Entanglement Spectrum Resolved by Loop Symmetries

TL;DR

This paper develops a mathematical framework to analyze the entanglement spectrum of quantum states with higher-form symmetries, revealing how topology influences entanglement structures in topological gauge theories.

Contribution

It introduces a novel algebraic topology and category theory-based approach to determine entanglement spectra for states with non-invertible higher-form symmetries, including topological gauge theories.

Findings

Framework reproduces topological entanglement entropy.

Confirms Li-Haldane conjecture for entanglement spectrum.

Analyzes entanglement structure on various manifolds.

Abstract

A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally non-invertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly.