Many-body localization in large systems: Matrix-product-state approach
Elmer V. H. Doggen, Igor V. Gornyi, Alexander D. Mirlin, Dmitry G. Polyakov
MMany-body localization in large systems: Matrix-product-stateapproach
Elmer V. H. Doggen
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, GermanyInstitut f¨ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Igor V. Gornyi
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, GermanyInstitut f¨ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, GermanyIo ff e Institute, 194021 St. Petersburg, Russia Alexander D. Mirlin
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, GermanyInstitut f¨ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, GermanyL. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, RussiaPetersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia
Dmitry G. Polyakov
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
Abstract
Recent developments in matrix-product-state (MPS) investigations of many-body localiza-tion (MBL) are reviewed, with a discussion of benefits and limitations of the method. This ap-proach allows one to explore the physics around the MBL transition in systems much larger thanthose accessible to exact diagonalization. System sizes and length scales that can be controllablyaccessed by the MPS approach are comparable to those studied in state-of-the-art experiments.Results for 1D, quasi-1D, and 2D random systems, as well as 1D quasi-periodic systems arepresented. On time scales explored (up to t ≈
300 in units set by the hopping amplitude), a slow,subdi ff usive transport in a rather broad disorder range on the ergodic side of the MBL transitionis found. For 1D random spin chains, which serve as a “standard model” of the MBL transition,the MPS study demonstrates a substantial drift of the critical point W c ( L ) with the system size L : while for L ≈
20 we find W c ≈
4, as also given by exact diagonalization, the MPS results for L = L limit, at W c ≈ . ff ects are much weaker, which suggests that theycan be largely attributed to rare events. For quasi-1D ( d × L , with d (cid:28) L ) and 2D ( L × L ) randomsystems, the MPS data demonstrate an unbounded growth of W c in the limit of large d and L , inagreement with analytical predictions based on the rare-event avalanche theory.1 a r X i v : . [ c ond - m a t . d i s - nn ] J a n eywords: Low-dimensional systems, Many-body localization, Matrix product states,Time-dependent variational principle
1. Introduction
Philip W. Anderson considered what happens to quantum particles when they move througha disordered medium [1]. He showed that disorder can fully halt transport. According to thecelebrated scaling theory of localization [2], even weak disorder is su ffi cient to prevent trans-port in the thermodynamic limit in one-dimensional (1D) and two-dimensional (2D) geometries(more accurately, this applies to all Wigner-Dyson symmetry classes in 1D and to two of them—orthogonal and unitary—in 2D). For three-dimensional (3D) systems, a transition from the phaseof localized to the phase of delocalized states occurs, which can be driven by the strength of dis-order or another parameter of the system [3, 4]. Such localization has been observed in variousdisordered media, ranging from photonic lattices [5] and optical fibers [6] to dilute Bose-Einsteincondensates [7, 8].While the Anderson-localization problem is highly non-trivial and very rich by itself, thecomplexity of the problem is further strongly increased when the interaction between particles isincluded. The added complexity has several interrelated facets. One of them concerns the zero-temperature quantum phase transitions (and corresponding phase diagrams) in an interactingdisordered system. We will not touch this (also very rich) field in the present brief review whichfocuses on the physics of highly excited states (i.e., those at non-zero energy density). Thenatural expectation in such a situation is that interactions between particles disturb interferencee ff ects that are responsible for Anderson localization, thus leading to delocalization. Indeed,Fermi’s golden rule would suggest a non-zero quasiparticle width (i.e., a finite decay rate) atany non-zero temperature. However, with su ffi ciently strong disorder, the application of Fermi’sgolden rule (that assumes a continuous spectrum of final states) may be invalidated, leading toa breakdown of the above expectation and to the emergence of a localized phase of the many-body interacting system at non-zero temperature. This was in fact pointed out by Andersonhimself, together with Fleishman, in the seminal paper [9]. The field of many-body localization(MBL), as it is known today, has its roots in theoretical works from the mid-2000s examining thisscenario in detail [10, 11]. Experiments on MBL-type systems started in the 2010s, especiallyusing ultracold atoms [12, 13], where disorder is implemented using laser fields.With the rapid development of numerical algorithms and increasing numerical resources, theproblem of MBL has become within the reach of computational approaches, initially in one di-mension. Oganesyan and Huse [14] and ˇZnidariˇc et al. [15] pioneered the numerical study of theMBL problem using two complementary approaches. One of them [14], exact diagonalization,fully solves the many-body problem through computational brute force. This approach has noapproximations, but since the computational complexity of a many-body system increases expo-nentially with the system (e.g., ∼ L for a spin- chain of length L ), the method is limited to smallsystems (up to L ≈ et al. [16] used a particularly numerically e ffi cient implementationof exact diagonalization and found evidence for an MBL transition—i.e., a transition from thethermalizing (ergodic) to the localized phase. At the same time, the apparent critical behaviourprovided by the corresponding scaling analysis turned out to be inconsistent with the Harris cri-terion, which is an indication of the fact that the system sizes reached by exact diagonalization( L ≈
20) are too small for investigation of the critical properties.
