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.
Highlights
Understanding ergodicity in quantum many-body systems remains a fundamental task of mathematical physics
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
Summary
Understanding ergodicity in quantum many-body systems remains a fundamental task of mathematical physics. It is known that speci c kinds of symmetries called strong symmetries [20] lead to degeneracy of the (time-independent) stationary state to which the open system evolves in the long-time limit. In this article we derive a lower bound for the stationary-state degeneracy of an open quantum system with multiple, non-abelian strong symmetries—symmetries whose corresponding operators do not necessarily commute. We achieve this through decomposing the matrix representation of these symmetries into a series of irreducible representations. L which de nes the adjoint Liouvillian superator L†
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