninteracting atomic quantum gases exposed to three-dimensional (3D) speckle potentials.Theoretical predictions for the mobility edge of atomshave also been reported and compared with the ex-perimental data.Interactions can nevertheless significantly perturb thesingle-particle picture of Anderson localization. Puzzlingmetal-insulator transitions , discovered in high-mobility2D electron systems in silicon, were later interpretedtheoretically in terms of a two-parameter scaling the-ory which combines disorder and strong electron-electroninteraction . In more recent years a great interesthas emerged around the concept of many-body localiza-tion (MBL), namely the generalization of Andersonlocalization to disordered interacting quantum systemsat finite particle density (for recent reviews see ). Inanalogy with the single-particle problem, MBL phasesare separated from (ergodic) thermal phases by criticalpoints located at finite energy density, known as many-body mobility edges.While MBL has been largely explored in one di- mensional disordered systems with short range interac-tions, both experimentally and theoretically , itsvery existence in systems with higher dimensions re-mains somewhat unclear. In particular it has been sug-gested that the MBL is inherently unstable againstthermalization in large enough samples. This predictioncontrasts with subsequent experimental and numeri-cal studies of 2D systems of moderate sizes, show-ing evidence of a many-body mobility edge. It must beemphasized that thorough numerical investigations, in-cluding a finite-size scaling analysis, are computationallychallenging beyond one spatial dimension .In the light of the above difficulties, it is interesting toconsider the zero density limit of the many-body prob-lem, focusing on the localization properties of few in-teracting particles in large (ideally infinite) disorderedlattices. Although these systems may represent overlysimplified examples of MBL states, they can show sim-ilar effects, including interaction-induced delocalizationtransitions with genuine mobility edges .In a seminal paper , Shepelyansky showed that twoparticles moving in a one-dimensional lattice and coupledby contact interactions can travel over a distance muchlarger than the single-particle localization length, beforebeing localized by the disorder. This intriguing effectwas confirmed by several numerical studies , tryingto identify the explicit dependence of the pair localiza-tion length on the interaction strength. Quantum walkdynamics of two interacting particles moving in a dis-ordered one-dimensional lattice has also been explored,revealing subtle correlation effects . Interacting few-body systems with more than two particles have also beenstudied numerically in one dimension, confirming the sta-bility of the MBL phase. For instance Ref. investigateda model of up to three bosonic atoms with mutual contactinteractions and subject to a spatially correlated disor-der generated by laser speckles, while Ref. addressedthe localization in the few-particle regime of the XXZspin-chain with a random magnetic field.The few-body problem in higher dimensions has beenmuch less explored. Based on analytical arguments, ithas been suggested that all two-particle states arelocalized by the disorder in two dimensions, whereas inthree dimensions a delocalization transition for the paircould occur even if all single-particle states are localized.In contrast, subsequent numerical investigations ofthe two-particle system in two dimensions reported ev-idence of an Anderson transition of the pair, providingexplicit results for the position of the mobility edge andthe value of the critical exponent.Using large-scale numerics, we recently investi-gated Anderson transitions for a system of two in-teracting particles (either bosons or fermions with op-posite spins), obeying the three-dimensional Anderson-Hubbard model. We showed that the phase diagramin the energy-interaction-disorder space contains multi-ple metallic and insulating regions, separated by two-body mobility edges. In particular we did find metal-lic pair states in regions where all single-particle stateswere localized, which can be thought of as a proxy forinteraction-induced many-body delocalization. Impor-tantly, our numerical data for the metal-insulator