Preprint submitted to Annals of Physics Special Issue: Localisation 2020 January 15, 2021 he second approach, which was used in Ref. [15] to demonstrate MBL at strong disorder,belongs to a class of numerical methods based on matrix product states (MPS) [17]. An MPS is atype of tensor network that encodes a many-body wave function in terms of a variational ansatz.A key feature of the MPS approach is that the accuracy of the method can be controlled throughthe bond dimension χ , which can be tuned to interpolate between a product state, where there isno entanglement, and a maximally entangled state (i.e., it allows for a description of the wavefunction without approximations). Not surprisingly, the computational e ffi ciency of describingmaximally entangled states using MPS is not better than that of exact diagonalization, so that theMPS approach is only useful for systems that are relatively weakly entangled. In practice, oneoften starts in an (unentangled) product state, and computes the dynamics until the entanglementgrows out of control. Thus, the approach is usually limited to finite times because of the growthof entanglement. We note that MPS approaches can also be used to probe the static properties,especially those of ground states that are often weakly entangled, but here we focus primarily ondynamics.The rate of the entanglement growth is therefore of significant importance for the applicabil-ity of MPS approach. In the MBL phase, for strong disorder, analytical and numerical studiesfind a slow, logarithmic growth of entanglement [18]. This is very beneficial for the investiga-tion of the MBL phase by the MPS method. On the other hand, in the opposite limit of weakdisorder, the system quickly thermalizes and shows ergodic behaviour [16], in agreement withthe eigenstate thermalization hypothesis (ETH) [19]. Of much physical interest is the behaviourof the system in a numerically broad range of intermediate disorder strengths, i.e, around theMBL transition, with the ergodic side of the MBL transition turning out to be particularly in-triguing. What is the behaviour of the entanglement in this regime? The answer to this questionis of paramount importance for the applicability of the MPS framework to the study of the MBLproblem around the transition point.Another key question concerns the role of the type of disorder in the system. Numerous ex-periments deal with interacting quasiperiodic (rather than truly random) systems. While suchsystems have no innate randomness, experimentally they also show an MBL transition [13]. Thedistinction between the two is crucial from a theoretical perspective because the physics of ran-dom systems is expected to be strongly a ff ected by rare regions, whereas such regions do notoccur in quasiperiodic systems. In particular, the type of disorder is expected to qualitativelyinfluence the type of dynamics, in the sense that a subdi ff usive dynamics is expected [20] on thebasis of Gri ffi ths e ff ects. Yet, the experiment indicates subdi ff usive behaviour also for quasiperi-odic systems.One more question concerns the case of higher-dimensional systems (i.e., beyond 1D), wherethe e ff ects of rare, weakly disordered regions should become stronger, which was predicted [21,22, 23] to destabilize the MBL phase in the thermodynamic limit at any fixed strength of disorder W . Yet, also in this case cold-atom experiments show evidence for an MBL transition at finitedisorder [24]. Importantly, however, the experiments explore finite systems, with the number ofatoms being of the order of 100, while the thermodynamic limit in the rare-region analysis isreached very slowly. In order to verify the predictions of the rare-region theory and to compareto the experiment, it is thus of crucial importance to perform computations in a broad range ofsystem sizes, at least up to those studied experimentally. This goal is by far out of reach of exactdiagonalization, which again demonstrates an important advantage of the MPS approach.These questions have been addressed in a series of papers by the present authors [25, 26, 27],where it was demonstrated that the MPS approach is very useful for the investigation of thevicinity of the MBL transition, including its ergodic side, in large systems and in a variety of3ettings. The key results, along with related advances by other researchers, are reviewed in thisarticle. For more general reviews of MBL, the reader is referred to Refs. [28, 29, 30, 31].
2. Analytical background
Analytical approaches to MBL-type problems were pioneered by Fleishman and Andersonin Ref. [9], where it was argued that short-range interactions in a system with localized single-particle states do not necessarily destroy localization at finite temperature T . The argumentwas based on the perturbative analysis of the single-particle decay rate caused by the excitationof localized particle-hole pairs. Later, higher-order electron-electron scattering processes wereaddressed in terms of an Anderson-localization problem on a certain graph in Fock space inRef. [32], where the broadening of hot single-particle states in a quantum dot (at zero tempera-ture) was studied.More recently, Refs. [10, 11] have studied the e ff ect of a weak short-ranged interaction indisordered systems where all states are localized in the absence of the interaction (in particular,in 1D geometry). These works posed the following question: How does the system evolves froma metal (at high temperature T ) to an insulator (at low T ), assuming no coupling to an externalbath? The scaling of the ratio of a higher-order matrix element to the corresponding many-bodylevel spacing with the order of the perturbation theory was obtained by means of an approximatemapping on the Cayley-tree in Fock space in Ref. [10] and a self-consistent resummation of theperturbation series in Ref. [11]. Both these works concluded that there is a finite- T localization-delocalization transition manifested in the temperature dependence of the quasiparticle decayrate, and thus of the conductivity, which is zero for T < T c and finite for higher T . The criticaltemperature T c of the MBL transition was found to be T c ∼ δ ξ α ln(1 /α ) , (1)where δ ξ is the single-particle level spacing within the localization volume and α (cid:28) ff ect of spectral di ff usion driven by the diagonal interaction matrix elements(discarded in Refs. [10, 11]) was taken into account. The analysis of the corresponding higher-order matrix elements resulted in a di ff erent scaling of the transition temperature for α (cid:28) T c ∼ δ ξ α / ln µ (1 /α ) , (2)with 0 ≤ µ ≤ /
2. The essence of the e ff ect is that the interaction-induced resonant transitionsbetween quasiparticle states in a many-body system shift single-particle energy levels, which4nhances the occurrence of new resonances, thus favoring delocalization. As a result, it turns outto be su ffi cient to find a single resonance in a certain spatial region to trigger further transitions,instead of requiring each single-particle state to have a resonant three-particle partner. Conse-quently, for α (cid:28)
1, delocalization persists down to lower temperatures than that given by Eq.(1). We note, however, that the di ff erence between Eqs. (1) and (2) is not essential for su ffi cientlystrong interaction, α ∼
1, which is used in most of the numerical simulations. When translatedinto the language of the XXZ spin chain at infinite temperature—the paradigmatic model usedfor numerical studies of MBL [15], see Eq. (11) below—both the MBL thresholds (1) and (2)become the conditions on the disorder strength, W = O (1) for ∆ ∼ J ∼ A closely related direction of the analytical studies of MBL, inspired by the ideas of Ref. [32],is based on the connection of MBL to Anderson localization on tree-like structures such as ran-dom regular graphs (RRG) and their relatives. An RRG is a finite graph with randomly connectedvertices (hence “random”), but with the same branching number—the number of legs at each ver-tex (hence “regular”). Locally, an RRG has the structure of a Cayley tree, but, in contrast to it,all sites of the RRG have the same connectivity, i.e., it has no boundary. The Anderson model onsuch a graph can be viewed as a toy model of MBL. In this analogy, RRG vertices play the role ofbasis many-body states (i.e., eigenstates of the non-interacting Hamiltonian), while the hoppingbetween them the interaction matrix elements. This model thus conforms to the situation whenthe energies of the basis many-body states are uncorrelated. The true many-body problem canbe exactly represented by a model on a certain graph in the Fock space with tree-like propertiesbut with non-trivial correlations. The e ff ect of spectral di ff usion partly reshu ffl es energies ofmany-body states connected by interaction matrix elements, thus rendering the structure closerto (although still di ff erent from) that of the RRG model. Despite the di ff erence, the RRG modelcaptures many key properties of the genuine many-body problem. For a more detailed review ofthe RRG model and its relations to the MBL problem, we refer the reader to Ref. [36].An important advantage of the RRG model is that it can be studied analytically in a fully con-trollable way by means of a field-theoretical approach and the saddle-point analysis justified forlarge system size [37, 38]. The resulting analytical predictions, including, in particular, the er-godicity of the delocalized phase, the localized character of the critical point, and critical scalingat the transition, have been supported by numerical simulations [37, 38, 39, 40, 41, 42, 43, 44].The fact that the RRG model can be solved analytically is very useful since it can be used totest performance of exact diagonalization. The maximum Hilbert-space size of the RRG systemthat can be studied realistically via exact diagonalization is ∼ vertices, which corresponds toa spin chain of the length ≈
