Abstract

Topological band theory was developed to predict and explain robust features in the ground state electronic structure of insulators and superconductors. Are these ideas applicable to open quantum systems? In this talk, I will discuss the topology of “non-Hermitian Hamiltonians” which evolve systems non-unitarily in time or space. While much of this formalism has been successfully applied to dissipative classical photonics, my focus will be on its relevance to open quantum many-body theory. Specifically, I will discuss progress to achieve “topologically-degenerate steady states,” in analogy with the equilibrium paradigm of topologically-degenerate ground states. I provide an example where a non-Hermitian Majorana zero mode is responsible for this behavior in a partially-projected atomic system. Current efforts are aimed at realizing robust degeneracies in the complex spectrum of the (Lindblad) master equation. From a mathematical point of view, non-Hermiticity of the Hamiltonian results in a richer set of topological symmetry classes. I highlight the importance of the “Bernard-LeClair” classes which generalize the Altland-Zirnbauer (AZ) classes in the absence of Hermiticity. These classes represent a new direction in symmetry-protected topological phases which go beyond the AZ and crystalline symmetries.