transi-tion were found to be consistent with the (orthogonal)universality class of the noninteracting model (single-particle excitations in a disordered electronic system withCoulomb interaction also undergo a metal-insulator tran-sition which belongs to the noninteracting universalityclass ).In this work we revisit the Shepelyansky problem intwo dimensions and shed light on the controversy. Wefind that no mobility edge exists for a single pair in an in-finite lattice, although interactions can dramatically en-hance the pair localization length. In particular we showthat previous claims of 2D interaction-driven Ander-son transitions were plagued by strong finite-size effects.The paper is organized as follows. In Sec. II we re-visit the theoretical approach based on the exact map-ping of the two-body Schrodinger equation onto an ef-fective single-particle problem for the center-of-mass mo-tion. The effective model allows to recover the entireenergy spectrum of orbitally symmetric pair states andis therefore equivalent to the exact diagonalization of thefull Hamiltonian in the same subspace; an explicit prooffor a toy Hamiltonian is given in Sec. III. In Sec. IVwe present the finite-size scaling analysis used to discardthe existence of the 2D Anderson transition for the pair,while in Sec. V we discuss the dependence of the two-body localization length on the interaction strength. Fi-nally in Sec. VI we provide a summary and an outlook. II. EFFECTIVE SINGLE-PARTICLE MODELFOR THE PAIR
The Hamiltonian of the two-body system can be writ-ten as ˆ H = ˆ H + ˆ U , whose noninteracting part ˆ H canbe decomposed as ˆ H sp ⊗ ˆ + ˆ ⊗ ˆ H sp . Here ˆ refers tothe one-particle identity operator, while ˆ H sp denotes the single-particle Anderson Hamiltonian: ˆ H sp = − J X h n , m i | m ih n | + X n V n | n ih n | , (1)where J is the tunnelling amplitude between nearestneighbor sites m and n , whereas V n represents the valueof the random potential at site n . In the following weconsider a random potential which is spatially uncorre-lated h V n V n ′ i = h V n i δ nn ′ and obeys a uniform on-sitedistribution, as in Anderson’s original work : P ( V ) = 1 W Θ( W/ − | V | ) , (2)where Θ( x ) is the Heaviside (unit-step) function and W is the disorder strength. The two particles are coupledtogether by contact (Hubbard) interactions described by ˆ U = U X m | m , m ih m , m | , (3)where U represents the corresponding strength. Westart by writing the two-particle Schrödinger equationas ( E − ˆ H ) | ψ i = ˆ U | ψ i , where E is the total energyof the pair. If U | ψ i = 0 , then E must belong to theenergy spectrum of the noninteracting Hamiltonian ˆ H .This occurs for instance if the two-particles correspondto fermions in the spin-triplet state, as in this case theorbital part of the wave-function is antisymmetric andtherefore h m , m | ψ i = 0 .Interactions are instead relevant for orbitally symmet-ric wave-functions, describing either bosons or fermionswith opposite spins in the singlet state. In this case fromEq. (3) we find that the wave-function obeys the follow-ing self-consistent equation | ψ i = X m U ˆ G ( E ) | m , m ih m , m | ψ i , (4)where ˆ G ( E ) = ( E ˆ I − ˆ H ) − is the non-interacting two-particle Green’s function. Eq. (4) shows that for contactinteractions the wave-function of the pair can be com-pletely determined once its diagonal amplitudes f m = h m , m | ψ i are known. By projecting Eq.(4) over thestate | n , n i , we see that these terms obey a closed equa-tion : X m K nm f m = 1 U f n , (5)where K nm = h n , n | ˆ G ( E ) | m , m i . Eq.(5) is then inter-preted as an effective single-particle problem with Hamil-tonian matrix K and pseudo-energy λ = 1 /U , corre-sponding to the inverse of the interaction strength. Inthe following we will address the localization propertiesof this effective model for the pair.The matrix elements of K are unknown and must becalculated explicitly in terms of the eigen-basis of thesingle-particle model, ˆ H sp | φ r i = ε r | φ r i , as K nm = N X r,s =1 φ n r φ ∗ m r φ n s φ ∗ m s E − ε r − ε s , (6)where N is the total number of lattice sites in the gridand φ n r = h n | φ r i are the amplitudes of the one-particlewave-functions. III. EQUIVALENCE WITH EXACTDIAGONALIZATION OF THE FULL MODEL
