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arXiv:2204.08835 · Physical review. B./Physical review. B

Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction

Safa Adami, Mohsen Amini, Morteza Soltani

Published Aug 9, 2022• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.106.054202

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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 \ensuremathν\ensuremath{\nu}\ensuremathν 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.
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.
We study the many-body localization (MBL) transition for interacting fermions subject to quasiperiodic potentials by constructing the local integrals of motion (LIOMs) in the MBL phase as time-averaged local operators. We study numerically how these time-averaged operators evolve across the MBL transition. We find that the norm of such time-averaged operators drops discontinuously to zero across the transition; as we discuss, this implies that LIOMs abruptly become unstable at some critical localization length of order unity. We analyze the LIOMs using hydrodynamic projections and isolating the part of the operator that is associated with interactions. Equipped with these data we perform a finite-size scaling analysis of the quasiperiodic MBL transition. Our results suggest that the quasiperiodic MBL transition occurs at considerably stronger quasiperiodic modulations and has a larger correlation-length critical exponent than previous studies had found.
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.
Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.
We develop a procedure which systematically generates all conserved operators in the disordered models of interacting fermions. Among these operators, we identify and count the independent and local integrals of motion (LIOM), which represent the hallmark of the many-body localization (MBL). The method is tested first on the prototype disordered chain of interacting spinless fermions. As expected for full MBL, we find for large enough disorder ${N}_{M}={2}^{M}\ensuremath{-}1$ independent and quasilocal LIOM with support on MMM consecutive sites. On the other hand, the study of the disordered Hubbard chain reveals that ${3}^{M}\ensuremath{-}1<{N}_{M}\ensuremath{\lesssim}{4}^{M}/2$, which is less than required for full MBL but much more than in the case of spinless fermions.
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