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arXiv:1906.00971

Strong and Weak Many-Body Localizations

Shi‐Xin Zhang, Hong Yao

Published Jun 3, 2019• 4 citations
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Abstract

Many aspects of many-body localization (MBL), including dynamic classification of MBL phases, remain elusive. Here, by performing real-space renormalization group (RSRG) analysis we propose that there are two distinct types of MBL phases: "strong MBL" induced by quasiperiodic (QP) potential and "weak MBL" induced by random potential. Strong and weak MBL phases can be distinguished by their different probability distributions of thermal inclusion and entanglement entropy: exponential decay in strong MBL phases but power-law decay in weak MBL. We further discuss underlying mechanisms as well as experimental implications of having two distinct types of MBL phases. Strong MBL induced by QP potential may provide a more robust and promising platform for quantum information storage and processing.

DOI
10.48550/arxiv.1906.00971

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Many aspects of many-body localization (MBL), including dynamic classification of MBL phases, remain elusive. Here, by performing real-space renormalization group (RSRG) analysis we propose that there are two distinct types of MBL phases: "strong MBL" induced by quasiperiodic (QP) potential and "weak MBL" induced by random potential. Strong and weak MBL phases can be distinguished by their different probability distributions of thermal inclusion and entanglement entropy: exponential decay in strong MBL phases but power-law decay in weak MBL. We further discuss underlying mechanisms as well as experimental implications of having two distinct types of MBL phases. Strong MBL induced by QP potential may provide a more robust and promising platform for quantum information storage and processing.
The precise nature of many-body localization (MBL) transitions in both random and quasiperiodic (QP) systems remains elusive so far. In particular, whether MBL transitions in QP and random systems belong to the same universality class or two distinct ones has not been decisively resolved. Here, we investigate MBL transitions in one-dimensional (d=1) QP systems as well as in random systems by state-of-the-art real-space renormalization group (RG) calculation. Our real-space RG shows that MBL transitions in 1D QP systems are characterized by the critical exponent ν≈2.4, which respects the Harris-Luck bound (ν>1/d) for QP systems. Note that ν≈2.4 for QP systems also satisfies the Harris-Chayes-Chayes-Fisher-Spencer bound (ν>2/d) for random systems, which implies that MBL transitions in 1D QP systems are stable against weak quenched disorder since randomness is Harris irrelevant at the transition. We shall briefly discuss experimental means to measure ν of QP-induced MBL transitions.
Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang et al. [Phys. Rev. B 93, 224201 (2016)] and Goremykina et al. [Phys. Rev. Lett. 122, 040601 (2019)], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length and an internal decay length $\ensuremath{\zeta}$ for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.
We develop a real-space renormalization-group (RSRG) scheme by appropriately inserting the long-range hopping t∼r^{-α} with nearest-neighbor interaction to study the entanglement entropy and maximum block size for the many-body localization (MBL) transition. We show that for α<2 there exists a localization transition with renormalized disorder that depends logarithmically on the finite size of the system. The transition observed for α>2 does not need a rescaling in disorder strength. Most important, we find that even though the MBL transition for α>2 falls in the same universality class as that of the short-range models, the transition for α<2 belongs to a different universality class. Because of the intrinsic nature of the RSRG flow toward delocalization, MBL phase for α>2 might suffer an instability in the thermodynamic limit while the underlying systems support algebraic localization. Moreover, we verify these findings by inserting microscopic details to the RSRG scheme where we additionally find a more appropriate rescaling function for disorder strength; we indeed uncover a power-law scaling with a logarithmic correction and a distinctly different stretched exponential scaling for α<2 and α>2, respectively, by analyzing system with finite size. This finding further suggests that microscopic RSRG scheme is able to give a hint of instability of the MBL phase for α>2 even considering systems of finite size.
We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior.
Localization in one-dimensional interacting systems can be caused by disorder potentials or non-Hermiticity. The former phenomenon is the many-body localization (MBL), and the latter is the many-body non-Hermitian skin effect (NHSE). In this work, we numerically investigate the interplay between these two kinds of localization, where the energy-resolved MBL arises from a deterministic quasiperiodic potential in a fermionic chain. We propose a set of eigenstate properties and long-time dynamics that collectively distinguish the two localization mechanisms in the presence of non-Hermiticity. By computing the proposed diagnostics, we show that the thermal states are vulnerable to the many-body NHSE while the MBL states remain resilient up to a strong non-Hermiticity. Finally, we discuss experimental observables that probe the difference between the two localizations in a non-Hermitian quasiperiodic fermionic chain. Our results pave the way toward experimental observations on the interplay of interaction, quasiperiodic potential, and non-Hermiticity.
The nature of the many-body localization (MBL) transition and even the existence of the MBL phase in random many-body quantum systems have been actively debated in recent years. In spatial dimension d>1d>1d>1, there is some consensus that the MBL phase is unstable to rare thermal inclusions that can lead to an avalanche that thermalizes the whole system. In this note, we explore the possibility of MBL in quasiperiodic systems in dimension d>1d>1d>1. We argue that (i) the MBL phase is stable against ``avalanches'' at strong enough quasiperiodic modulations for d=2d=2d=2, and (ii) the possibility of an avalanche strongly constrains the finite-size scaling behavior of the MBL transition. We present a suggestive construction that MBL is unstable for $d\ensuremath{\ge}3$.