q-Gaussian Crossover in Overlap Spectra towards 3D Edwards-Anderson Criticality

Abstract

We introduce a spectral approach to characterizing the three-dimensional Edwards-Anderson spin glass. By analyzing the eigenvalue statistics of overlap matrices constructed from two-dimensional cross-sections, we identify a crossover from the Wigner semicircle law at high temperatures towards a Gaussian distribution, which is consistently attained near the spin-glass critical point. Visible for different distributions of the random coupling, the Gaussian distribution can potentially serve as a robust spectral indicator of criticality. Remarkably, the spectral density is well-described by Tsallis statistics, with the entropic index $q$ evolving from $q = -1$ (semicircle, $T=\infty$) to $q = 1$ (Gaussian) at $T_c$, revealing a statistical structure inside the paramagnetic phase. We find $q\le 1$ within numerical precision. While the local level statistics remain consistent with GOE statistics, reflecting standard level repulsion, the temperature dependence appears mainly in the global spectral density. Our results present spectral statistics as a computationally efficient complement to multi-replica correlator methods and provide a new perspective on cooperative and critical phenomena in disordered systems.

Figures (7)

• Figure 1: Scaled spectral density $P(\lambda)/P(0)$ of bulk eigenvalues for the overlap matrix $\mathcal{M}$, shown across temperatures for system size $L = 22$. Spectra are rescaled along both axes by $P(0)$ to account for the peak height and the temperature-dependent spectral width, enabling direct shape comparison across temperatures. A smooth crossover is observed from the Wigner semicircle (dashed green) at high temperatures to a Gaussian form (dashed magenta) near the critical point $\beta_c \simeq 1.05$ (i.e., $T_c \simeq 0.952$; black dashed line). Right inset illustrates the close agreement of the spectral shape with a Gaussian at $\beta=\beta_c$. Left inset: the matrix built from a single replica, $\mathcal{M}^{\prime s}_{ij}=s_{ij}$, yields a semicircle at all $T$.
• Figure 2: Temperature dependence of $D_{\rm KL}$ between the numerically obtained bulk spectrum and a variance-matched Gaussian for system sizes $L=16$ to $L=100$ (for large $L$, data are shown over a restricted temperature range). For $\beta/\beta_c \gtrsim 1$ (vertical red dashed line), $D_{\rm KL}$ decreases with $L$ and tends toward zero, indicating convergence of the bulk spectral density to a Gaussian.
• Figure 3: Bulk eigenvalue distributions $P(\lambda)$ for system size $L = 36$ at selected temperatures, empirically fitted to the $q$-Gaussian form \ref{['eqn:qgaussian']} (red curves). At high temperatures, the spectral density is close to the Wigner semicircle law ($q = -1$), and smoothly crosses over to a Gaussian distribution ($q = 1$) at criticality $T \approx T_c$ (i.e., $\beta \approx \beta_c$).
• Figure 4: Temperature dependence of the Tsallis entropic index $q$, extracted by minimizing the KL divergence between the empirical spectral density and the $q$-Gaussian fit, Eq. \ref{['eqn:qgaussian']}, for system sizes $L = 18$ to $L = 100$. The curves begin to depart from the semicircle value $q = -1$ at progressively lower $\beta$ for larger systems.
• Figure A1: Scaled spectral density $P(\lambda)/P(0)$ of bulk eigenvalues for the overlap matrix $\mathcal{M}$ of the 3D EA model with $\pm J$ couplings, shown for system size $L=24$ across temperatures $\beta=1/T$. A smooth crossover is observed from the Wigner semicircle (dashed green) at high $T$ to a Gaussian form (dashed magenta) near the critical point $\beta_c \approx 0.9075$ (i.e., $T_c \approx 1.102$). Inset: at $\beta=\beta_c$, the bulk density matches a Gaussian within statistical uncertainty.