# Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians

## TL;DR

This paper demonstrates that charge and dipole conservation in fracton systems cause Hilbert space fragmentation, leading to non-thermal behavior and ergodicity breaking in one-dimensional spin models, with implications for understanding thermalization failure.

## Contribution

It reveals how charge and dipole conservation induce Hilbert space fragmentation and ergodicity breaking in dipole-conserving Hamiltonians, supported by concrete models and comparisons to random circuit systems.

## Key findings

- Finite auto-correlation saturates at finite value, indicating non-thermal behavior.
- Strong fragmentation into exponentially many invariant subspaces prevents thermalization.
- Weak fragmentation persists with four-site terms, but allows thermalization for typical states.

## 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 $S^z$) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature auto-correlation saturates to a finite value---showcasing non-thermal 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 $S^z$ 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 non-thermal 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.