20. Comparing exact-diagonalization results with analytical predic-tions in the limit of infinite system size, one finds rather strong finite-size e ff ects for the criticaldisorder and the scaling at criticality. In particular, the apparent critical point drifts substan-tially towards stronger disorder with increasing system size. As we describe in this review, avery similar trend is observed in genuine interacting systems. The analysis of RRG saddle-pointequations in Ref. [38] by population dynamics has allowed one to reach e ff ectively the systemwith Hilbert-space size of nearly 10 , which corresponds to spin chains of length larger than L =
60. It was found that such system sizes are indeed needed to reach a good convergence ofthe numerically found critical behaviour to analytical predictions. In true many-body systems,finite-size e ff ects are expected to be still stronger than on RRG, due to rare-region e ff ects. Wethus come again to the conclusion that numerical studies of systems with ≈
100 sites are of greatneed for computational investigation of the MBL physics.5 .3. Rare regions and avalanches
The perturbative analysis of MBL focuses on typical disorder configurations, thus neglect-ing Gri ffi ths-type e ff ects associated with exponentially rare events. Rare regions with locallyanomalously strong or anomalously weak disorder may potentially a ff ect the localization andtransport properties in large systems, where the probability of finding at least one such rareregion becomes substantial. In particular, strong-disorder spots on the ergodic side of the MBLtransition were argued to slow down the dynamics in the system, leading to subdi ff usive transport[45, 46, 47, 48]. As another manifestation of a potential impact of rare regions, the MBL phasecan be destabilized through the growth of “ergodic spots,” which naturally arise in the system inregions with anomalously weak disorder. Such ergodic spots can grow over time, by absorbingthe neighboring localized regions—the process that has been termed an avalanche [21, 22, 23].The many-body delocalization point is then the point where this growth becomes unbounded, sothat the whole system eventually thermalizes. The avalanche description has specific implica-tions for the qualitative behaviour of MBL-type systems, and on the way this behaviour changesfor various geometries and types of disorder, as we are going to detail.The avalanche picture assumes the presence of a rare region with anomalously weak disorder,such that, in the absence of couplings to the rest of the system, this “seed” would be ergodic. Theexact many-body states in this ergodic (thermal) seed are assumed to be described by randommatrix theory, while the rest of the system, excluding the ergodic seed, is considered to be many-body localized. The Hamiltonian of the initially localized part of the system is then described interms of local integrals of motion (LIOM)—“spins,” characterized by the localization length ξ that depends on the stength of disorder. Switching on the coupling between the subsystems, theratio of the coupling matrix element for hybridizing a spin located at distance r from the spot tothe many-body level spacing inside the ergodic spot should satisfy e − r /ξ N − / s / N − s (cid:38) . (3)Here, N s is the dimension of the space of many-body states inside the spot and the factor N − / s in the matrix element accounts for the ergodic (random-matrix) scaling of the local operators. Ina 1D spin chain, N s = (cid:96) for an ergodic spot of size (cid:96) (in units of the lattice constant). It followsthen that, once ξ > / ln 2 , (4)an emergent avalanche in the chain cannot be stopped [21, 22]. This condition translates into thecondition on the disorder strength W , which does not depend on the strength of the interaction,in contrast to the conditions of the type (1) and (2).The probability of finding a rare thermal seed of the required size depends on the systemdimensions and dimensionality: the larger the system, the higher this probability, and hencedelocalization is more probable than in smaller systems. Thus, in numerical studies of MBL, theestimated position W c of the MBL transition is expected to drift towards stronger disorder withincreasing system size. At variance with 1D systems, this growth of W c with the system size aspredicted by the avalanche theory turns out to be unbounded in systems of higher dimensionality.This is because the size N s of the Hilbert space associated with the ergodic seed increases, then,with increasing its linear dimension faster than an exponential [21]. In particular, avalanches in2D systems of size L × L give rise to the following dependence of the critical disorder strength W c ( L , L ) on L [49]: ln W c ( L , L ) ∼ ln / L . (5)6urther, the avalanche-induced scaling of the critical disorder strength W c ( L , d ) in a quasi-one-dimensional strip of width d (cid:29) L ≥ d is found tobe [27] W c ( L , d ) ∼ exp (cid:104) c ln / ( Ld ) (cid:105) , d ≤ L < L ∗ ( d ) , d , L > L ∗ ( d ) . (6)with an estimate c ≈ .
57 for the numerical coe ffi cient c . Here the crossover length L ∗ ( d ) isgiven by L ∗ ( d ) ∼ d − exp (cid:32) d ln 2 c (cid:33) . (7)We see that in the 1D limit ( d = W c ( ∞ , ∼
1, as has been already stated above. In the quasi-1D regime, i.e., for increasing L (cid:29) d at fixed d and, we also have a finite value of W c ( ∞ , d ), which, however, increases expo-nentially with d . Finally, in the 2D regime, d ∼ L , Eq. (6) takes the form of Eq. (5), predictingan unbounded growth of W c with L . Thus, according to the avalanche theory, the MBL getsdestabilized in the 2D case by rare events in the thermodynamic limit if the disorder strength W is kept fixed. At the same time, for large but finite L , there is a well-defined MBL transition atan L -dependent critical disorder W c ( L ).It is worth emphasizing that the avalanche theory has a rather phenomenological character.Its conclusions are crucially dependent on the validity of the central assumption that the growingseed retains its “most ergodic” character, i.e., that it can be modeled by random matrix theory.While numerical support for this assumption has been obtained in simplified models, it is ofcrucial importance to compare the predictions of the avalanche theory to numerical simulationsof genuine MBL systems of su ffi ciently large size—which points to the importance of MPS-based computational investigations of the MBL transition beyond 1D. Below, we will overviewnumerical results obtained by the MPS approach that do provide support for the avalanche theory.Utilizing the ideas of the strong-randomness renormalization-group (RG) framework, severalphenomenological RG theories have been proposed to describe the MBL transition. Recentworks have expanded on the idea of the competition of localizing and delocalizing rare-evente ff ects, suggesting that the MBL transition is of the Berezinskii-Kosterlitz-Thouless (BKT) type[50, 51, 52, 53]. A key prediction from this class of theories is that the critical point is itselflocalized (similar to the critical point in RRG models) and, therefore, only weakly entangled.In this brief review, we discuss evidence for such a scenario from the viewpoint of the MPSapproach. The localized character of the critical point is highly beneficial for the investigationof the critical region and of a part of the ergodic phase (with W comparable to W c ), because of arelatively slow growth of the entanglement.
