Lie-algebraic approach to one-dimensional translationally invariant free-fermionic dissipative systems

TL;DR

This paper introduces a Lie-algebraic method to exactly solve the dynamics of one-dimensional translationally invariant free-fermionic dissipative systems, revealing spectral properties and phase transition criteria with experimental relevance.

Contribution

It develops a novel Lie-algebraic framework to solve the Lindblad equation for these systems and analyzes their spectral and phase transition properties.

Findings

Exact solutions for the density matrix at all times.

Generic criterion for dissipative gap closure.

Identification of gapless modes with linear spectrum.

Abstract

We study dissipative translationally invariant free-fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach, we solve the Lindblad equation and find the density matrix at all times for arbitrary time dependence of the Liouvillian. We then investigate the Liouvillian spectral properties and derive a generic criterion for the closure of the dissipative gap, which is believed to be linked with nonequilibrium dissipative phase transitions. We illustrate our findings with a few exotic examples. Particularly, we show the presence of gapless modes with a linear spectrum for fermions with long-range hopping, which might be related to nonunitary conformal field theories. The predicted effects can be probed in experiments with ultracold atomic and quantum-optical systems using currently available experimental facilities.