UniqueNESS: Graph Theory Approach to the Uniqueness of Non-Equilibrium Stationary States of the Lindblad Master Equation

TL;DR

This paper addresses the problem of determining when a Lindblad-driven quantum system possesses a unique non-equilibrium stationary state (NESS) and, in particular, when the steady state is strictly positive. It introduces a graph-theoretic framework that encodes generator sets as adjacency matrices of directed graphs and tests for strong connectivity to ensure the generated operator algebra is the full $\,\mathcal{B}(\mathcal{H})$, thereby guaranteeing a unique, faithful NESS. The method unifies Jordan and Lie algebra perspectives via transitivity/completeness and reveals scale-invariant, self-similar connectivity patterns (cobwebs) in driven-dissipative spin lattices, capturing how dissipative and coherent parts cooperate to reach the full algebra. The framework provides a practical criterion that complements existing algebraic conditions (e.g., the Kossakowski kernel, bicommutant, and Yoshida criteria) and offers scalable tooling for analyzing NESS in complex many-body settings with potentially multiple stationary states in parameter regions.

Abstract

The dimensionality of kernels for Lindbladian superoperators is of physical interest in various scenarios out of equilibrium, for example in mean-field methods for driven-dissipative spin lattice models that give rise to phase diagrams with a multitude of non-equilibrium stationary states in specific parameter regions. We show that known criteria established in the literature for unique fixpoints of the Lindblad master equation can be better treated in a graph-theoretic framework via a focus on the connectivity of directed graphs associated to the Hamiltonian and jump operators.

Figures (14)

• Figure 1: Web of Connectivity for $\{L\}$ with 2 Sites
• Figure 2: Web of Connectivity for $\{L\}$ with 3 Sites
• Figure 3: Web of Connectivity for $\{L\}$ with 4 Sites
• Figure 4: Web of Connectivity for $H-i\sum L^*L/2$ with 2 Sites
• Figure 5: Web of Connectivity for $H-i\sum L^*L/2$ with 3 Sites