Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

TL;DR

The paper investigates how integrability and chaos manifest in dissipative open quantum systems by analyzing the complex spectrum of the Liouvillian. It proposes a unifying picture in which the full complex-spacing statistics are described by a static 2D Coulomb gas with inverse temperature $β∈[0,2]$, bridging Poisson and Ginibre limits. Using boundary-driven XXZ spin chains, it provides numerical evidence for a continuous transition from integrable to chaotic statistics and demonstrates universality of the $β=2$ bulk distribution across Ginibre ensembles. The results extend Grobe-Haake-Sommers’ small-spacing cubic repulsion to full distributions and suggest a broad applicability of 2D Coulomb-gas statistics in non-Hermitian quantum systems.

Abstract

We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance $s$ is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature $β\in[0,2]$. Here, $β=0$ yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and $β=2$ equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalise the results of Grobe, Haake and Sommers who derived a universal cubic level repulsion for small spacings $s$. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at $β=2$. It holds for all three Ginibre ensembles of random matrices with independent real, complex or quaternion matrix elements.

Figures (2)

• Figure 1: Comparison of the level spacing distributions for various Liouville operators \ref{['Liouvillian']}, the analytical spacing distributions \ref{['Poisson-2d']} (Poisson $\beta=0$, dotted) and \ref{['GinibreSpacing']} (Ginibre $\beta=2$, dashed) as well as fits to general Coulomb gas \ref{['2DCoulomb']} simulations (Figures (b) and (d), solid). Unfolding \ref{['unfold']} is used with the smearing parameter $\sigma=4.5\,\bar{s}$, see \ref{['unfold']}, where the mean spacing varies from $\bar{s}=0.0036$ to $0.0045$ for the data sets (a) - (d). The first moment of all spacings is normalised to unity.
• Figure 2: Comparison of the spacing distribution \ref{['GinibreSpacing']} with normalised first moment and those of the GinOE (blue crosses), GinUE (red squares), and GinSE (purple triangles) in the bulk of the spectrum. For the latter we use the standard $2N$- dimensional representation of an $N$-dimensional quaternionic matrix, making the complex eigenvalues unique, cf. Ginibre. An ensemble of $1000$$500\times 500$ matrices has been generated in a Monte Carlo simulation. Here, the unfolding is trivial due to a uniform density of all three ensembles.