3. MBL problem and matrix product states
Here we briefly describe the MPS concept and several key MPS algorithms that have beenused to describe the MBL transition. The reader is referred to the extensive general reviews onthe density matrix renormalization group (DMRG) [17], 1D and 2D tensor networks [54], andthe computation of dynamics using MPS [55] for more in-depth details.An MPS describes the many-body wave function in the following way. For simplicity, con-sider a 1D system on a lattice of length L with site index i = , . . . , L , with a single, binary degree7f freedom σ i on each site. This could be a spin-1 / | Ψ (cid:105) = (cid:88) σ c σ ...σ L | σ , . . . , σ L (cid:105) . (8)With the binary degree of freedom per site, the dimension of the Hilbert space (and thus the num-ber of coe ffi cients c σ ...σ L ) is 2 L , so that the full diagonalization of the problem is exponentiallyexpensive in terms of computational e ff ort. Certain gains can be made by restricting the Hilbertspace to, say, a specific spin or particle number sector [56], but the exponential scaling is un-avoidable and eventually makes it prohibitively expensive to compute the eigenspectrum beyond L ≈
22 and dynamics beyond L ≈
28. MPS-based algorithms avoid this problem by restrictingthe number of the coe ffi cients, at the cost of approximating the exact many-body state, in thefollowing way (for open boundary conditions): | Ψ (cid:105) = (cid:88) { σ i } A σ A σ · · · A σ L | σ , . . . , σ L (cid:105) . (9)Here, each A σ i is a matrix (whose dimension can in principle depend on i ), whence the name ma-trix product state emerges. In the case of open boundary conditions, the left and right matricesin the product are row and column vectors respectively, so that a scalar coe ffi cient is obtained,while in the case of periodic boundary conditions one typically takes the trace of the product ofmatrices of fixed dimensions. The name matrix product states is a bit of a misnomer: we arein fact talking about a tensor as there are three indices per A . Mathematicians therefore preferthe name tensor train . In principle, the full many-body state can, without loss of generality,be decomposed in such an MPS. We seem to have gained nothing, but the main benefit of thisapproach lies in a key approximation that can be made with the help of singular value decompo-sitions. The result of this approximation is that the dimension of each matrix A σ i is restricted tobe (at most) χ × χ , where χ is the bond dimension. During the singular value decomposition, wecan discard the smallest singular values, retaining only the χ largest ones. As it turns out, thisapproximation corresponds physically to considering only a low-entanglement subsector of theHilbert space.What makes the MPS approach to be especially suitable for studying the physics around theMBL transition? As we have pointed out in Sec. 2, it is expected on the basis of analyticalarguments that the critical point is essentially localized and therefore exhibiting low entangle-ment. By continuity, this is expected to be applicable also to moderately large systems on theergodic side close to the MBL transition. Such low-entanglement states are well-captured bytensor networks like MPS. The MPS Ansatz (9) describes the wave function, but it does not prescribe how the coe ffi -cients ought to be obtained. For this purpose there are two classes of algorithms: those targetingstatics (the eigenstates of the system and their properties) and those targeting dynamics startingfrom some arbitrary state.Lim and Sheng [57] considered the MBL problem using DMRG-X, an algorithm based onWhite’s original DMRG algorithm [58] which targets ground states. DMRG-X instead targetshighly excited states. Normally, DMRG would be unable to distinguish such states since thelevel spacing becomes too small. DMRG-X exploits the fact that, on the localized side of the8ransition, di ff erent eigenstates di ff er in both energy and spatial extent, boosting the resolutionof the algorithm. In this way, the authors of Ref. [57] were able to capture the MBL transitionnumerically. An alternative quantum-circuit approach that can be applied to both 1D and 2Dsystems to target highly excited eigenstates was pioneered by Wahl et al. [59, 60], building onan earlier work by Bauer and Nayak [61].In this review, we mainly focus on MPS investigation of quantum dynamics. Early studiesof MBL using MPS [15, 18] employed time-dependent DMRG (tDMRG) or a closely relatedmethod of time-evolving block decimation (TEBD) [62, 63, 64]. The limiting factor in terms ofthe applicability of the method is the accumulation of errors at each time step. As one crossesthe MBL transition from the MBL side to the ergodic side, the entanglement grows with timefaster. Therefore, for a given bond dimension χ and desired error (cid:15) , the maximum time that canbe reached decreases with decreasing disorder strength W . For TEBD and t-DMRG, at eachtime step there is a truncation error that results from repeated truncations of the entanglementspectrum.Recently, a novel MPS-based algorithm was proposed, which does not su ff er from thesetruncation errors: the time-dependent variational principle (TDVP) [65]. Instead of a truncation,there is a projection onto the variational subspace described by the MPS: ∂∂ t | ψ (cid:105) = − i P MPS H| ψ (cid:105) , (10)where H is the Hamiltonian of the system and P MPS is the projection operator. Of course, it isnot a priori given that the projection errors induced by the time evolution according to Eq. (10)are smaller than those induced in other MPS methods. In fact, this depends on the problemat hand [55]. For the MBL problem, the TDVP approach turns out to be advantageous as willbe discussed in more detail below. This is related to the fact that the projection conserves theglobally conserved quantities, in particular, the energy. The MPS-TDVP approach permits tocontrollably explore the time evolution up to su ffi ciently long times, t ∼
100 – 300 in units of thelattice hopping times for disorder W around W c . This is the same time range as the one accessedin cold-atom experiments.Tensor networks permitting to tackle problems beyond 1D geometry have also been devel-oped. In particular, an approach based on projected entangled pair-states (PEPS) has been appliedto the 2D MBL problem [66, 67]. It is also possible to generalize the MPS ansatz to di ff erentlattice geometries by introducing longer-range terms in the Hamiltonian, by using matrix prod-uct operators (MPOs) [68, 69]. It is not a priori clear which approach is the most suitable forthe MBL problem beyond 1D geometry. As we discuss below, however, the MPS-TDVP-basedapproach provides a particularly e ffi cient method of simulation also in this situation.A distinct MPS-based approach for simulating dynamics was recently developed by ˇZnidariˇc et al. [70]. Their approach uses the Lindblad master equation to simulate the current driventhrough the system. In a certain sense, this approach is the “opposite” of the usual MPS-basedapproaches as it works most e ffi ciently in the ergodic limit of low disorder. The authors ofRef. [70] have successfully used this approach to capture the crossover from di ff usive behaviourat very weak disorder to subdi ff usive behaviour at moderate disorder. However, as the criticalpoint has essentially the localized character, this method loses its e ffi ciency when one movestowards W c from the ergodic side. Specifically, as one approaches the MBL transition, the timescales required for the current to saturate diverge.9 . Model and observables We will consider the following models: (i) a 1D chain with random disorder, (ii) a 1Dquasiperiodic chain, and (iii) a 2D square lattice with random disorder.
