Super-operator structures and no-go theorems for dissipative quantum phase transitions

TL;DR

This paper analyzes the structure of Liouville super-operators in open quantum systems, demonstrating that many dissipative phase transitions are impossible and that convergence to steady states is exponential, with implications for various models.

Contribution

It introduces a super-operator framework that bounds spectral gaps and proves no-go theorems for dissipative phase transitions in many quantum systems.

Findings

Dissipative phase transitions are often impossible in large classes of systems.

Spectral gaps can be bounded from below using super-operator structures.

Convergence to steady states is exponential in these systems.

Abstract

In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville super-operator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The super-operator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems, and are also demonstrated for various spin models.