Topology and many-body localization

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Abstract

We discuss the problem of localization in two dimensional electron systems in the quantum Hall (single Landau level) regime. After briefly summarizing the well-studied problem of Anderson localization in the non-interacting case, we concentrate on the problem of disorder induced many-body localization (MBL) in the presence of electron–electron interactions using numerical exact diagonalization and eigenvalue spacing statistics as a function of system size. We provide evidence showing that MBL is not attainable in a single Landau level with short range (white noise) disorder in the thermodynamic limit. We then study the interplay of topology and localization, by contrasting the behavior of topological and nontopological subbands arising from a single Landau level in two models — (i) a pair of extremely flat Hofstadter bands with an optimally chosen periodic potential, and (ii) a Landau level with a split-off nontopological impurity band. Both models provide convincing evidence for the strong effect of topology on the feasibility of many-body localization as well as slow dynamics starting from a nonequilibrium state with charge imbalance.

Introduction

It took nearly two decades after Anderson’s seminal 1958 paper [1] establishing the existence of electron localization due to disorder to uncover the nature of the transition [2], and thereby establish that all states in the non-interacting Anderson model with potential disorder were localized for dimensions two and below, using the scaling theory of localization [3]. Since then, the study of Anderson localization has spawned a veritable industry, consisting of experiments, analytic theory and numerical approaches. What emerged was a beautiful story of the interplay of dimension and universality classes of Hamiltonians in the phenomenon of Localization. A small selection of review articles describing the progress in the twentieth century is in Refs. [4], [5], [6], [7], [8], [9], [10].

Soon after the scaling theory of localization [3] was formulated, experiments on two-dimensional electron gases in semiconductors in a high magnetic field [11] uncovered the phenomenon known as the integer quantum Hall effect (IQHE) — the quantization of the Hall conductance in two-dimensional systems in integer multiples of the fundamental unit e2/h. The theoretical explanation which soon followed [12], showed that localization played a central role in the quantization of the Hall conductance over much of the phase diagram. However, the existence of a nonzero Hall conductivity pointed to a new result — that in such systems, the localization length diverged at certain critical energies [13], in contrast to the situation in zero magnetic field. We now recognize this as being due to the topological nature [14], [15] of Landau levels, which have given rise to a much richer set [16] of universality classes of disordered systems, beyond the original Wigner–Dyson classes [4]. This divergence has been much studied by diverse numerical methods [17], [18], [19], [20], though the precise critical exponent [21], [22], [23], and even the nature [24], [25] of the divergence is still being debated.

The integer quantum Hall effect was followed a few years later by the even more surprising discovery [26] and theoretical understanding [27] of the fractional quantum Hall effect (FQHE), where the quantization of the Hall conductance was at rational fractions (in units of e2/h), and the explanation relied entirely on electron–electron interactions. The initial result of Laughlin [27] was soon generalized to a hierarchy of fillings [28]; several field-theoretic approaches were attempted using the concept of composite particles, the most successful of which was the composite fermion approach of Jain [29], [30]. Jain’s approach gave rise to a hierarchy of FQHE states which was in amazing agreement1 with experiment, both in identification, as well as strength (magnitudes of the excitation gap whose existence is responsible for the FQHE). The FQHE naturally gave rise to the question of localization due to disorder in such systems, such as would disorder destroy the topological nature of the ground state by closing the excitation gap? Though only a few quantitative numerical results exist to date, tracking both the topological character [31], [32] and the entanglement entropy [33], [34] in the ground state shows that such a FQH Hall to non-topological insulator transition does take place.

The phenomenon of spin localization was originally in Anderson’s mind when he wrote the 1958 paper [1], motivated by the beautiful magnetic resonance experiments on doped semiconductors by Feher [35], [36]. Research in spin localization saw significant progress in the last two decades of the twentieth century, motivated by experiments on magnetic properties — again in doped semiconductors [37], [38] as well as disordered quasi-1D organic salts [39], [40]. Numerical and theoretical investigations [41], [42], [43], [44], [45], [46] led to the conclusion that disorder leads to spin localization at low energies, not only in one-dimension, but also in disordered electronic systems in higher dimensions. This could happen even when charge localization is absent, so the system is a metal at low temperatures with a diverging magnetic susceptibility over a finite range of the phase diagram!

