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arXiv:1608.01328 · Physical review. B./Physical review. B

Emergent local integrals of motion without a complete set of localized eigenstates

Scott Geraedts, R. N. Bhatt, Rahul Nandkishore

Published Feb 13, 2017• 41 citations
arXivHTMLPDF

Abstract

Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a set of localized eigenstates to be complete. Given a set of localized eigenstates comprising a nonzero fraction $(1\ensuremath{-}{p}_{f})$ of the full many-body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have nonzero weight in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a ``global dressing'' whose weight vanishes as $\ensuremath{\sim}{p}_{f}$ as ${p}_{f}\ensuremath{\rightarrow}0$. In this sense, the existence of a nonzero fraction of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability.

DOI
10.1103/physrevb.95.064204

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Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a set of localized eigenstates to be complete. Given a set of localized eigenstates comprising a nonzero fraction (1\ensuremath−pf)(1\ensuremath{-}{p}_{f})(1\ensuremath−pf​) of the full many-body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have nonzero weight in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a ``global dressing'' whose weight vanishes as \ensuremath∼pf\ensuremath{\sim}{p}_{f}\ensuremath∼pf​ as pf\ensuremath→0{p}_{f}\ensuremath{\rightarrow}0pf​\ensuremath→0. In this sense, the existence of a nonzero fraction of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability.
Many properties of a quantum system can be obtained from just a single eigenstate of its Hamiltonian. For example, a single eigenstate can be used to determine whether a system is integrable or chaotic and, in the latter case, to establish its thermal properties. Focusing on the XXZ model, we show that the local integrals of motion, which lie at the heart of integrability, can also be estimated from a small number of eigenstates. Moreover, as the system size increases, fewer eigenstates are required, so that in the thermodynamic limit, the integrals of motion can be obtained from a vanishingly small fraction of all eigenstates. Interestingly, this property does not extend to integrals of motion arising solely from Hilbert space fragmentation, as found in the folded XXZ model, where the majority of eigenstates has to be used. This represents one of the few fundamental differences known between integrability and Hilbert space fragmentation.
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
We prove the existence of extensive many-body Hamiltonians with few-body interactions and a many-body mobility edge: all eigenstates below a nonzero energy density are localized in an exponentially small fraction of "energetically allowed configurations" within Hilbert space. Our construction is based on quantum perturbations to a classical low-density parity check code. In principle, it is possible to detect this eigenstate localization by measuring few-body correlation functions in efficiently preparable mixed states.
Many-body localized (MBL) systems are often described using their local integrals of motion, which, for spin systems, are commonly assumed to be a local unitary transform of the set of on-site spin-z operators.We show that this assumption cannot hold for topologically ordered MBL systems.Using a suitable definition to capture such systems in any spatial dimension, we demonstrate a number of features, including that MBL topological order, if present, (i) is the same for all eigenstates, (ii) is robust in character against any perturbation preserving MBL, and (iii) implies that on topologically nontrivial manifolds a complete set of integrals of motion must include nonlocal ones in the form of local-unitary-dressed noncontractible Wilson loops.Our approach is well suited for tensor-network methods and is expected to allow these to resolve highly excited finite-size-split topological eigenspaces despite their overlap in energy.We illustrate our approach on the disordered Kitaev chain, toric code, and X-cube model.
The presence and character of local integrals of motion---quasilocal operators that commute with the Hamiltonian---encode valuable information about the dynamics of a quantum system. In particular, strongly disordered many-body systems can generically avoid thermalization when there are extensively many such operators. In this work, we explicitly construct local conserved operators in one-dimensional spin chains by directly minimizing their commutator with the Hamiltonian. We demonstrate the existence of an extensively large set of local integrals of motion in the many-body localized phase of the disordered XXZ spin chain. These operators are shown to have exponentially decaying tails, in contrast to the ergodic phase where the decay is (at best) polynomial in the size of the subsystem. We study the algebraic properties of localized operators and confirm that in the many-body localized phase, they are well described by ``dressed'' spin operators.
We show how the thermodynamic properties of large many-body localized systems can be studied using quantum Monte Carlo simulations. We devise a heuristic way of constructing local integrals of motion of high quality, which are added to the Hamiltonian in conjunction with Lagrange multipliers. The ground state simulation of the shifted Hamiltonian corresponds to a high-energy state of the original Hamiltonian in the case of exactly known local integrals of motion. The inevitable mixing between eigenstates as a consequence of nonperfect integrals of motion is weak enough such that the characteristics of many-body localized systems are not averaged out, unlike the standard ensembles of statistical mechanics. Our method paves the way to study higher dimensions and indicates that a fully many-body localized phase in 2D, where (nearly) all eigenstates are localized, is likely to exist.