Emergent Wigner-Dyson Statistics and Self-Attention-Based Prediction in Driven Bose-Hubbard Chains

TL;DR

The paper addresses predicting chaotic spectral statistics in a driven Bose-Hubbard lattice without external disorder. It introduces modulable hidden variables and a Gaussian-based self-attention framework to map the many-body spectrum into a high-dimensional feature space and implement a thermodynamic feedback loop that matches variance. The main finding is that the driven, strongly interacting regime with $U \gg J$ yields Wigner-Dyson-like statistics, with an effective Dyson index that interpolates between GUE and GSE, and a variance-stabilizing RG flow connecting the Hilbert-space cutoff to chaos. The approach provides automatic spectrum prediction with high accuracy and reveals non-Fermi-liquid-like behavior in strongly interacting bosonic phases.

Abstract

We propose an algorithm based on modulable hidden variables and adaptive step lengths, inspired by heuristic statistical physics and the replica method, to study the effect of mutual correlations and the emergent Wigner-Dyson distribution in a driven many-body system. Specifically, we apply this method to the driven Bose-Hubbard chain to illustrate the competition between coherent driving, hopping, and on-site interactions. Unlike the asymptotic high-dimensional statistics regime in random systems, here the randomness emerges dynamically from the interplay between the driving field $F$ and the nonlinearity $U$. We reveal the relation between the UV cutoff of the effective momentum space (related to the particle number truncation) and the system's chaotic behavior (SYK-like features). The inverse of the effective Hilbert space cutoff, acting as an essential degree-of-freedom (DOF) other than the bosonic modes, relates to the distribution and statistical variance of the interaction-induced coupling. By mapping the 1D chain to a high-dimensional feature space via a Gaussian-based self-attention mechanism, we replace the direct diagonalization of the full Hamiltonian with a predictive algorithm where the flavor number $O(M)$ is determined by the local potential difference generated by the Kerr non-linearity $\frac{1}{2}U$. The resulting system follows statistics intermediate between the Gaussian Symplectic Ensemble (GSE) and Gaussian Unitary Ensemble (GUE), contingent on the ratio $U/J$. Our algorithm allows for the automatic optimization and prediction of the resulting many-body spectrum to arbitrary accuracy, revealing non-Fermi liquid-like behavior in the strongly interacting bosonic phase.

Figures (5)

• Figure 1: Simulation result base on Algorithm 1, where we show the weight as a function of the hidden variable (the interaction energy). One can notice that the highest energy always corresponds to the lowest number of states (faintest color of blue). We set $L=4$ and LearnRate=0.1. The energy axes are expressed in units of the hopping amplitude $J$.
• Figure 2: The same with Fig.1 but we set $L=6$ and LearnRate=0.8.
• Figure 3: Visualization of the adaptive thermodynamic feedback potential $E_i(M)$ acting on the spectral weights. The feedback potential modulates the probability distribution based on the instantaneous variance mismatch $\Delta_{\sigma^2} = \sigma_{target}^2 - \sigma_t^2$. The red curve corresponds to the heating regime. When the current variance is insufficient ($\Delta_{\sigma^2} > 0$), the algorithm applies an inverted Gaussian potential (positive parabola). This potential is maximal at the spectral edges (far from the mean $\mu_t$), amplifying the weights of the tail states via $w_{new} \propto w_{old} e^{+E_i}$, thereby expanding the distribution width. The blue curve corresponds to the cooling regime. When the current variance exceeds the target ($\Delta_{\sigma^2} < 0$), a confining harmonic potential (negative parabola) is applied. This imposes a strong negative feedback $E_i \ll 0$ at the boundaries exponentially suppressing the tail weights via $w_{new} \propto w_{old} e^{-|E_i|}$ and forcing the distribution to contract towards the mean, reducing the variance until equilibrium with the target Wigner-Dyson statistics is re-established. The gray dashed line indicates the current weighted mean energy $\mu_t$, where the feedback force vanishes.
• Figure 4: The same as Fig.2 but with $\sigma_{noise}=0.08$.
• Figure 5: particle number correlations for $L=6$ with $\sigma_{noise}=0$ (a) and $\sigma_{noise}=0.08$ (b). Nearest neighbor spacing distribution $P(s)$ for $\sigma_{noise}=0$ (c) and $\sigma_{noise}=0.08$ (d).