We first discuss the most widely studied model: the XXZ Heisenberg chain of length L withon-site random disorder: H = (cid:88) i (cid:20) J (cid:16) S + i S − i + + S − i S + i + (cid:17) + ∆ S zi S zi + + h i S zi (cid:21) , (11)where S i are spin- operators, ∆ / J is the anisotropy, and the on-site field h i is taken from auniform distribution h i ∈ [ − W , W ], with W denoting the strength of disorder. By virtue of Jordan-Wigner transformation, this Hamiltonian is equivalent to that of interacting fermions (or hard-core bosons): H = (cid:88) i (cid:20) J b † i b i + + H . c . ) + ∆ ˆ n i ˆ n i + + h i ˆ n i (cid:21) . (12)Here b i , b † i are the annihilation and creation operators for a spinless fermion or a hard-core boson,ˆ n i = b † i b i − /
2, and J and ∆ are the hopping amplitude and the nearest-neighbor interaction,respectively. Typically, numerical studies choose J = ∆ =
1, although sometimes values of ∆ / J di ff erent from unity are chosen in order to avoid possible peculiarities of the isotropic ∆ / J = ffi ciently strong disorder, those peculiarities should not beof importance. We will focus on the J = ∆ = An alternative, which has been investigated in experiments [13], is to take a quasi-periodic potential, which is not truly random. Specifically, one typically considers the many-body gen-eralization of the Aubry-Andr´e-Harper problem [71, 72], wherein the potential experienced byparticles is given by h i = W π Φ i + φ ) , φ ∈ [0 , π ) . (13)Here W again represents the strength of the potential, while Φ is the period of the potential. Ifone chooses Φ irrational, then the potential periodicity is incommensurate with that of the lattice.Further, φ is a random phase; one averages over a uniform distribution of φ over [0 , π ).It is well known that the non-interacting ( ∆ =
0) Aubry-Andr´e model undergoes a local-ization transition at W = W > ff erences between the random and quasiperiodic problems. In particular, while arandom potential has extended rare regions where the potential is anomalously weak or anoma-lously strong, a quasiperiodic potential cannot have such regions. Therefore, one should expectqualitative di ff erences in the behaviour of those observables that are essentially related to rareevents. On the experimental side, quasiperiodic systems are easier to study, at least in cold-atomexperiments, which is why the earliest MBL experiments [13] considered this type of systems.10 .1.3. Two dimensions, random disorder The Hamiltonian, in the hard-core boson language of section 4.1.1, can be straightforwardlygeneralized to a 2D (square) lattice: H = (cid:88) (cid:104) i j ; i (cid:48) j (cid:48) (cid:105) (cid:20) − J b † i j b i (cid:48) j (cid:48) + H . c . ) + U ˆ n i j ˆ n i (cid:48) j (cid:48) (cid:21) + (cid:88) i j h i j ˆ n i j . (14)Here b † i j is again the creation operator for a hard-core boson at the site given by indices i , j , U isthe nearest-neighbor interaction, J is the energy associated with particle hopping to neighboringsites, and h i j is the magnitude of the on-site potential, with ˆ n i j ≡ b † i j b i j . We define this model ona square lattice of size L × d , with i ∈ [1 , L ] and j ∈ [1 , d ]. Numerical studies—as well as experiments—often consider the density imbalance . It is usedas the key observable characterizing the dynamics in the MPS-TDVP data presented below.For a 1D geometry, the idea is that one initially prepares the system in a charge-density-wave(N´eel) state: ψ ( t = = { , , , , . . . , , } , (15)where 0 or 1 indicates the local occupation and corresponds to, respectively, − / / S z -basis in the spin language. The imbalance is then defined as the memory of this state at latertimes: I ( t ) = n odd − n even n odd + n even , (16)where n odd and n even are the densities at odd and even sites, respectively. By definition, I (0) = columns (fixed i ) will be takento be occupied for odd i and unoccupied for even i .The appeal of the imbalance lies in the fact that, first, it is relatively easily accessed in exper-iments and, second, it relaxes to zero rapidly in the absence of disorder. Further, the density canbe easily computed using MPS approaches [17], as opposed to the level statistics of eigenstates,which requires exact diagonalization [16].Inspection of the decay of the imbalance provides a way for determining the transition fromthe ergodic to the localized phase. In the ergodic phase, W < W c , one expects a power-law decay,[20]: I ( t ) ∝ t − β . (17)On the other hand, in the MBL phase, W > W c (and at the critical disorder W = W c ), oneexpects β = I ( t ), as obtained in finite-time simulations, does not exactly follow thepower-law decay (17). One can generalize Eq. (17) by defining a running power-law exponent β ( t ) = − ∂ ln I ( t ) /∂ ln t . One then expects lim t →∞ β ( t ) = W > W c and lim t →∞ β ( t ) > W < W c . (Of course, the t → ∞ limit should be taken after the limit of an infinite system size.)Clearly, when the position of W c is extracted from finite-time simulations, one has to rely on anextrapolation to long times. Let us note that not only the limiting value of β at t → ∞ but alsothe whole time dependence β ( t ) is of interest, since it contains important information about thedynamics of many-body localization and delocalization. It is also worth emphasizing that, sincethe characteristic times in MPS-TDVP numerical simulations presented below are of the sameorder as in experiments, one can directly compare β ( t ) to experimental measurements.11 .3. Entanglement entropy Another popular measure for the analysis of MBL-type systems is the entropy S , by whichone usually means the bipartite von Neumann entropy of entanglement [73]. The entropy corre-sponding to a subsystem A of the system can be obtained by tracing out the degrees of freedomof the exterior B : S A = − Tr( ρ A ln ρ A ) , ρ A ≡ Tr B | Ψ (cid:105)(cid:104) Ψ | . (18)Most commonly, the bipartite entropy for a division of the system in two halves is used. Thisquantity, too, can be obtained straightforwardly by MPS methods—in fact, it is obtained “forfree” during the course of MPS algorithms [17]. The entropy grows logarithmically in the lo-calized phase [18], and faster than logarithmically on the ergodic side. Because a logarithmicgrowth is di ffi cult to distinguish from a weak power-law growth, this makes the transition moredi ffi cult to pinpoint using entropy-related measures than using measures derived from particletransport—which freezes on the MBL side of the transition. For this reason, we will focus onparticle transport—quantified by the imbalance—in the remainder of this review.
5. Numerical results from matrix product states
We will now present the results for the MBL problem as obtained from MPS simulations.
We begin with the case of a random 1D system. The model defined on lattice of length L (with sites labeled by i = , . . . , L ) was defined in Sec. 4.1.1. This model (or equivalent models)was recently considered using the TDVP [25, 74, 75]. In Ref. [74], Kloss et al. argue that theTDVP can be used to target times up to t ∼
100 even for moderate disorder W = .
