Abstract
We show that the combination of charge and dipole conservation—characteristic of fracton systems—leads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total ) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value—showcasing nonthermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.
- Received 12 April 2019
- Accepted 23 January 2020
DOI:https://doi.org/10.1103/PhysRevX.10.011047
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Regardless of initial form, ice cream always melts on a warm summer day, an example of “reaching thermal equilibrium”—the melted ice cream retains no information about its original state. Analogously, we also expect to find these thermal states in ensembles of many quantum particles, with the initial local information scrambled all over the place. However, in certain situations, these expectations fail because of additional restrictions on the dynamics of the system, thus avoiding thermalization in the usual way. Here, we introduce a general mathematical framework for describing such counterintuitive quantum many-body systems.
Our framework is called Hilbert space fragmentation, wherein the space of configurations of a many-body system falls apart into exponentially many disconnected subsectors. These sectors are not labeled by usual conserved quantities, such as energy or particle number, but with the help of emergent symmetries.
As a concrete example, we consider 1D systems conserving charge and its associated dipole and study the structure of these emergent subspaces. In the most constrained case, we show that, evolving from a generic initial state, the system retains memory of the initial local structure, thus failing to thermalize. It is as if different parts of our “ice cream” possess unrelated temperatures, with no clear relation to its initial shape or any other property. Moreover, we show that this fragmentation occurs for generic dipole-conserving chains with finite-range couplings.
Future work might consider whether we can find general recipes for constructing quantum many-body systems with fragmented Hilbert spaces, identify the conservation laws that label the many invariant subspaces, or discover how the Hilbert space fragmentation manifests in experimental systems.
