Criticality and Phase Classification for Quadratic Open Quantum Many-Body Systems

TL;DR

This paper investigates the steady states and phase transitions of quadratic open quantum many-body systems, revealing conditions for criticality and phase classification in fermionic and bosonic models.

Contribution

It provides a comprehensive analysis of steady state properties, correlation decay, and phase structure in quadratic Lindblad systems, including bounds on correlation lengths and criteria for criticality.

Findings

Steady states in 1D systems exhibit exponential decay of correlations.

Fermionic systems with finite-range interactions are noncritical in any dimension.

Bosonic systems can be critical in dimensions greater than one.

Abstract

We study the steady states of translation-invariant open quantum many-body systems governed by Lindblad master equations, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian. These systems are called quasifree and quadratic, respectively. We find that steady states of one-dimensional systems with finite-range interactions necessarily have exponentially decaying Green's functions. For the quasifree case without quadratic Lindblad operators, we show that fermionic systems with finite-range interactions are noncritical for any number of spatial dimensions and provide bounds on the correlation lengths. Quasifree bosonic systems can be critical in $D>1$ dimensions. Last, we address the question of phase transitions in quadratic systems and find that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped Liouvillians belong to the same phase.