5. This valueis significantly below the estimate for the critical disorder W c ≈ . ffi cient also rather deep into the ergodic regime.In Ref. [25], by the present authors together with Schindler, Tikhonov, and Neupert, the slowtransport around the MBL transition was explored and the position of the transition was estimatedby the MPS-TDVP method, in systems up to L =
100 and at times up to t = β ( W ) is a monotonically de-creasing function of W , as expected. In the case that the system is localized, β ( W > W c ) → t → ∞ . As was already mentioned above, since only finite times are available, an extrapolationis needed, and the error made during the course of this extrapolation has to be estimated. InRef. [25], this error is taken as bounded from below by the residual decay of the non-interactingsystem (triangle symbols in Fig. 1). The critical disorder W c is then estimated by adding statis-tical errors resulting from the finite number of di ff erent disorder realizations. This leads to theestimate W c ≈ . L =
16 yield, within the same procedure, asubstantially smaller estimate, W c ≈
4, which is in good agreement with the results of exact di-agonalization for systems of comparable size ( W c ≈ . W c ≈ . W c on the basis of imbalance dynamics. As the system size increasesfrom L =
16 to L =
50, a substantial drift of β is observed, leading to a substantially higher valueof W c ≈ .
5. This drift towards delocalization with increasing system size is similar to that foundin the RRG model but is still more pronounced. Indeed, in the present case of a spin chain, oneexpects an additional contribution to the drift that originates from rare ergodic spots discussed in12 W − − β L = 16 L = 50 L = 100 L = 100 (TEBD)non-interacting Figure 1: Decay of the imbalance exponent β as a function of the disorder strength W for system sizes L =
16, 50, and100, as computed from the time window t ∈ [50 , © Sec. 2. At the same time, the values for L =
50 and L =
100 agree within error bars, indicatingthat system sizes of L ≈
50 are su ffi cient to essentially reach the L → ∞ saturation. This suggeststhat the estimate W c ≈ . L → ∞ . This estimate was further supported in Ref. [25] by a machinelearning analysis [77] of the data obtained from the TDVP, which yielded essentially the samevalue of W c .Chanda et al. studied the same problem using both TDVP and time-dependent DMRG [75].They considered longer times (up to t =
500 for L =
50) and characterized the decay of theimbalance by β ( t ). The data of Ref. [75] support the conclusion of Ref. [25] about a substantialdrift of the critical point towards stronger disorder with increasing L . The authors of Ref. [75]used, however, a significantly higher value for the error, β error = .
02, than the error estimateof Ref. [25], thus attributing all values β < β error to the localized phase. Clearly, increasing thecuto ff β error reduces the estimate for W c . As a result, Ref. [75] came to an estimate W c ≈ .
2. It isworth pointing out that a very recent work of the same group, using an advanced version of theexact-diagonalization approach for up to L =
24 in combination with an extrapolation to L → ∞ ,obtained the estimate W c ≈ . et al. [79] considered the same model by tDMRG.Inspired by the experiment [24], they used a di ff erent initial state characterized by a domain wallbetween two oppositely polarized halves of the system (empty and filled, in the fermionic lan-guage), and studied the dynamics of the domain wall melting. Qualitatively, the results are inagreement with those found for the N´eel state. At the same time, the authors of Ref. [79] did notattempt an accurate analysis of transport close to the transition point, of the value of W c , and ofits drift with the system size.Recent works [80, 81] have emphasized a sizeable drift of the critical point W c ( L ) as foundin exact diagonalization (i.e. for L (cid:46)
20) of the 1D model (11). Specifically, their results13 W − . . . . . . β L = 50 , Φ = ( √ − / L = 16 , Φ = ( √ − / L = 50 , Φ = √ / L = 16 , Φ = √ / Figure 2: Imbalance exponent β in the case of the quasiperiodic potential (13) for various system sizes L , field strengths W and periodicity Φ , over the time window t ∈ [50 , ≈ φ . Figure adapted from Ref. [26], © for the drift can be cast in the form ∂ W c /∂ L ≈ . W c stays finite in the L → ∞ limit in thismodel. It is worth mentioning that a hypothesis of W c growing linearly with L is in dramaticdisagreement with the existing analytical results on the MBL transition. Indeed, such a fastgrowth cannot be related to any kind of rare events; if for real, it should be seen on the level ofthe perturbative treatment. However, no such analytical evidence exists, see Sec. 2.1. We believethat the numerical results of Refs. [80, 81] actually do not provide a ground for the conclusionon the asymptotic growth of W c ; they rather demonstrate strong finite-size corrections. Indeed,the magnitude of the drift is in very good agreement with our result that W c reaches the valueof W c ≈ . L =
50. Of course, using only the exact-diagonalization numerical data (whichcorrespond necessarily to a relatively narrow range of quite small L ), it is extremely di ffi cult tojudge on a saturation of this drift at large L . It was emphasized in Refs. [82, 83] that the results ofˇSuntajs et al. are due to finite-size e ff ects and that a similar, approximately linear, drift of W c isobserved at small L also in models for which it is known analytically that W c is finite in the imit L → ∞ . An important example is the RRG model discussed above. It was shown in Ref. [78]that an assumption of 1 / L finite-size e ff ects yields a good fit to the exact-diagonalization data,resulting in the estimate W c ≈ .
4. Moreover, a recent work by Panda et al. [84] argues thatsystem sizes of at least around L ≈
50 are needed to assess the existence of MBL. This perfectlyagrees with our finding that a substantial drift of the apparent W c is observed between L ≈ L =
50, while the system sizes L =
50 and L =
100 yield nearly identical results.
