Bridging Commutant and Polynomial Methods for Hilbert Space Fragmentation

TL;DR

This paper establishes a theoretical connection between two methods for identifying Hilbert space fragmentation in quantum models, potentially unifying their approaches and improving understanding of quantum dynamics.

Contribution

The paper proves a theorem linking the commutant algebra and integer characteristic polynomial factorization methods for HSF, applicable to models with rational matrix representations.

Findings

Most known HSF models satisfy the theorem's conditions

The theorem shows ICPF HSF is equal or finer than CA HSF in certain cases

Discussion of models where ICPF and CA methods differ

Abstract

A quantum model exhibits Hilbert space fragmentation (HSF) if its Hilbert space decomposes into exponentially many dynamically disconnected subspaces, known as Krylov subspaces. A model may however have different HSFs depending on the method for identifying them. Here we establish a connection between two vastly distinct methods recently proposed for identifying HSF: the commutant algebra (CA) method and integer characteristic polynomial factorization (ICPF) method. For a Hamiltonian consisting of operators admitting rational number matrix representations, we prove a theorem that, if its center of commutant algebra have all eigenvalues being rational, the HSF from the ICPF method must be equal to or finer than that from the CA method. We show that this condition is satisfied by most known models exhibiting HSF, for which we demonstrate the validity of our theorem. We further discuss representative models for which ICPF and CA methods yield different HSFs. Our results may facilitate the exploration of a unified definition of HSF.