Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$

TL;DR

This work develops a fermionic replica nonlinear sigma model approach to non-Hermitian random matrices in symmetry classes AI$^{\dag}$ and AII$^{\dag}$. By deriving $n$-fold replica expressions for the $n$-th moments of $k$-point characteristic polynomials, the authors compute the density of states and two-point spectral correlations in the Gaussian ensembles, using saddle-point analyses and cosine-sine decompositions over the relevant groups (Sp, O). The key results include exact finite-$N$ representations for $Z_n^{(1)}$ and large-$N$ expressions for $Z_n^{(2)}$, leading to analytical forms for $R_1(z,\bar{z})$ and $R_2(z_1,\bar{z}_1,z_2,\bar{z}_2)$, with explicit leading behaviors such as uniform disk spectra and corrected two-point correlations. While the DoS agrees reasonably with numerics in the applicable regimes, the replica-limit-derived two-point functions show deviations near small eigenvalue spacings, indicating the need for more systematic replica treatments or supersymmetric approaches. Overall, the paper advances analytical control over universal bulk spectral statistics in non-Hermitian RMT and lays groundwork for extensions to non-Gaussian ensembles and physical contexts like open quantum systems and Lindblad dynamics.

Abstract

Symmetry of non-Hermitian matrices underpins many physical phenomena. In particular, chaotic open quantum systems exhibit universal bulk spectral correlations classified on the basis of time-reversal symmetry$^{\dagger}$ (TRS$^{\dagger}$), coinciding with those of non-Hermitian random matrices in the same symmetry class. Here, we analytically study the spectral correlations of non-Hermitian random matrices in the presence of TRS$^{\dagger}$ with signs $+1$ and $-1$, corresponding to symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$, respectively. Using the fermionic replica non-linear sigma model approach, we derive $n$-fold integral expressions for the $n$th moment of the one-point and two-point characteristic polynomials. Performing the replica limit $n\to 0$, we qualitatively reproduce the density of states and level-level correlations of non-Hermitian random matrices with TRS$^{\dagger}$.

Figures (10)

• Figure 1: Numerical calculations for one-point (left) and two-point (right) functions for the three universality classes. We sample matrices of size $10^3\times 10^3$ for classes A and AI$^\dagger$ and size $\left( 2 \times 10^3 \right) \times \left( 2 \times 10^3 \right)$ for class AII$^\dagger$. To numerically obtain $R_1$, we simply compute the histogram of the sampled spectra in each class. The numerical computation of $R_2$ is more subtle. To analyze bulk correlations, we must choose eigenvalues away from the edge of the spectrum. Our prescription is to choose eigenvalues within a disc centered at the origin with $2/3^{\text{rd}}$ the radius of the spectrum. In this region, $R_2$ depends only on the distance $|z_1-z_2|$. Therefore, the Mathematica library function PairCorrelationG can be used, which gives $\rho^{-2} R_2(\sqrt \rho |z_1 - z_2|)$, where $\rho$ is the density of eigenvalues in this region. We then average this quantity over $10^3$ samples in each class. In the above plot, to compare correlations across different classes, we have scaled the data such that $\rho$ is identical for each class. For comparison, we also plot the known analytical result for the bulk two-point correlation function in class A mehta2004random.
• Figure 2: Plot of $\frac{Z^{(1)}_n(z, \bar{z})}{Z^{(1)}_n(0,0)}e^{-2ng^{-1}|z|^2}$ as a function of $\left| z \right|$ for class AI$^\dagger$. For each value of $N$, we sample $10^5$ non-Hermitian random matrices according to Eq. \ref{['eq:AIdag-gaussian']} with $g=2$. The solid markers show the ensemble-averaged characteristic polynomials. The black dashed curves show the same quantity calculated by the NL$\sigma$M in Eq. \ref{['eq:AI-dag-1pt-microscopic']}.
• Figure 3: Comparison between the analytical result in Eq. \ref{['AId DOS analytical']} and the density of states obtained from numerical calculations. The numerical results are obtained using $2 \times 10^4$ realizations of $10^3 \times 10^3$ non-Hermitian random matrices in class AI$^\dagger$ sampled according to Eq. \ref{['eq:AIdag-gaussian']} with $g=2$. Note that this is a logarithmic plot. This might cause the spread of the numerics in the plot.
• Figure 4: Logarithmic plot of the two-point characteristic polynomial in class AI$^\dagger$. We plot the expression $\frac{Z^{(2)}_n(0,0,z_2, \bar{z}_2)}{Z^{(2)}_n(0,0,0,0)}e^{-2ng^{-1}|z_2|^2}$ instead of $Z^{(2)}_n(0,0,z_2, \bar{z}_2)$ for better visualization. For each value of $N=2$ and $N=5$, we collect $10^6$ realizations of non-Hermitian random matrices, while for $N=10$ and $N=50$ we collect $5\times 10^6$ realizations. In all cases, we use $g=2$. The solid markers show the numerically computed ensemble-averaged moments of characteristic polynomials. The black dashed curves show the same quantity calculated with the NL$\sigma$M for large $N$ in Eq. \ref{['eq:AI-dag-2pt-microscopic']}.
• Figure 5: Comparison between the analytical result in Eq. \ref{['two-pt AIdag']} and the two-point correlation function obtained from numerical calculations. The numerical results are obtained by $2 \times 10^4$ realizations of $10^3 \times 10^3$ non-Hermitian random matrices in class AI$^\dagger$ sampled according to Eq. \ref{['eq:AIdag-gaussian']} with $g=2$.