We now consider the case of a quasiperiodic potential (13). This problem was studied, usingthe TDVP-MPS approach, in Ref. [26], for system sizes up to L =
50 and times up to t = t − . . . . . . . . fi tt e dp o w e r l a w β ( t ) TEBD χ = 128TEBD χ = 256TDVP χ = 64TDVP χ = 96TDVP χ = 128 Figure 3: Comparison of the imbalance decay exponent β ( t ) using two di ff erent MPS methods, TDVP and TEBD, usingvarious choices of the convergence parameter χ . Parameters are: the strength of the quasiperiodic field W =
4, systemsize L =
50, and periodicity
Φ = ( √ − /
2. Di ff erent disorder realizations were used for each choice. From Ref. [26], © imbalance dynamics provide a clear evidence of the MBL transition and allow one to estimate W c in a large system.One important result of Ref. [26] is a strong dependence of the critical disorder W c on theperiodicity Φ . In particular, for two values of Φ , data for which are shown in Fig. 2, the resultsare W c ≈ . Φ = ( √ − / W c ≈ . Φ = √ /
2. This strong dependencecan be qualitatively explained as resulting from a di ff erence in the statistics of the potential atneighbouring sites [26, 85]. This a ff ects properties of localized single-particle states, which inturn influence susceptibility of the system to interaction-induced many-body delocalization.While there are similarities in the qualitative behaviour of β ( W ) in the random and quasiperi-odic cases, there are also significant di ff erences. First, the finite-size e ff ects are substantiallysmaller in the quasiperiodic case, as compared to the purely random case: the estimate of W c de-pends only weakly on the system size. This is in agreement with the theoretical predictions [86],where it is argued that the disordered and quasiperiodic cases are essentially di ff erent. Second,the data on the ergodic side of the MBL transition in a quasiperiodic system indicate an increaseof β ( t ) with time (see Fig. 3). On the time scale probed by MPS-TDVP this increase is rathermodest (see Sec. 5.2.1), in agreement with experiments that reported a slow decay that can bewell fitted by a power law. At the same time, the increase of β ( t ) is consistent with the absenceof the e ff ects of rare regions for the quasiperiodic potential, in contrast to the case of a randompotential, where these e ff ects are expected to lead to subdi ff usive transport in the long-time limit,as was discussed above. Recent studies by time-dependent Hartree-Fock approximation [87](that allows one to reach much longer times at the expense of losing the full control of accuracyprovided by the MPS-TDVP method) confirm the qualitative di ff erence in the behaviour of β ( t )for random and quasi-periodic systems. 15 .2.1. Comparison of MPS methods We have also compared [26], in the quasiperiodic case, the time-dependent behaviour of theimbalance decay, quantified using β ( t ), for two di ff erent MPS methods: the TDVP and TEBD.The results are shown in Fig. 3. The following qualitative features are observed. First of all,and most reassuringly, we observe a clear convergence of both methods to the same curve withincreasing bond dimension χ . Second, the convergence with χ is superior in the case of theTDVP (at least from the perspective of the imbalance dynamics): at fixed time t ≈ ff erence between χ =
96 and χ =
128 in TDVP is smaller than the deviation of the TEBD datacomputed with even higher bond dimension χ = ff erencebetween the two methods: in the case of TDVP, when convergence is lost, the algorithm appearsto overestimate the decay, whereas in the case of TEBD the opposite is true. Specifically, theTEBD results show a downturn in β ( t ) towards zero reflecting the spurious saturation of theimbalance. With increasing χ , the time at which this downturn starts becomes longer. Thesuperior performance of TDVP with respect to TEBD on the ergodic side of the MBL transitionwas attributed [26] to di ff erent ways of truncation and, in particular, to non-conservation ofenergy within TEBD. A related analysis was performed in Ref. [75] for the truly random case,with similar findings.In the case of TDVP, convergence is reached up to t ≈
180 using χ = β of roughly 20% is observed, whereas in the case of randomdisorder, the power law appears to be robust over such time scales, at least on the ergodic side ofthe transition. This di ff erence between the quasiperiodic and random cases was already pointedout in Sec. 5.2. In Ref. [27], the MPS-TDVP approach was extended to study the model on a 2D squarelattice, see Sec. 4.1.3. More specifically, two types of geometry were investigated: quasi-1Dladders of L × d sites, with d = , , and 4, and L (cid:29) d , and 2D samples with d = L . The modelis determined by the Hamiltonian (14), with i ∈ [1 , L ] and j ∈ [1 , d ]. In the initial state, columns with odd index i are occupied, similar to the experiment [88]. The results for the exponent β ofthe columnar imbalance decay are shown in Fig. 4. In analogy with the 1D case, they were usedto determine the critical disorder W c ( L , d ). The resulting estimates are shown in Fig. 5 for thequasi-1D (left panel) and 2D (right panel) cases, in comparison with analytical predictions basedon the avalanche theory, Eq. (6).In the quasi-1D case, the second line of Eq. (6) predicts an exponential increase of W c ( L , d )with increasing number of legs d in the ladder, W c ( L , d ) ∼ d , under the condition L > L ∗ ( d ),where L ∗ ( d ) is given by Eq. (7). Importantly, L ∗ ( d ) increases very fast with d according to thisformula: it is rather small for d = , , and 3, is moderately large for d = L ∗ (4) ≈ d ≥
5. The numerical results in the left panel of Fig. 5 are in very goodagreement with these predictions: at d =
4, the increase of W c ( L , d ) deviates downwards fromthe exponential curve [corresponding to the large- L limit of Eq. (6)].In the 2D case, the analytically expected increase of W c ( L , L ) is given by the first line ofEq. (6). While it is asymptotically slower than a power law, it yields a rather big increase overthe available interval of system sizes from L = L =
8. As is seen in the right panel of Fig. 5the values of W c ( L , L ) provided by the TDVP analysis do show a strong increase, which is in facteven somewhat faster. The agreement is rather good, taking into account that the system sizes L are moderately large, while the formula has been derived for the asymptotic limit of large L .16
10 15disorder W d ec a y β ( a ) ( b )( c ) ( d ) L = 8 ,d = 2 L = 20 ,d = 2 L = 40 ,d = 2 L = 20 ,d = 2 (H)
10 15 20 25 30 35disorder W L = 8 ,d = 3 L = 20 ,d = 3 L = 8 ,d = 3 (H) L = 20 ,d = 3 (H)
15 20 25 30 35disorder W d ec a y β L = 8 ,d = 4 L = 8 ,d = 4 (H) L = 20 ,d = 4 (H)
10 15 20 25 30 35 40 45 50disorder W L = 4 ,d = 4 (H) L = 6 ,d = 6 (H) L = 8 ,d = 8 (H) L = 10 ,d = 10 (H) Figure 4: Imbalance decay exponent β for quasi-1D and 2D systems with random disorder, with length L and width d ,and various disorder strengths W , as computed for the time window t ∈ [50 , d =
2, b) three-legladder, d =
3, c) four-leg ladder, d =
4, d) 2D case, d = L . Di ff erent colours in the same panel indicate di ff erentimplementations of the TDVP. From Ref. [27], © d c r i t i c a l d i s o r d e r W c ( L , d ) ( a ) ( b ) quasi-1D ( d < L ) L = 8 L = 20 L = 50analytical L c r i t i c a l d i s o r d e r W c ( L , L )
2D ( L = d ) analyticalL = 4L = 6L = 8L = 10 Figure 5: Critical disorder W c ( L , d ) for (a) quasi-1D and (b) 2D cases, compared to analytical predictions for (a) W c ( ∞ , d )with d (cid:29) W c ( L , L ) with L (cid:29) © et al. [79] have earlier considered this model in the case of a two-leg ladder ( d = d = L as 8 (cid:46) W c (cid:46)
10, their Fig. 4 shows that in fact the dynamics of melting has notsaturated, either for W = W =
10. Thus, the data of Ref. [79] indicate a critical disorder W ( ∞ ,
2) higher than 10, in consistency with the result W c ( ∞ , ≈
13 of Ref. [27].
