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arXiv:1901.00034 · Physical review. B./Physical review. B

Comparing many-body localization lengths via nonperturbative construction of local integrals of motion

Pai Peng, Zeyang Li, Haoxiong Yan, Ken Xuan Wei, Paola Cappellaro

Published Dec 13, 2019• 19 citations
arXivHTMLPDF

Abstract

Many interesting properties of many-body localization (MBL) can be understood using a phenomenological model, involving local integrals of motion (LIOMs). Usually, LIOMs can only be computed approximately. Here, the authors explicitly compute a complete set of LIOMs nonperturbatively by maximizing their overlap with physical spin operators. The scheme is immune to accidental resonances and can be applied even at the phase transition point, providing a powerful tool to study microscopic features of MBL as characterized by various localization lengths.

DOI
10.1103/physrevb.100.214203

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Many interesting properties of many-body localization (MBL) can be understood using a phenomenological model, involving local integrals of motion (LIOMs). Usually, LIOMs can only be computed approximately. Here, the authors explicitly compute a complete set of LIOMs nonperturbatively by maximizing their overlap with physical spin operators. The scheme is immune to accidental resonances and can be applied even at the phase transition point, providing a powerful tool to study microscopic features of MBL as characterized by various localization lengths.
Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body-localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin-flip operator), evolved with the system's Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, in both experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin-chain model. We show that the resulting LIOMs are quasilocal and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.
Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such LIOMs, we explore and study some of their structural properties across the MBL transition. We first consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a nonperturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but, rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert space. Our cluster-size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\ensuremath{\nu}$ both in the presence and absence of interactions. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.
Conservation laws are of paramount importance in our understanding of classical and quantum dynamics. Here, we present a general framework for constructing exact quantum integrals of motion with the desired locality and quantum numbers, which will be illustrated for the case of many-body-localization (MBL). The latter has been understood theoretically in terms of the existence of an extensive number of local integrals of motion (LIOMs). Using our approach, we show that one can express the task of finding LIOMs as an optimization problem. For some specifications, this problem surprisingly connects to the question of finding classical ground states of spin-glass models. Our work unifies previous results obtained in the MBL context and reveals intriguing connections between many-body localization, spin-glass physics, and constrained optimization problems.
Abstract The emergent integrability in a many-body localized (MBL) system can be well characterized by the existence of the complete set of local integrals of motion (LIOMs). Such exactly conserved and exponentially localized operators are often understood as quasiparticle operators which can be expanded in terms of single-particle operators dressed with different numbers of particle-hole pairs. Here, we consider a one-dimensional XXZ spin- <?CDATA 12\frac{1}{2}21​?> 1 2 Heisenberg chain in the presence of a random field and try to quantify the corrections needed to be considered in the picture of quasiparticles associated with LIOMs due to the presence of particle-hole excitations. To this end, we explicitly present the multibody expansion of LIOM creation operators of the system in the MBL regime. We analytically obtain the coefficients of this expansion and discuss the effect of higher-order corrections associated with different numbers of particle-hole excitations. Our analysis shows that depending on the localization length of the system, there exist a regime in which the contributions that come from higher-order terms can break down the effective one-particle description of the LIOMs and such quasiparticles become essentially many-body-like.
A complete set of local integrals of motion (LIOM) is a key concept for describing many-body localization (MBL), which explains a variety of intriguing phenomena in MBL systems. For example, LIOM constrain the dynamics and result in ergodicity violation and breakdown of the eigenstate thermalization hypothesis. However, it is difficult to find a complete set of LIOM explicitly and accurately in practice, which impedes some quantitative structural characterizations of MBL systems. Here we propose an accurate numerical method for constructing LIOM, discover through the LIOM an interaction-induced characteristic length ξ + , and prove a ‘quasi-product-state’ structure of the eigenstates with that characteristic length ξ + for MBL systems. More specifically, we find that there are two characteristic lengths in the LIOM. The first one is governed by disorder and is of Anderson-localization nature. The second one is induced by interaction but shows a discontinuity at zero interaction, showing a nonperturbative nature. We prove that the entanglement and correlation in any eigenstate extend not longer than twice the second length and thus the eigenstates of the system are the quasi-product states with such a localization length .
We review the current (as of Fall 2016) status of the studies on the emergent integrability in many‐body localized models. We start by explaining how the phenomenology of fully many‐body localized systems can be recovered if one assumes the existence of a complete set of (quasi)local operators which commute with the Hamiltonian (local integrals of motions, or LIOMs). We describe the evolution of this idea from the initial conjecture, to the perturbative constructions, to the mathematical proof given for a disordered spin chain. We discuss the proposed numerical algorithms for the construction of LIOMs and the status of the debate on the existence and nature of such operators in systems with a many‐body mobility edge, and in dimensions larger than one. image