Abstract
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; (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.
Highlights
Systems displaying many-body localization [1,2] (MBL) violate the eigenstate thermalization hypothesis [3] and do not thermalize
This tacitly assumes the absence of topological order: It implies that Fully MBL (FMBL) eigenstates arise from product states of this spin-z basis via local unitary transformation, which guarantees [37,38] that they are topologically trivial
We propose the following definition of FMBL and (t)LIOMs to capture Abelian topological order on topologically trivial manifolds: Definition 1
Summary
Systems displaying many-body localization [1,2] (MBL) violate the eigenstate thermalization hypothesis [3] and do not thermalize (see Refs. [4,5] for some recent reviews). LIOMs are commonly assumed to form a complete set arising as a local unitary transform of the set of on-site spin-z operators This tacitly assumes the absence of topological order: It implies that FMBL eigenstates arise from product states of this spin-z basis via local unitary transformation, which guarantees [37,38] that they are topologically trivial. The concept of tLIOMs can be illuminated by placing LIOMs into the broader context of the stabilizer formalism [39] From this perspective, one sees the on-site spin-z operators as just one choice of local stabilizers, namely those of product states in this spin-z basis. Topological LIOMs must correspond to a different set, namely the local stabilizers in the commuting projector limits of topological phases These tLIOMs are closely related to the approach of Ref. This tLIOM–tensor-network combination is the perspective through which we shall seek new avenues for the numerical characterization of topological FMBL systems
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