The effective single-particle model of the pair, Eq. (5),allows to reconstruct the entire energy spectrum of or-bitally symmetric states for a given interaction strength U . At first sight this is not obvious because the matrix K is N × N , and therefore possesses N eigenvalues, whilethe dimension of the Hilbert space of orbitally symmet-ric states is N ( N + 1) / , which is much larger. The keypoint is that one needs to compute the matrix K andthe associated eigenvalues λ r = λ r ( E ) , with r = 1 , ...N ,for different values of the energy E . The energy levelsfor fixed U are then obtained by solving the equations λ r ( E ) = 1 /U using standard root-finding algorithms.Let us illustrate the above point for a toy model with N = 2 lattice sites in the absence of disorder. In thiscase the Hilbert space of symmetric states is spannedby the three vectors | , i , | , i and ( | , i + | , i ) / √ .The corresponding energy levels of the pair can be foundfrom the exact diagonalization of the × matrix of theprojected Hamiltonian: H ed = U −√ −√ −√ −√ U . (7)An explicit calculation yields E = U and E = ( U ±√ U + 16) / . Let us now show that we recover exactlythe same results using our effective model. The single-particle Hamiltonian is represented by the matrix H sp = (cid:18) − − (cid:19) , (8)whose eigenvalues are given by ε = − and ε = 1 .The associated wavevectors are | φ i = ( | i + | i ) / and | φ i = ( | i − | i ) / . From Eq.(6) we immediately find K = (cid:18) A BB A (cid:19) , (9)where A = ( E/ ( E − /E ) / and B = ( E/ ( E − − /E ) / . The corresponding eigenvalues of K are given by λ ( E ) = A − B = 1 /E and λ ( E ) = A + B = E/ ( E − .The condition λ = 1 /U yields E = U , while λ = 1 /U admits two solutions, E = ( U ± √ U + 16) / , allowingto recover the exact-diagonalization energy spectrum. In -4 -2 0 2 4 E -10-50510 λ λ=1 FIG. 1. Eigenvalues of the matrix K of the effective modelof the pair, Eq. (5) for a toy model of N = 2 coupled siteswith no disorder, plotted as a function of the energy E of thepair (blues data curves). For a given interaction strength U ,the entire spectrum of N ( N + 1) / energy levels of orbitallysymmetric states of the pair can be obtained by intersectingthe data curves with the horizontal line, λ = 1 /U , here shownfor U = 1 (dashed red line). The corresponding three energylevels are E = − . , E = 1 and E = 2 . . Fig.1 we plot the energy dependence of the two eigen-values of K for our toy model. Intersecting the curveswith the horizontal line λ = 1 /U (dashed red line) yieldsvisually the three sought energy levels for the orbitallysymmetric states.We stress that extracting the full energy spectrum ofthe pair based on the effective model, for a fixed value ofthe interaction strength U , is computationally demand-ing as N becomes large. The effective model is insteadvery efficient, as compared to the exact diagonalization,when we look at the properties of the pair as a functionof the interaction strength U , for a fixed value of the to-tal energy E . This is exactly the situation that we willbe interested in below. IV. ABSENCE OF 2D DELOCALIZATIONTRANSITIONS FOR THE PAIR
Numerical evidence of 2D Anderson transition fortwo particles obeying the Anderson-Hubbard model intwo dimensions was first reported on the basis oftransmission-amplitude calculations performed on rect-angular strips of length L = 62 and variable width up to M = 10 . For a pair with zero total energy and for inter-action strength U = 1 , the delocalization transition wasfound to occur for W = 9 . ± . . The result was also con-firmed from the analysis of the energy-level statistics,although with slightly different numbers.The existence of a 2D mobility edge for the pair wasalso reported in Ref. , where a decimation method wasemployed to compute the critical disorder strength as afunction of the interaction strength U , based on latticesof similar sizes. For U = 1 . , a pair with zero totalenergy was shown to undergo an Anderson transition at W = 9 ± . .Below we verify the existence of the 2D delocaliza-tion transition of the pair based on the transmission-amplitude computations . for the effective model (5),following the procedure developed in Ref. . In order tocompare with the previous theoretical predictions, we set E = 0 and W = 9 . We consider a rectangular strip ofdimensions L, M , with L ≫ M , containing N = M L lattice sites. In order to minimize finite-size effects, theboundary conditions on the single-particle Hamiltonian H sp are chosen periodic in the orthogonal direction ( y )and open along the transmission axis ( x ). We rewrite therhs of Eq. (6) as K nm = X r =1 φ n r φ ∗ m r h n | G sp ( E − ε r ) | m i , (10)where G sp ( ε ) = ( εI − H sp ) − is the Green’s function (e.g.the resolvent) of the single-particle Anderson Hamilto-nian (1), I being the identity matrix. Due to the openboundary conditions along the longitudinal direction,the Anderson Hamiltonian possesses a block tridiagonalstructure, each block corresponding to a transverse sec-tion of the grid. This structure can be exploited to effi-ciently compute the Green’s function G sp ( ε ) in Eq. (10)via matrix inversion. In this way the total number ofelementary operations needed to compute the matrix K scales as M L , instead of M L , as naively expectedfrom the rhs of Eq. (6).Once computed the matrix K of the effective model,we use it to evaluate the logarithm of the transmissionamplitude between two transverse sections of the strip asa function of their relative distance n x : F ( n x ) = ln X m y ,n y |h , m y | G p ( λ ) | n x , n y i| . (11)In Eq. (11) G p ( λ ) = ( λI − K ) − is the Green’s functionassociated to K with λ = 1 /U and the sum is taken overthe sites m y , n y of the two transverse sections.For each disorder realization, we evaluate F ( n x ) at reg-ular intervals along the bar and apply a linear fit to thedata, f fit ( n x ) = pn x + q . For a given value of the in-teraction strength, we evaluate the (disorder-averaged)Lyapunov exponent γ = γ ( M, U ) as γ = − p/ , where p is the average of the slope. We then infer the localizationproperties of the system from the behavior of the reducedlocalization length, which is defined as Λ = (
M γ ) − . Inthe metallic phase Λ increases as M increases, whereasin the insulating phase the opposite trend is seen. Atthe critical point, Λ becomes constant for values of M sufficiently large. Hence the critical point U = U c ofthe Anderson transition can be identified by plotting thereduced localization length versus U for different valuesof the transverse size M and looking at their commoncrossing points. U Λ M=8 M=10M=12M=16 M=20
FIG. 2. Reduced localization length of the pair plottedas a function of the interaction strength for increasing val-ues of the transverse size M = 8 , , , , of the grid.The results are obtained by averaging over N tr different dis-order realizations, varying from N tr = 600 ( M = 8) to N tr = 1000 ( M = 20) . The disorder strength is fixed to W = 9 and the pair has zero total energy, E = 0 , imply-ing that Λ( − U ) = Λ( U ) . The different curves cross in theinterval . < U < . , indicating a possible 2D delocal-ization transition, as claimed in previous investigations .The 2D Anderson transition is actually a finite-size effect, asthe crossing points disappear for larger values of M , see Fig.3. In Fig. 2 we show the reduced localization length Λ asa function of the interaction strength for increasing valuesof the strip width, ranging from M = 8 to M = 20 . Thelength of the grid is fixed to L = 400 . Notice that, since E = 0 , the reduced localization length is an even functionof the interaction strength, Λ( − U ) = Λ( U ) . We see that Λ exhibits a nonmonotonic dependence on U , as previ-ously found in the one and in the three-dimensional versions of the Anderson-Hubbard model. In particular,interactions favor the delocalization of the pair, the effectbeing more pronounced near U = 6 . We also notice fromFig. 2 that the curves corresponding to different valuesof M intersect each others around U = 1 , suggesting in-deed a possible phase transition, as previously reportedin Ref. . A closer inspection of the data, however,reveals that the crossing points are spread out in the in-terval . . U . . ; in particular, they drift to strongerinteractions as the system size increases, in analogy withthe three-dimensional case studied earlier .A key question is whether a further increase of thestrip’s width M will only cause a (possibly large) shiftof the critical point, or rather, the localized phase willultimately take over for any value of the interactionstrength. To answer this question, we have performed ad-ditional calculations using larger grids, corresponding to M = 30 , , . In order to guarantee a sufficiently largeaspect ratio, the length of the bar was fixed to L = 500 .The obtained results are displayed in Fig.3. We noticethat the crossing points have completely disappeared and U Λ M=30 M=40M=50
