Solutions for dissipative quadratic open systems: part II - fermions

TL;DR

This paper extends methods for solving Lindblad master equations to fermionic systems, providing analytical solutions for normal modes, relaxation rates, and steady states in non-interacting fermionic chains.

Contribution

It introduces a reduction technique to diagonalize a non-Hermitian matrix for fermionic systems, paralleling previous bosonic work, with key differences for fermions.

Findings

Analytical expressions for normal master modes and rapidities

Explicit construction of non-equilibrium steady states

Applicable to boundary dissipative fermionic chains

Abstract

This is the second part of a work in which we show how to solve a large class of Lindblad master equations for non-interacting particles on $L$ sites. Here we concentrate on fermionic particles. In parallel to part I for bosons, but with important differences, we show how to reduce the problem to diagonalizing an $L \times L$ non-Hermitian matrix which, for boundary dissipative driving of a uniform chain, is a tridiagonal bordered Toeplitz matrix. In this way, both for fermionic and spin systems alike, we can obtain analytical expressions for the normal master modes and their relaxation rates (rapidities) and we show how to construct the non-equilibrium steady state.