Threefold Way for Typical Entanglement

TL;DR

This paper explores the statistical properties of entanglement spectra in quantum systems with various symmetries, establishing a threefold classification analogous to Dyson's threefold way, and introduces fractionalization of time reversal symmetry to extend the framework.

Contribution

It introduces a novel framework connecting symmetry classes in quantum entanglement spectra to Dyson's threefold way, including fractionalized symmetries and their spectral implications.

Findings

Entanglement spectra follow Laguerre ensembles depending on symmetry class.

Fractionalized time reversal symmetry leads to Laguerre symplectic ensemble (LSE).

Spectral degeneracies depend on non-Abelian symmetry and cohomology class.

Abstract

A typical quantum state with no symmetry can be realized by letting a random unitary act on a fixed state, and the subsystem entanglement spectrum follows the Laguerre unitary ensemble (LUE). For integer-spin time reversal symmetry, we have an analogous scenario where we prepare a time-reversal symmetric state and let random orthogonal matrices act on it, leading to the Laguerre orthogonal ensemble (LOE). However, for half-integer-spin time reversal symmetry, a straightforward analogue leading to the Laguerre symplectic ensemble (LSE) is no longer valid due to that time reversal symmetric state is forbidden by the Kramers' theorem. We devise a system in which the global time reversal operator is fractionalized on the subsystems, and show that LSE arises in the system. Extending this idea, we incorporate general symmetry fractionalization into the system, and show that the statistics of the entanglement spectrum is decomposed into a direct sum of LOE, LUE, and/or LSE. Here, various degeneracies in the entanglement spectrum may appear, depending on the non-Abelian nature of the symmetry group and the cohomology class of the non-trivial projective representation on the subsystem. Our work establishes the entanglement counterpart of the Dyson's threefold way for Hamiltonians with symmetries.