Higher-order gap ratios of singular values in open quantum systems

TL;DR

This work establishes a universal link between higher-order gap ratios of singular values in open quantum systems and the nearest-neighbor spacing statistics of a log-gas in a harmonic trap, with an effective inverse temperature $\beta'(k)=\frac{k(k+1)}{2}\beta+(k-1)$. The key insight is that $P^{(k)}(r,\beta)=P(r,\beta'(k))$, enabling robust discrimination of symmetry classes across a range of non-Hermitian settings, including GinOE/UE/SE random matrices, random Liouvillians, Lindblad baths, and non-Hermitian Hamiltonians. The authors validate the universality using the Kullback–Leibler divergence and show explicit results for several platforms, as well as a Poisson and a dissipative XXZ model demonstrating ergodic-localized transitions. Overall, higher-order singular-value gap ratios provide a powerful, universal diagnostic of long-range spectral correlations and symmetry classes in open quantum systems, beyond what complex spectra alone offer.

Abstract

Understanding open quantum systems using information encoded in its complex eigenvalues has been a subject of growing interest. In this paper, we study higher-order gap ratios of the singular values of generic open quantum systems. We show that $k$-th order gap ratio of the singular values of an open quantum system can be connected to the nearest-neighbor spacing ratio of positions of classical particles of a harmonically confined log-gas with inverse temperature $β'(k)$ where $β'(k)$ is an analytical function that depends on $k$ and the Dyson's index $β=1,2,$ and $4$ that characterizes the properties of the associated Hermitized matrix. Our findings are crucial not only for understanding long-range correlations between the eigenvalues but also provide an excellent way of distinguishing different symmetry classes in an open quantum system. To highlight the universality of our findings, we demonstrate the higher-order gap ratios using different platforms such as non-Hermitian random matrices, random dissipative Liouvillians, Hamiltonians coupled to a Markovian bath, and Hamiltonians with in-built non-Hermiticity.

Figures (12)

• Figure 1: Plots for $k$-th order level spacing ratios [Eq. \ref{['k-th-ratio']}] for non-Hermitian random matrix (NHRM) that belongs to GinOE class. (a) and (b) represents $k=3$ and $k=4$, respectively (black-solid). We notice remarkable agreement between the higher order level spacing ratio and the nearest neighbour spacing ratio of Dyson's log gas (red-solid) with an effective inverse temperature given in Eq. \ref{['scaling']} for $\beta=1$ and holds for any $k\geq 1$. We choose NHRM of size $10^4 \times 10^4$ and obtain the statistics over 500 realizations. We further report the value of KL divergence [Eq. \ref{['KL-divergence']}] to be $0.0088$ and $0.0005$, for (a) and (b), respectively, thereby cementing the remarkable agreement.
• Figure 2: Plots for $k$-th order level spacing ratios [Eq. \ref{['k-th-ratio']}] for the random Liouvillian given in Eq. \ref{['eq:RL']}. Here (a) and (b) represents $k=3$ and $k=4$, respectively (black-solid). The singular value statistics follows the scaling relation [Eq. \ref{['scaling-distribution']}] for $\beta =1$ (red-solid). Similar to Fig. \ref{['fig:GinOE-NHRM']}, we once again observe a remarkable agreement. The results here are obtained for system size $M=4$ for $50$ realizations. In this case we find the KL divergence [Eq. \ref{['KL-divergence']}] to be $0.020$ and $0.0019$, for (a) and (b), respectively.
• Figure 3: Plots for $k$-th order level spacing ratios [Eq. \ref{['k-th-ratio']}] for the physical Lindbladian given in Eq. \ref{['eq:PL']} with Hamiltonian given in Eq. \ref{['spin-ising']} and the jump operators given by $L_i = \sqrt{\gamma} \, \sigma_i^{-}$. Here (a) and (b) represents $k=2$ and $k=4$, respectively (black-solid). Similar to Fig. \ref{['fig:GinOE-NHRM']}, we once again observe a remarkable agreement with $\beta=1$ (red-solid). The results here are obtained for system size $M=6$ averaged over $50$ different realizations. We choose $J=1$, $h_x= -1.05$, $h_z = 0.2$, and $\gamma=0.77$. In this case we find the KL divergence [Eq. \ref{['KL-divergence']}] to be $0.008$ and $0.032$, for (a) and (b), respectively.
• Figure 4: Plots for $k$-th order level spacing ratios [Eq. \ref{['k-th-ratio']}] for the non-Hermitian Hamiltonian [Eq. \ref{['eq:GinUE_H']}] deep in the localized regime ($h=14$). Here (a) and (b) represents $k=1$ and $k=4$, respectively (black-solid). We observe remarkable agreement with Eq. \ref{['singular-poisson']} (red-solid). The corresponding KL diverges are $0.1016$ and $0.0098$, respectively. We consider $M=16$, $J=1$, $g=0.1$, $V=2$, and $\gamma=0.1$ and the lattice is half filled. The spacing ratio distributions are obtained by performing an average over $50$ different realizations.
• Figure 5: Plots for $k$-th order level spacing ratios [Eq. \ref{['k-th-ratio']}] for the non-Hermitian Hamiltonian [Eq. \ref{['eq:GinUE_H']}] deep in the ergodic phase ($h=2$). Here (a) and (b) represents $k=2$ and $k=3$, respectively (black-solid). The corresponding KL diverges are $0.0048$ and $0.0098$, respectively. All other parameters are same as in Fig. \ref{['fig:nhh2p']}. The spacing ratio distributions are obtained by performing an average over $50$ different realizations. The numerical results are in excellent agreement with Eq. \ref{['P_r_beta']} setting $\beta=2$ (red-solid).