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

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 $\frac{1}{2}U$. 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.