6. Summary
In this article, we have reviewed recent results on application of the MPS-based approachto the investigation of quantum dynamics in interacting disordered systems around the MBLtransition. The focus was put on the advances achieved in the framework of the MPS-TDVPmethod [25, 26, 27] and a comparison to the results obtained by other methods. The most salientconclusions of these studies are as follows:1. The MPS-based framework, and in particular the MPS-TDVP approach, is a powerful toolfor the investigation of MBL, complementary to exact diagonalization. It allows one tostudy controllably large systems, with ∼
50 – 100 “qubits,” up to times t ∼
300 close to thecritical region. These system sizes and time scales are essentially the same as probed inthe state-of-the-art experiments. It is demonstrated that the method works successfully for1D, quasi-1D, and 2D random systems as well as 1D quasi-periodic systems. It is shownthat the imbalance decay, which is also studied experimentally and can be characterizedby a power-law exponent β , is a convenient observable for monitoring the dynamics andlocating the MBL transition within the MPS approach.2. On the explored time scales, the systems show a slow, subdi ff usive transport in a ratherbroad range of the disorder strength W on the ergodic side of the MBL transition. Thisagrees with analytical expectations with respect to the localized character of the criticalpoint of the MBL transition and of the e ff ect of rare regions. In connection with this,one finds also a relatively slow growth of the entanglement in a su ffi ciently broad windowaround the transition point. This is highly favorable for the MPS-based approaches: oneobtains convergence with modest values of the bond dimension χ up to rather long times.3. For 1D random spin chains, which serve as a “standard model” of the MBL transition, theMPS study demonstrates a substantial drift of the critical point W c ( L ) with the system size L . This drift is similar to that found for the RRG model but is even more pronounced,which can be attributed to the e ff ect of rare regions. Specifically, while for L ≈
20 we find W c ≈
4, as also given by exact diagonalization, the MPS-TDVP results for L =
50 and L =
100 yield W c ≈ .
5. The agreement with exact diagonalization for small system sizecorroborates the validity of the determination of the critical point on the basis of quantumdynamics at intermediate time scales. The fact that the values of W c obtained for L = L =
100 systems are nearly equal suggests that W c ≈ . L → ∞ .4. For quasi-periodic (Aubry-Andr´e) systems, the finite-size e ff ects are much weaker, whichis consistent with the absence of rare regions with anomalously large or weak disorder inthe quasi-periodic setting. Further, a growth of β ( t ) with t is found in the quasi-periodiccase, again consistent with the di ff erence in comparison with random system in what con-cerns the rare regions. The MBL transition point W c is found to depend strongly on the18eriod Φ of the quasi-periodic potential, which is attributed to properties of correspondingsingle-particle localized states in the absence of interaction.5. For quasi-1D ( d × L , with d (cid:28) L ) and 2D ( L × L ) random systems, the MPS-TDVPdata indicate an unbounded growth of W c in the limit of large d and L . The results arein good agreement with analytical predictions (6) for the critical disorder W c ( L , d ) basedon the rare-event avalanche theory. Thus, the MPS results support the prediction of theavalanche theory that the MBL phase is destabilized in 2D in the thermodynamic limit ifconsidered at fixed disorder W . At the same time, there is a well-defined MBL transitionat a size-dependent disorder W c ( L , d ).The MPS approach is thus a very fruitful avenue for analyzing MBL-type systems. Its keyadvantage over exact diagonalization is a possibility to study the quantum dynamics in large sys-tems, with ≈
100 “qubits,” as in current (and presumably near-future) experiments with quantumdevices. The step from ≈
20 (as accessible to exact diagonalization) to 100 “qubits” is of greatimportance. However, not unexpectedly, the MPS approach also has its limitations. The maindrawback of exact diagonalization—the limitation to relatively small sizes —is replaced by a cut-o ff in time. Longer simulation times can ameliorate this issue somewhat, especially at strongdisorder, but eventually the growth of entanglement will become an insurmountable obstacle.This means that conclusions about the critical disorder are drawn based on an assumption thatthe time scale accessed by the method is su ffi cient to judge about the long-time limit. As wasemphasized, the agreement between W c obtained from the MPS approach and from the exactdiagonalization for relatively small systems clearly supports this assumption. Nevertheless, it isworth emphasizing the following “disclaimer” that applies quite generically to investigations ofphase transitions: numerical results concerning the phase transition are most useful when theysupplement analytical (and, if available, experimental) results; by themselves, they provide norigorous proof concerning the ultimate phase diagram, position of the critical point, and criticalbehaviour.Before closing the paper, we list a few perspective directions for future research.1. One important direction concerns a possible optimization of the MPS-type approachesbeyond 1D geometry. In Ref. [27], a mapping of the 1D structure of an MPS to a “snake”-like structure was used. It is not obvious a priori that this is the most e ffi cient way, since itdoes not fully take advantage of the local structure of the lattice in 2D geometry. While itappears that the MPS mapping used in Ref. [27] is more numerically e ffi cient than a PEPSimplementation [67] that employs a 2D structure into the tensor network, it is conceivablethat a still more e ffi cient approach might be devised, perhaps on the basis of tree tensornetworks (TTN) [89] or the multiscale entanglement renormalization ansatz (MERA) [90].2. Most of the research of the quantum dynamics within the MPS approach up to now hasaddressed the imbalance dynamics. Other initial states and other observables—especiallythose that can be studied in experiment—are also of interest. In particular, it would beinteresting to develop further the investigations of the domain-wall melting started inRef. [79].3. Another very intriguing direction is a study of the dynamics of ergodicity avalanches inspecially devised settings, see, e.g., a recent experimental work [91].4. A very interesting—and also very challenging—prospect is the investigation of the criticalbehaviour at the MBL transition (in particular, verification of RG theories predicting theBKT-like scaling near the transition). 19 . Acknowledgments We acknowledge collaboration with T. Neupert, F. Schindler, and K. Tikhonov on Ref. [25],which was the paper that started our activity on the MPS-based studies of the MBL problem.In course of these studies, we enjoyed useful discussions with many colleagues, including F.Alet, Y. Bar Lev, I. Bloch, F. Evers, M. H. Fischer, S. Gopalakrishnan, S. Goto, M. Heyl, C.Karrasch, M. Knap, N. Laflorencie, D. Luitz, S. R. Manmana, M. M¨uller, R. M. Nandkishore,A. Polkovnikov, S. Rex, A. Scardicchio, B. I. Shklovskii, K. Tikhonov, T. Wahl, J. Zakrzewski,and M. ˇZnidariˇc. The authors acknowledge support by the state of Baden-W¨urttemberg throughbwHPC.
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