The proper generalization of the story of localization to interacting electron systems came in the first decade of the twenty-first century, with the formulation and first true understanding of many-body localization. While several attempts had been made to generalize Anderson’s idea to interacting many-body systems, the crucial breakthrough [47], [48] came in the period 2005–6, particularly with the comprehensive paper by Basko, Aleiner and Altshuler [47] using perturbative approaches. In a few years, complete (or infinite temperature) many-body localization (MBL) came to be understood [49], [50] as a breaking of the eigenstate thermalization hypothesis [51], [52], and the subject underwent an explosive expansion, see e.g. [53], [54], [55], [56], [57], [58]. More details and references can be found in many excellent reviews [59], [60], [61], [62], [63]. Currently, MBL seems to be well established in random field spin chains, a one-dimensional spin model, whereas for spin models with randomness in higher dimensions (d=2 and above), strong arguments have been made to support the contention that MBL is destroyed by rare fluctuation effects. Recent numerical studies of one-dimensional quasiperiodic systems [64] show that MBL is more robust than in the disordered case, indirectly supporting the above contention. Nevertheless, it should be recognized that because the Hilbert space grows exponentially with the total size (Ld, where L is the linear dimension), numerical studies, especially in d>1, are limited to rather small sizes.

Here we report the results of numerical studies of MBL in systems in the quantum Hall regime. Such studies are interesting on several fronts. First, the system possesses only the charge degree of freedom — the spin degree of freedom is quenched because of the high magnetic field; consequently, the Hilbert space does not grow as fast as in the case of fermions with both charge and spin degrees of freedom. Secondly, the model, while not one-dimensional, has the next lowest dimensionality (d=2), thereby keeping the rate of growth of Hilbert space with size manageable. Thirdly, one can formulate it in a continuum version, and thereby avoid discrete lattice effects present in lattice models in d>1. Finally, it allows one to study directly the effect of topology on MBL, something not studied in spin models. A direct motivation for studying this effect numerically comes from an earlier analytical study [65] of MBL in Landau levels, which concluded that MBL was not possible in a Landau level broadened by disorder because of the diverging localization length at the single particle level. A first numerical study [66] confirmed that qualitative result, but found a much larger effect of topology than predicted by the analytic considerations.

Section snippets

The quantum Hall regime and the QH Hamiltonian

We consider a generic system of N electrons of mass m and charge e moving in the xy plane in the presence of a magnetic field, whose Hamiltonian is the sum of three terms: Hˆ=Tˆ+Uˆ+Uˆint=i=1Nπi22m+i=1NV(ri)+ijNVint(ri,rj).Here πp+eA is the dynamical momentum operator, V(r) is an arbitrary single-particle potential and Vint(r,r) is the electron–electron interaction. When the magnetic field B is constant and perpendicular to the plane, the free-electron part of the Hamiltonian turns into a

Models of Landau level subbands

These results beg the question — what is the primary cause for the absence of MBL in the LLL? Is it primarily topology, or dimensionality? Would a 2-D system with no topological character be more amenable to an MBL phase at large disorder? In this context, we tried to uncouple the peculiar non-zero Chern character of the Landau level wave functions from their 2-D nature in a series of studies [73], [74].

The aim was to split the LLL into subbands, each subband having a separate Chern character.

Disorder in the non-interacting system

We first briefly comment on the nature of single-particle localization in the periodic potential model of Section 3.1. In this model, we set the disorder to be a random short-range Gaussian correlated potential of the form Vdis(r)Vdis(r)=W2σ2e|rr|2/2σ2The length scale σ is a tunable length scale parameter. We found that this type of disorder led to more localized wave functions than the delta-correlated disorder [74].

In order to quantify the spatial footprint of the wave functions, we

Conclusions

In the one and a half decades that have passed since the original paper by Basko et al. [47] establishing the existence of many-body localization, starting from a completely Anderson localized band and including electron–electron interactions in a perturbative manner, it has become clear that MBL is much harder to achieve in interacting electron systems than Anderson localization in noninteracting models. The latter is ubiquitous in one dimension, and also in two dimensions with potential

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Parts of the research described in this work were done in collaboration with Scott D. Geraedts and Matteo Ippoliti. Their contributions, and those of Rahul Nandkishore, to our overall understanding of many body localization in the quantum Hall regime are gratefully acknowledged. In addition, RNB acknowledges the profound influence of the late P.W. Anderson on his research, particularly on disordered electronic systems, for over four decades. This research was supported by the US Department of

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