Spectral fluctuations of multiparametric complex matrix ensembles: evidence of a single parameter dependence

Abstract

We numerically analyze the spectral statistics of the multiparametric Gaussian ensembles of complex matrices with zero mean and variances with different decay routes away from the diagonals. As the latter mimics different degree of effective sparsity among the matrix elements, such ensembles can serve as good models for a wide range of phase transitions e.g. localization to delocalization in non-Hermitian systems or Hermitian to non-Hermitian one. Our analysis reveals a rich behavior hidden beneath the spectral statistics e.g. a crossover of the spectral statistics from Poisson to Ginibre universality class with changing variances for finite matrix size, an abrupt transition for infinite matrix size and the role of complexity parameter, a single functional of all system parameters, as a criteria to determine critical point. We also confirm the theoretical predictions in \cite{psgs, psnh}, regarding the universality of the spectral statistics in non-equilibrium regime of non-Hermitian systems characterized by the complexity parameter.

Figures (9)

• Figure 1: Local ergodicity of the spectral fluctuations on the complex plane: The figure displays a comparison of $P(S) \equiv P_e(S, e; \Lambda_e)$ obtained by ensemble averaging with that of spectral-ensemble averaging for three different $b$-values for each ensemble (i.e BE, PE and EE). The ensemble averaging is obtained by choosing a single spacing (equivalently at a single spectral point) near $z \sim 0$ and averaging over an ensemble of $2500$ matrices each of size $N=1024$. The spectral-ensemble averaging is carried out by averaging over a few spacings ($\sim 100$) within a range $\Delta z$ around $z \sim 0$ from each matrix as well as over the ensemble of $25$ matrices; the total number of spacings is same for both the cases. As the visuals indicate, the two averages are almost analogous.
• Figure 2: Ergodicity of the spectral fluctuations on the complex plane for Ginibre ensemble : The figure displays a comparison of $P(S) \equiv P_e(S, e; \infty)$ obtained by ensemble averaging with that of spectral averaging, for different choice of single spacings i.e different $e$ values. The details of the two averaging procedure are same as in figure 1. The two averages almost overlap in each case, irrespective of the choice of the single spacing used for the ensemble averaging.
• Figure 3: Single parameter governed evolution of the fluctuation measures: The figure displays the spectral-ensemble averaged nearest neighbour spacing distribution $P(S) \equiv P_e(S, e; \Lambda_e)$ of the eigenvalues on the complex plane ($10 \%$ taken from the neighborhood for $e \sim 0$) for many $b$ values while $N$ is kept fixed ($N=1024)$ for the three ensembles, each consisting of $2500$ matrices. As can be seen from the figure, a smooth crossover from Poisson to Ginibre limit occurs for each case as $b$ and thereby $\Lambda_e$ varies. A slight shift of the peak towards left for BE cases seems to be an artefact of the unfolding issues. A good agreement of $P(S)$ for finite $\Lambda_e$ with eq.(\ref{['psfit']}) is displayed in figure \ref{['psfit1']} (the comparison displayed in a separate figure for the purpose of clarity).
• Figure 4: $P_e(S, e; \Lambda_e)$ for finite $\Lambda_e$: comparison with theoretical conjecture eq.(\ref{['psfit']}): The figure displays a comparison with eq.(\ref{['psfit']}) of the $P(S) \equiv P_e(S, e; \Lambda_e)$ many $b$ values and a fixed $N=1024$ for the three ensembles, each consisting of $2500$ matrices. As given in table 1, the fitted parameters $A, B, C$ can be recast as the functions of $\Lambda_e$. Here again the definition $\Lambda_e=\log b-C$ as the crossover parameter seemingly apply quite well.
• Figure 5: Radial and angular dependence of spacing ratios: Figure illustrates the radial $\rho(|z|)$ and angular parts $\rho(\theta)$ of the spectral-ensemble averaged nearest neighbour spacing distribution $P_z(z)$ with $z=|z| {\rm e}^{i \theta}$ on the complex plane for many $b$ values for fixed system size $N=1024$ and ensembles size $M=5000$. As shown in left panels of figure, while $\rho(|z|)$ for intermediate $b$ values lies between Poisson and Ginibre limit in both regions $|z| < 0.5$ and $|z| > 0.5$, they seemingly converge to same point in the neighborhood of $|z| \sim 0.5$. The display in right panel confirm the almost homogeneous $\rho(\theta)$ distribution in Poisson limit for each ensemble but it rapidly changes with increasing $b$. Indeed $\rho(\theta)$ approaches a minimum for $\theta=0$ as $b$ approaches Ginibre limit. This indicates that increasing level repulsion among consecutive eigenvalues with increasing $b$ causes them to lie at large angular separations.