Constructing local integrals of motion in the many-body localized phase

Abstract

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, both in 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 quasi-local, and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.Comment: 8 pages, 3 figure

Many-body-localization transition: strong multifractality spectrum for matrix elements of local operators

“…At the other end of the spectrum, the criterion concerning the minimal exponent α loc q=+∞ = 0 (Eq. 38 and the corresponding discussion) can be related to the existence of an extensive number of emergent localized conserved operators in the Many-Body-Localized phase [33][34][35][36][37][38][39][40][41].…”

“…The Strong Disorder Real-Space RG approach (see [18,19] for reviews) developed initially by Ma-Dasgupta-Hu [20] and Daniel Fisher [21][22][23] to construct the ground states of random quantum spin chains, has been extended for the whole Many-Boy-Localized phase into the Strong Disorder RG procedure for the unitary dynamics [24,25], and into the RSRG-X procedure to construct the whole set of excited eigenstates [26][27][28][29][30]. This construction is possible only because the Many-Body-Localized phase is characterized by an extensive number of emergent localized conserved operators [33][34][35][36][37][38][39][40][41]. As a consequence, the Strong Disorder RG breaks down when the MBL transition towards delocalization is approached, and other types of real-space RG have been proposed for the critical point, based on the notion of insulating or thermalizing blocks of various sizes [42][43][44] and are in favor of a direct transition towards the ergodic phase.…”

Many-body-localization transition: sensitivity to twisted boundary conditions

“…can be understood in terms of the complete set of the Local Integrals of Motions (LIOMS) [62][63][64][65][66][67][68][69][70][71] : two nearby eigenstates in energy have typically different LIOMS everywhere in the sample, so that the off-diagonal matrix element of the local current operator involves some tunneling through the entire system and is thus exponentially suppressed (see [26] for a much more detailed discussion). The variance V in Eq.…”

Many-Body Localization Characterized from a One-Particle Perspective

“…This manybody localization (MBL) transition occurs at finite energy densities and is not a conventional thermodynamic transition [24,25]. Instead, it can be understood as a dynamical phase transition, associated with the emergence of a complete set of local conserved quantities in the localized phase, which thus behaves as an integrable system [26][27][28][29][30]. This restricts the entanglement entropy of the eigenstates to an area law [31], in contrast to the volume law predicted by the eigenstate thermalization hypothesis for the ergodic delocalized phase [32][33][34].…”