Stationary State Degeneracy of Open Quantum Systems with Non-Abelian Symmetries

TL;DR

This paper analyzes how non-Abelian symmetries in open quantum systems determine the degeneracy of stationary states, providing a theoretical framework and examples that reveal the structure and scaling of these degeneracies.

Contribution

It introduces a method to decompose the Hilbert space for systems with non-Abelian symmetries, deriving bounds on stationary state degeneracy and simplifying Liouvillian analysis.

Findings

Derived a tight lower bound for stationary state degeneracy.

Bound scales cubically with system size in SU(2) symmetric cases.

Theoretical framework reduces computational complexity in analyzing steady states.

Abstract

We study the null space degeneracy of open quantum systems with multiple non-Abelian, strong symmetries. By decomposing the Hilbert space representation of these symmetries into an irreducible representation involving the direct sum of multiple, commuting, invariant subspaces we derive a tight lower bound for the stationary state degeneracy. We apply these results within the context of open quantum many-body systems, presenting three illustrative examples: a fully-connected quantum network, the XXX Heisenberg model and the Hubbard model. We find that the derived bound, which scales at least cubically in the system size the $SU(2)$ symmetric cases, is often saturated. Moreover, our work provides a theory for the systematic block-decomposition of a Liouvillian with non-Abelian symmetries, reducing the computational difficulty involved in diagonalising these objects and exposing a natural, physical structure to the steady states - which we observe in our examples.