Absence of dissipation-free topological edge states in quadratic open fermions

TL;DR

This paper proves that generic quadratic open fermionic systems governed by Lindblad equations cannot host dissipation-free topological edge states due to their topological triviality, establishing a fundamental limit on topological phenomena in such systems.

Contribution

It provides a no-go theorem showing the absence of dissipation-free topological edge states in quadratic open fermionic systems, extending topological classification to open quantum systems.

Findings

Dissipation-free topological edge states do not exist in generic quadratic open fermionic systems.

The Lindblad generator can be mapped to a non-Hermitian matrix that is always deformable to a trivial Hermitian matrix.

The results apply broadly to systems with a gapped bulk and bounded spectrum.

Abstract

We prove a no-go theorem: generic quadratic open fermionic systems governed by Lindblad master equations do not host dissipation-free topological edge states protected by the dissipation gap. By analogy with topological insulators and superconductors, we map the Lindblad generator to a first-quantized non-Hermitian matrix representation that plays the role of a band Hamiltonian. Edge modes of this matrix with vanishing real part are exactly dissipation-free. We show that this matrix is always adiabatically deformable, through a symmetry-preserving path, to a topologically trivial Hermitian matrix. Hence no symmetry-protected, dissipation-free edge modes exist in quadratic open fermions. Our results apply to generic quadratic fermionic Lindbladians and require only a gapped bulk and a bounded spectrum. They establish a clear boundary for robust topological phenomena in open fermionic systems.