Probing Hilbert space fragmentation and the block inverse participation ratio

TL;DR

This paper investigates how Hilbert space fragmentation affects quantum many-body Hamiltonians near exactly fragmented models, introducing a modified inverse participation ratio to identify transition boundaries and suggesting fragmentation as a phase.

Contribution

It develops a novel block inverse participation ratio to detect fragmentation transitions and demonstrates its effectiveness across different Hamiltonian families.

Findings

Block IPR predicts fragmentation boundaries consistent with level statistics and entanglement measures.

Approximate fragmentation influences system behavior even in the thermodynamic limit.

Fragmentation appears to constitute a distinct phase in quantum many-body systems.

Abstract

We consider a family of quantum many-body Hamiltonians that show exact Hilbert space fragmentation in certain limits. The question arises whether fragmentation has implications for Hamiltonians in the vicinity of the subset defined by these exactly fragmented models, in particular in the thermodynamic limit. We attempt to illuminate this issue by considering distinguishable classes of transitional behavior between fragmented and nonfragmented regimes and employing a set of numerical observables that indicate this transition. As one of these observables we present a modified inverse participation ratio (IPR) that is designed to capture the emergence of fragmented block structures. We compare this block IPR to other definitions of inverse participation ratios, as well as to the more traditional measures of level-spacing statistics and entanglement entropy. In order to resolve subtleties that arise in the numerics, we use perturbation theory around the fragmented limit as a basis for defining an effective block structure. We find that our block IPR predicts a boundary between fragmented and nonfragmented regimes that is compatible with results based on level statistics and bipartite entanglement. A scaling analysis indicates that a finite region around the exactly fragmented limit is dominated by effects of approximate fragmentation, even in the thermodynamic limit, and suggests that fragmentation constitutes a phase. We provide evidence for the universality of our approach by applying it to a different family of Hamiltonians, that features a fragmented limit due to emergent dipole conservation.