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arXiv:2111.11543 · Physical review. B./Physical review. B

Local integrals of motion detection of localization-protected topological order

Thorsten B. Wahl, Florian Venn, Benjámin Béri

Published Apr 29, 2022• 20 citations
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Abstract

Many-body-localized (MBL) phases can be topologically distinct, but distinguishing these phases using order parameters can be challenging. Here we show how topologically distinct local integrals of motion, variationally parametrized by quantum circuits, can be used to numerically demonstrate the topological inequivalence of MBL phases. We illustrate our approach on a fermionic chain where both topologically distinct MBL phases and benchmark comparisons to order parameters are possible. We also use our approach, augmented by the excited state density-matrix renormalization group algorithm, to extract high-energy topological doublets. We describe applying our methods to higher dimensions to identify MBL topological order and topological multiplets hidden by the dense many-body spectrum.

DOI
10.1103/physrevb.105.144205

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Many-body-localized (MBL) phases can be topologically distinct, but distinguishing these phases using order parameters can be challenging. Here we show how topologically distinct local integrals of motion, variationally parametrized by quantum circuits, can be used to numerically demonstrate the topological inequivalence of MBL phases. We illustrate our approach on a fermionic chain where both topologically distinct MBL phases and benchmark comparisons to order parameters are possible. We also use our approach, augmented by the excited state density-matrix renormalization group algorithm, to extract high-energy topological doublets. We describe applying our methods to higher dimensions to identify MBL topological order and topological multiplets hidden by the dense many-body spectrum.
Many-body localization (MBL) has been proposed to enable and protect topological order in all eigenstates, vastly expanding the traditional ground-state setting. However, for the most intriguing case of two-dimensional (2D) systems with anyons and topology-dependent degeneracies, the dense many-body spectrum challenges studying this MBL protection numerically. Here we use large-scale full-spectrum variational to demonstrate MBL-protected topological order in the disordered 2D toric code perturbed by magnetic fields. We show that the system has topological local integrals of motion (tLIOMs) for magnetic field strengths below h c ≈ 0.1 times the toric-code coupling scale. Combining tLIOMs with exact diagonalization, we also identify high-energy topological multiplets in the dense many-body spectrum. The phase diagram we find is consistent with toric-code and trivial MBL phases being separated by an intervening thermal phase. Published by the American Physical Society 2024
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.
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) addresses the absence of thermalization in interacting quantum systems, with nonergodic high-energy eigenstates behaving as ground states, only area-law entangled. However, computing highly excited many-body eigenstates using exact methods is very challenging. Instead, we show that one can address high-energy MBL physics using ground-state methods, which are much more amenable to many efficient algorithms. We find that a localized many-body ground state of a given interacting disordered Hamiltonian H0{\mathcal{H}}_{0}H0​ is a very good approximation for a high-energy eigenstate of a parent Hamiltonian, close to H0{\mathcal{H}}_{0}H0​ but more disordered. This construction relies on computing the covariance matrix, easily achieved using density-matrix renormalization group for disordered Heisenberg chains up to L=256L=256L=256 sites.
The entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here, we show that strongly disordered systems in the many-body localized phase have power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or free systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems.