Conserved Quantities from Entanglement Hamiltonian

TL;DR

This paper demonstrates that entanglement Hamiltonians of excited states in quantum many-body systems can be approximated by linear combinations of local conserved quantities, with the structure revealed through an entanglement Hamiltonian superdensity matrix.

Contribution

It introduces the EHSM as a tool to identify conserved quantities from entanglement Hamiltonians and explores their properties in free, integrable, and chaotic systems.

Findings

EHSM eigen-operators reveal conserved quantities.

Number of nonzero EHSM eigenvalues scales with subregion volume in free fermions.

Decay of EHSM eigenvalues indicates integrability or chaos.

Abstract

We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities, with relative commutation errors $\mathcal{O}\left(\frac{\text{subregion boundary area}}{\text{subregion volume}}\right)$. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigen-operators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume, and some of their EHSM eigen-operators can be rather nonlocal, although subregionally quasilocal. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly in power law if the system is integrable, with the exponent $s\approx 1$ ($s\approx 1.5\sim 2$) if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigen-operators of which correspond to the identity and the subregion Hamiltonian.