Tenfold Way for Quadratic Lindbladians

TL;DR

This paper develops a topological classification for open fermionic systems described by quadratic Lindblad equations, revealing edge states with finite lifetimes that depend on system-environment interactions.

Contribution

It extends topological classification to non-Hermitian Lindbladian systems, linking symmetry classes to physical edge excitations in open quantum systems.

Findings

Topological classification applies to quadratic Lindbladians with non-Hermitian matrices.

Edge modes with finite lifetimes are identified, independent of steady state.

Dissipators can preserve or destroy the topological classification in 1D models.

Abstract

We uncover a topological classification applicable to open fermionic systems governed by a general class of Lindblad master equations. These `quadratic Lindbladians' can be captured by a non-Hermitian single-particle matrix which describes internal dynamics as well as system-environment coupling. We show that this matrix must belong to one of ten non-Hermitian Bernard-LeClair symmetry classes which reduce to the Altland-Zirnbauer classes in the closed limit. The Lindblad spectrum admits a topological classification, which we show results in gapless edge excitations with finite lifetimes. Unlike previous studies of purely Hamiltonian or purely dissipative evolution, these topological edge modes are unconnected to the form of the steady state. We provide one-dimensional examples where the addition of dissipators can either preserve or destroy the closed classification of a model, highlighting the sensitivity of topological properties to details of the system-environment coupling.