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Bridging Commutant and Polynomial Methods for Hilbert Space Fragmentation

Bo-Ting Chen, Yu-Ping Wang, Biao Lian

TL;DR

The paper establishes a formal bridge between two approaches to Hilbert space fragmentation (HSF): the commutant-algebra (CA) method and the integer characteristic polynomial factorization (ICPF) method. It proves a general theorem: when the center of the commutant is rationally diagonalizable (all eigenvalues rational), ICPF yields a fragmentation that is equal to or finer than CA, and the ICPF basis can depend on Hamiltonian coefficients if the fragmentation is strictly finer. The authors verify the theorem across representative models (t-$J_z$, PF, TL, QB, and Fibonacci) and show that, in most known HSF systems, ICPF $\succeq$ CA holds, with equality in many cases; the Fibonacci model provides a counterexample where ICPF $\prec$ CA due to irrational center eigenvalues. Collectively, these results unify perspectives on HSF and suggest a pathway toward a single, coefficient-aware definition of fragmentation.

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.

Bridging Commutant and Polynomial Methods for Hilbert Space Fragmentation

TL;DR

The paper establishes a formal bridge between two approaches to Hilbert space fragmentation (HSF): the commutant-algebra (CA) method and the integer characteristic polynomial factorization (ICPF) method. It proves a general theorem: when the center of the commutant is rationally diagonalizable (all eigenvalues rational), ICPF yields a fragmentation that is equal to or finer than CA, and the ICPF basis can depend on Hamiltonian coefficients if the fragmentation is strictly finer. The authors verify the theorem across representative models (t-, PF, TL, QB, and Fibonacci) and show that, in most known HSF systems, ICPF CA holds, with equality in many cases; the Fibonacci model provides a counterexample where ICPF CA due to irrational center eigenvalues. Collectively, these results unify perspectives on HSF and suggest a pathway toward a single, coefficient-aware definition of fragmentation.

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.
Paper Structure (14 sections, 9 theorems, 62 equations, 1 table)

This paper contains 14 sections, 9 theorems, 62 equations, 1 table.

Table of Contents

  1. Basic properties of von Neumann algebras
  2. Formal proof of ICPF $\succeq$ CA
  3. ICPF $\succeq$ CA
  4. ICPF $\succ$ CA: coefficient-dependent basis
  5. $t$-$J_{z}$ model
  6. Model description
  7. Hilbert space fragmentation
  8. Pair-Flip model
  9. Model description
  10. Hilbert space fragmentation
  11. Temperley-Lieb Model
  12. Model description
  13. Hilbert space fragmentation
  14. The Quantum Breakdown Model

Key Result

Theorem 1

Assume $\mathcal{Z}$ is the center of commutant algebra of a quantum model. If every element $\hat{z}_k\in \bold{Basis}(\mathcal{Z})$ (which is a rational operator) has all its eigenvalues $c(\hat{z}_k)$ being rational (i.e. $c(\hat{z}_k)\in \mathbb{Q}(i)$), the model has ICPF $\succeq$ CA.

Theorems & Definitions (14)

  • Theorem 1: The main theorem
  • Corollary 1
  • Proposition 1
  • Definition 1
  • Definition 2
  • Theorem 2: The bi-commutant theorem
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 4 more