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

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.