Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking

TL;DR

The paper investigates how complex eigenvalue statistics in an open, dissipative quantum many-body system separate Hermiticity breaking from integrability breaking. Using a strongly interacting XXZ spin chain with purely imaginary disorder, the authors fit NN and NNN spacings to a $2$d Coulomb-gas description with inverse temperature $\beta$ and to $D$-dimensional Poisson processes, enabling a unified view of two distinct crossovers. At small disorder $\gamma$, Hermiticity breaks first, driving a transition from $1$-d Poisson to $2$-d Poisson (effective dimension $1\le D<2$); at intermediate $\gamma$, integrability breaks with $\beta \approx1.3$–$1.4$ consistent with AI$^{\dag}$ statistics; for large $\gamma$, spin alignment restores an integrable non-Hermitian regime with $2$d Poisson again. The work demonstrates that two separate mechanisms shape complex spectra in open many-body systems and shows how non-Hermitian RMT and Poisson processes can describe these regimes across scales.

Abstract

Open quantum systems have complex energy eigenvalues which are expected to follow non-Hermitian random matrix statistics when chaotic, or 2-dimensional (2d) Poisson statistics when integrable. We investigate the spectral properties of a many-body quantum spin chain, the Hermitian XXZ Heisenberg model with imaginary disorder. Its rich complex eigenvalue statistics is found to separately break both Hermiticity and integrability at different scales of the disorder strength. With no disorder, the system is integrable and Hermitian, with spectral statistics corresponding to the 1d Poisson point process. At very small disorder, we find a transition from 1d Poisson statistics to an effective $D$-dimensional Poisson point process, showing Hermiticity breaking. At intermediate disorder we find integrability breaking, as inferred from the statistics matching that of non-Hermitian complex symmetric random matrices in class AI$^†$. For large disorder, as the spins align, we recover the expected integrability (now in the non-Hermitian setup), indicated by 2d Poisson statistics. These conclusions are based on fitting the spin chain data of numerically generated nearest and next-to-nearest neighbour spacing distributions to an effective 2d Coulomb gas description at inverse temperature $β$. We confirm such an effective description of random matrices also applies in class AI$^†$ and AII$^†$ up to next-to-nearest neighbour spacings.

Figures (14)

• Figure 1: Schematic illustration of the spin chain model with local dissipative disorder described by Eq. \ref{['XXZ_iW_Hamiltonian']}.
• Figure 2: (a) Spacing distributions of the Poisson process, with dimension $1\leq D\leq 2$ varying in steps $0.1$ (red--grey). Notice that in both cases the maximum moves from right to left when decreasing dimension $D$. (b) The numerically generated NN and NNN spacing distributions of the 2dCG for $\beta=0-2$, increasing in steps $0.2$ (grey--blue). This also includes analytic formulas for 2d Poisson with $\beta=0$, \ref{['PoiD-NN']} and \ref{['PoiD-NNN']} with the lowest maxima, and random matrix class A with $\beta=2$, \ref{['ANN']} and \ref{['ANNN']}, with highest maxima. Note that the grey curves for $D=2$ in (a) are the same as the grey curves for $\beta=0$ in (b) since the two processes then coincide.
• Figure 3: Two transitions in the XXZ model with dissipative disorder. Hermiticity breaking (left): For $0.05\leq\gamma<0.25$ we find that the NN and NNN spectral statistics changes from those of 1d Poisson to 2d Poisson, thus Hermiticity is broken first. Shown is the fitted effective dimension $D$ for NN (full curve) and NNN (dashed curve). Data is shown for $L=16$ spins in the $\mathcal{S}_z = 0$ magnetization sector. Integrability breaking (right): For larger $\gamma$, i.e. $0.5<\gamma<20$ we find a very good fit to $\beta$ of the 2dCG. At $\gamma=2$ the NN (full curve) and NNN statistics (dashed curve) are very close to those of AI$^\dagger$ corresponding to $\beta\approx 1.3-1.4$ (horizontal band), representing a complete breaking of integrability. As $\gamma$ is increased further, there is a crossover to the statistics of 2d Poisson (horizontal line at $\beta=0$).
• Figure 4: The NN (top), NNN spacing distribution (middle) of at least 490 disorder realizations, and scatter plot with selected eigenvalues highlighted in red for a single realization of the spectrum (bottom) for $\gamma=0,0.05,0.1,0.15,0.2,0.25$. In red (full line) we show \ref{['PoiD-NN']} and \ref{['PoiD-NNN']} for the NN and NNN distribution, respectively, for the best fitted value for the effective dimension $D$, with corresponding standard deviation $\sigma$ (in units of $10^{-2})$. For comparison we also show the results for $D=1$ (blue dashed) and for $D=2$ (black dashed line) as an orientation. For pure XXZ, $\gamma=0.0$, the spectrum is real, and since there is no disorder the data shown is for a single realization. In addition, the full $\mathcal{S}_z = 0$ is degenerate and splits into two sectors: an even Parity sector and an odd Parity one; we show statistics only for the even Parity sector. Switching on $\gamma>0$ we see how these sectors start to mix, which may also be responsible for the disagreement of the NNN at the smallest values of $\gamma=0.05$.
• Figure 5: Cuts through the spectral density (for a single realization), in imaginary direction at $x_0\approx 0$, for $\gamma=0.1$ (left) and $\gamma=0.2$ (right).