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arXiv:1901.04505 · Physical review. B./Physical review. B

Instability of many-body localized systems as a phase transition in a nonstandard thermodynamic limit

Sarang Gopalakrishnan, David A. Huse

Published Apr 16, 2019• 70 citations
arXivHTMLPDF

Abstract

At the many-body localization transition, the dynamics of an isolated quantum system freezes out. In systems with long-range interactions (and possibly short-range systems in many spatial dimensions) this transition does not happen in the conventional thermodynamic limit: arbitrarily weak interactions prevent many-body localization. Here, the authors argue that even in these cases a sharp phase transition exists if one scales down the interaction strength with increasing system size in a specific nonstandard way.

DOI
10.1103/physrevb.99.134305

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At the many-body localization transition, the dynamics of an isolated quantum system freezes out. In systems with long-range interactions (and possibly short-range systems in many spatial dimensions) this transition does not happen in the conventional thermodynamic limit: arbitrarily weak interactions prevent many-body localization. Here, the authors argue that even in these cases a sharp phase transition exists if one scales down the interaction strength with increasing system size in a specific nonstandard way.
As disorder strength increases in quantum many-body systems a new phase of matter, the so-called many-body localization, emerges across the whole spectrum. This transition is energy dependent, a phenomenon known as mobility edge, such that the mid-spectrum eigenstates tend to localize at larger values of disorder in comparison to eigenstates near the edges of the spectrum. Many-body localization becomes more sophisticated in long-range interacting systems. Here, by focusing on several quantities, we draw the phase diagram as a function of disorder strength and energy spectrum, for a various range of interactions. Regardless of the underlying transition type, either second-order or Kosterlitz-Thouless, our analysis consistently determines the mobility edge, i.e., the phase boundary across the spectrum. We show that, long-range interaction enhances the localization effect and shifts the phase boundary towards smaller values of disorder. In addition, we establish a hierarchy among the studied quantities concerning their corresponding transition boundary and critical exponents. Interestingly, we show that deliberately discarding some information of the system can mitigate finite-size effects and provide results in line with the analytical predictions at the thermodynamic limit.
Recent theoretical and numerical evidence suggests that localization can survive in disordered many-body systems with very high energy density, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in extended quantum systems far from the zero-temperature limit. It separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (``ergodic'') phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present numerical evidence that many-body localization also occurs in models without disorder but rather a quasiperiodic potential. In one dimension, these systems already have a single-particle localization transition, and we show that this transition becomes a many-body localization transition upon the introduction of interactions. We also comment on possible relevance of our results to experimental studies of many-body dynamics of cold atoms and nonlinear light in quasiperiodic potentials.
This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a ddd-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].
Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators (l∗-bits)({\mathrm{l}}^{*}\text{-bits})(l∗-bits) underlie localization in such systems. In dimensions d>1d>1d>1, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits. Physical consequences of l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in d=1d=1d=1 and thermal eigenstates at all parameters in d>1d>1d>1. Near-term experiments in ultracold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits.
The authors study the phase transition between thermal and many-body localized phases in one-dimensional systems with quenched randomness and short-range interactions. The approach uses a renormalization group model, very similar to those of previous works, that puts the avalanche instability of the many-body localized phase at center stage. Within that model, the universal critical behavior is determined analytically, and the phase transition is found to belong to a new universality class that differs from the previously found Kosterlitz-Thouless result in important ways.
We investigate the many-body localization (MBL) transition in the quantum Ising model with long-range interactions. Unlike spin chains with short-range interactions, where the MBL transition point remains elusive due to strong finite-size effects and local fluctuations, long-range interactions suppress such fluctuations and enable clearer signatures of critical behavior. Using exact results from a related Bethe lattice localization problem, we estimate the MBL threshold within logarithmic accuracy and find consistency with exact diagonalization. Although the critical disorder diverges in the thermodynamic limit, our results demonstrate that the critical regime can still be probed and highlight the relevance of this model to systems with dipole-dipole, elastic, or indirect exchange interactions.