Many-body localization in large systems: Matrix-product-state approach

Abstract

Recent developments in matrix-product-state (MPS) investigations of many-body localization (MBL) are reviewed, with a discussion of benefits and limitations of the method. This approach allows one to explore the physics around the MBL transition in systems much larger than those accessible to exact diagonalization. System sizes and length scales that can be controllably accessed 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 are presented. On time scales explored (up to $t\approx 300$ in units set by the hopping amplitude), a slow, subdiffusive transport in a rather broad disorder range on the ergodic side of the MBL transition is 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 $W_c\left(L\right)$ with the system size $L$: while for $L\approx 20$ we find $W_c\approx 4$, as also given by exact diagonalization, the MPS results for $L=50$–100 provide evidence that the critical disorder saturates, in the large-$L$ limit, at $W_c\approx 5.5$. For quasi-periodic systems, these finite-size effects are much weaker, which suggests that they can be largely attributed to rare events. For quasi-1D ($d\times L$, with $d\ll L$) and 2D ($L\times L$) random systems, the MPS data demonstrate an unbounded growth of $W_c$ in the limit of large $d$ and $L$, in agreement with analytical predictions based on the rare-event avalanche theory.

Introduction

Philip W. Anderson considered what happens to quantum particles when they move through a disordered medium [1]. He showed that disorder can fully halt transport. According to the celebrated scaling theory of localization [2], even weak disorder is sufficient to prevent transport 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 phase of localized to the phase of delocalized states occurs, which can be driven by the strength of disorder or another parameter of the system [3], [4]. Such localization has been observed in various disordered media, ranging from photonic lattices [5] and optical fibres [6] to dilute Bose–Einstein condensates [7], [8].

While the Anderson-localization problem is highly non-trivial and very rich by itself, the complexity of the problem is further strongly increased when the interaction between particles is included. The added complexity has several interrelated facets. One of them concerns the zero-temperature quantum phase transitions (and corresponding phase diagrams) in an interacting disordered system. We will not touch this (also very rich) field in the present brief review which focuses on the physics of highly excited states (i.e., those at non-zero energy density). The natural expectation in such a situation is that interactions between particles disturb interference effects 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) at any non-zero temperature. However, with sufficiently strong disorder, the application of Fermi’s golden rule (that assumes a continuous spectrum of final states) may be invalidated, leading to a 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 Anderson himself, 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 this scenario in detail [10], [11]. Experiments on MBL-type systems started in the 2010s, especially using ultracold atoms [12], [13], where disorder is implemented using laser fields.

With the rapid development of numerical algorithms and increasing numerical resources, the problem of MBL has become within the reach of computational approaches, initially in one dimension. Oganesyan and Huse [14] and Žnidarič et al. [15] pioneered the numerical study of the MBL problem using two complementary approaches. One of them [14], exact diagonalization, fully solves the many-body problem through computational brute force. This approach has no approximations, but since the computational complexity of a many-body system increases exponentially with the system (e.g., $\sim 2^L$ for a spin-$\frac{1}{2}$ chain of length $L$), the method is limited to small systems (up to $L\approx 20$). Luitz et al. [16] used a particularly numerically efficient implementation of exact diagonalization and found evidence for an MBL transition—i.e., a transition from the thermalizing (ergodic) to the localized phase. At the same time, the apparent critical behaviour provided by the corresponding scaling analysis turned out to be inconsistent with the Harris criterion, which is an indication of the fact that the system sizes reached by exact diagonalization ($L\approx 20$) are too small for investigation of the critical properties.

The 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 a type 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 through the bond dimension $\chi$, which can be tuned to interpolate between a product state, where there is no entanglement, and a maximally entangled state (i.e., it allows for a description of the wave function without approximations). Not surprisingly, the computational efficiency of describing maximally entangled states using MPS is not better than that of exact diagonalization, so that the MPS approach is only useful for systems that are relatively weakly entangled. In practice, one often starts in an (unentangled) product state, and computes the dynamics until the entanglement grows out of control. Thus, the approach is usually limited to finite times because of the growth of 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 on dynamics.

The rate of the entanglement growth is therefore of significant importance for the applicability of MPS approach. In the MBL phase, for strong disorder, analytical and numerical studies find a slow, logarithmic growth of entanglement [18]. This is very beneficial for the investigation of the MBL phase by the MPS method. On the other hand, in the opposite limit of weak disorder, the system quickly thermalizes and shows ergodic behaviour [16], in agreement with the eigenstate thermalization hypothesis (ETH) [19]. Of much physical interest is the behaviour of the system in a numerically broad range of intermediate disorder strengths, i.e, around the MBL transition, with the ergodic side of the MBL transition turning out to be particularly intriguing. What is the behaviour of the entanglement in this regime? The answer to this question is of paramount importance for the applicability of the MPS framework to the study of the MBL problem around the transition point.

Another key question concerns the role of the type of disorder in the system. Numerous experiments deal with interacting quasiperiodic (rather than truly random) systems. While such systems have no innate randomness, experimentally they also show an MBL transition [13]. The distinction between the two is crucial from a theoretical perspective because the physics of random systems is expected to be strongly affected by rare regions, whereas such regions do not occur in quasiperiodic systems. In particular, the type of disorder is expected to qualitatively influence the type of dynamics, in the sense that a subdiffusive dynamics is expected [20] on the basis of Griffiths effects. Yet, the experiment indicates subdiffusive behaviour also for quasiperiodic systems.

One more question concerns the case of higher-dimensional systems (i.e., beyond 1D), where the effects 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 finite disorder [24]. Importantly, however, the experiments explore finite systems, with the number of atoms being of the order of $100$, while the thermodynamic limit in the rare-region analysis is reached very slowly. In order to verify the predictions of the rare-region theory and to compare to the experiment, it is thus of crucial importance to perform computations in a broad range of system sizes, at least up to those studied experimentally. This goal is by far out of reach of exact diagonalization, 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 the vicinity of the MBL transition, including its ergodic side, in large systems and in a variety of settings. The key results, along with related advances by other researchers, are reviewed in this article. For more general reviews of MBL, the reader is referred to Refs. [28], [29], [30], [31].