Integer characteristic polynomial factorization and Hilbert space fragmentation

TL;DR

This paper links integer polynomial factorization to Hilbert space fragmentation, providing a new method to identify Krylov subspaces and analyze thermalization properties in Hamiltonians with integer representations.

Contribution

It introduces a novel approach connecting polynomial factorization to Krylov subspace structure, aiding the detection of Hilbert space fragmentation in Hamiltonians.

Findings

Integer polynomial factorization corresponds to Krylov subspace existence.

Method demonstrated on pair-hopping model to reveal Hilbert space fragmentation.

Generalization to other rings of integers discussed.

Abstract

Models with Hilbert space fragmentation are characterized by (exponentially) many dynamically disconnected subspaces, not associated with conventional symmetries but captured by nontrivial Krylov subspaces. These subspaces usually exhibit a whole range of thermalization properties, from chaotic to integrable, to quantum many-body scars. However, so far, they have not been properly defined, nor can they be easily found, given a Hamiltonian. In this work, we consider Hamiltonians that have integer representations, a common feature of many (most) celebrated models in condensed matter. We show the equivalence of the integer characteristic polynomial factorization and the existence of Krylov subspaces generated from integer vectors. Considering the pair-hopping model, we illustrate how the factorization property can be used as a method to unveil Hilbert space fragmentation. We discuss the generalization over other rings of integers, for example those based on the cyclotomic field which are relevant when working in a given ($\ne 0, \pi$) momentum sector.