FIG. 3. Same plot as in Fig.2 but for larger grids withtransverse sizes M = 30 , , obtained by averaging over N tr = 3600 ( M = 30) , M = 40) N tr = 2850 ( M = 50) different disorder realizations. Notice that all crossing pointshave disappeared, indicating that the pair is ultimately local-ized by the disorder for any value of the interaction strength. the pair is ultimately localized by the disorder, irrespec-tive of the value of the interaction strength. This leadsus to conclude that the results of Refs. were plaguedby severe finite-size effects and no Anderson transitioncan actually take place for a pair in a disordered latticeof infinite size. V. PAIR LOCALIZATION LENGTH
Although the pair cannot fully delocalize in two di-mensions, interactions can lead to a drastic enhance-ment of the two-particle localization length. This quan-tity can be estimated using the one-parameter scalingansatz
Λ = f ( ˜ ξ/M ) , stating that the reduced localizationlength depends solely on the ratio between two quanti-ties: the width M of the strip and a characteristic length ˜ ξ = ˜ ξ ( U, W, E ) , which instead depends on the model pa-rameters and on the total energy of the pair (but not onthe system sizes L, M ). This latter quantity coincides, upto a multiplicative numerical constant a , with the soughtpair localization length, ξ = a ˜ ξ .We test the scaling ansatz for our effective model (5)using the numerical data for M = 30 , , displayedin Fig.3, corresponding to the largest system sizes. Let U j , with j = 1 , ..N U , be the values of the interactionstrength used to compute the reduced localization length(in our case N U = 44 ). We then determine the cor-responding unknown parameters ˜ ξ ( U = U j ) through aleast squares procedure, following the procedure devel-oped in Ref. . Plotting our data in the form ln Λ( M, U ) vs ln M results in multiple data curves, each of themcontaining three data points connected by straight lines(corresponding to linear interpolation). Let Λ i be one -2 -1 ξ /M Λ ~ FIG. 4. Double logarithmic plot of the reduced localiza-tion length as a function of the ratio ˜ ξ/M , where ˜ ξ is theunnormalized localization length obtained from the solutionof Eq. (12) and M is the width of the strip. The differentsymbols correspond to the data for M = 30 (up triangles), M = 40 (circles) and M = 50 (diamonds), shown in Fig. 3.All data approximately collapse on a single curve, verifyingthe scaling ansatz Λ = f (˜ ξ/M ) . of the (3 N U ) numerical values available for the reducedlocalization length. The horizontal line ln Λ = ln Λ i willgenerally intersect some of these curves. We find conve-nient to introduce a matrix η which keeps track of suchevents: if the curve U = U j is crossed, we set η ij = 1 and call ln M ij the corresponding point; otherwise we set η ij = 0 .The unknown parameters are then obtained by solvingthe following set of equations: X j (X i η ij (cid:18) N i − δ jk N i (cid:19)) ln ˜ ξ ( U j ) == X j (X i η ij (cid:18) N i − δ jk N i (cid:19)) ln M ij , (12)where N i = P j η ij is the total number of crossingpoints obtained for each Λ i value. Eq.(12) is of the form AX = B and can easily be solved. Notice however thatthe solution is not unique because the matrix A is singu-lar. Indeed the correlation length ˜ ξ ( U ) is defined up toa multiplicative constant, ˜ ξ → a ˜ ξ , implying that ln ˜ ξ isdefined up to an additive constant, ln ˜ ξ → ln ˜ ξ + ln a .In Fig.4 we verify the correctness of the scaling ansatz,by plotting the reduced localization length as a functionof the ratio ˜ ξ/M , where ˜ ξ is obtained from the solutionof Eq. (12). We see that our numerical data for differ-ent values of the interaction strength and system size docollapse on a single curve, thus confirming the scalinghypothesis.In the main panel of Fig. 5 we plot the unnormalizedlocalization length of the pair as a function of the in- U ξ U a e s t ~ FIG. 5. Unnormalized localization length ˜ ξ of the pair plot-ted as a function of the interaction strength. Notice the log-arithmic scale in the y axis, showing that interactions canenhance the 2D localization length of the pair by more thanthree orders of magnitude. The inset displays the estimateof the multiplicative constant a , fixing the absolute scale ofthe localization length, plotted as a function of the interac-tion strength. The estimate is obtained by fitting the numer-ical data in Fig.3 corresponding to weak interactions usingEq. (13), from which we extract a est = ξ/ ˜ ξ . This quantitykeeps increasing as U diminishes, signaling that the stronglylocalized regime is not fully reached in our simulations. teraction strength. We see that ˜ ξ varies over more thanthree orders of magnitude in the interval of U consid-ered. In particular, for weak interactions the growth isapproximately exponential in U , as highlighted by thesemi-logarithmic plot. Based on analytical arguments,Imry suggested that the localization length of the pairin the weakly interacting regime should obey the relation ξ ∝ ξ sp e b ( Uξ sp ) , where ξ sp is the single-particle localiza-tion length of the Anderson model and b is a numericalfactor. A possible reason of the discrepancy is that thecited formula might apply only for relatively modest val-ues of the interaction strength, which were not exploredin our numerics. Further work will be needed to addressthis point explicitly.The constant a , allowing to fix the absolute scale ofthe localization length of the pair, is independent of theinteraction strength. Its numerical value can in principlebe inferred by fitting the data in the strongly localizedregime, according to Λ = ξM + c (cid:18) ξM (cid:19) , (13)where c is a number. In our case the most localized statesare those at weak interactions, where the reduced local-ization length takes its minimum value. For each val-ues U = U j falling in this region, we fit our numericaldata according to Eq. (13), yielding ξ = ξ ( U ) . The es-timate of the multiplicative constant, which is definedas a est = ξ ( U ) / ˜ ξ ( U ) , is displayed in the inset of Fig. 5. Since the estimate of a does not saturates for small U , weconclude that, even for the weakest interactions and thelargest system sizes considered, the pair has not yet en-tered the strongly localized regime underlying Eq. (13).The latter is typically achieved for Λ . . , whereasour smallest value of the reduced localization length is Λ( M = 50 , U = 0 . ≃ . . From the inset of Fig. 5we also see that a est increases as U diminishes, suggestingthat the result obtained for U = 0 . actually provides alower bound for the multiplicative constant. This allowsus to conclude that a ≥ . . VI. CONCLUSION AND OUTLOOK
Based on an efficient mapping of the two-bodySchrodinger equation, we have addressed the localizationproperties of two bosons or two fermions in a spin-singletstate obeying the 2D Anderson-Hubbard model. We havefound that no interaction-induced Anderson transitionoccurs for disordered lattices of infinite size in contrastwith previous numerical works, which we have shown tobe biased by finite-size effects. In this way we reconcilethe numerics with the one-parameter scaling theory oflocalization, predicting the absence of one-particle An-derson transition in two dimensions, in the presence ofboth time reversal and spin rotational symmetries. Thelocalization length exhibits a non nonmonotonic behav-ior as a function of U , characterized by an exponentialgrowth for weak interactions.We point out that the absence of the 2D mobility edgefor the two-particle system has been proven for the caseof contact interactions; similar conclusions should applyalso for short but finite-range interactions. The case oftrue long-range (e.g Coulomb) interactions is conceptu-ally different and can lead to opposite conclusions .A compelling problem is to investigate the implicationsof our results for a 2D system at finite density of particles,where many-body delocalization transitions have insteadbeen observed both numerically and experimentally inthe strongly interacting regime. We expect that, in thezero density limit, the many-body mobility edge disap-pears, irrespective of the bosonic or fermionic statisticsof the two particles. ACKNOWLEDGEMENTS
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