Emergent Wigner-Dyson statistics and self-attention-inspired prediction in driven Bose-Hubbard chains.

We propose an algorithm based on modulable hidden variables and adaptive step lengths, inspired by statistical physics and renormalization group principles, to study the emergent Wigner-Dyson level statistics in a driven many-body system. We apply this method to the boundary-driven Bose-Hubbard chain to illustrate the competition between coherent driving, hopping, and on-site interactions. Unlike random matrix models where chaos is induced through quenched disorder, here ergodicity emerges intrinsically from many-body configuration hybridization induced by the interplay between the driving field F and the nonlinearity U. The resulting system follows statistics approaching the Gaussian orthogonal ensemble (GOE) limit. Using a self-attention-inspired Gaussian relevance map, the algorithm organizes the discrete site-occupation features and iteratively redistributes a coarse-grained spectral measure. This reduces the computational overhead relative to calculating the full eigenspectrum, although the truncated Hilbert-space dimension D itself still grows exponentially with system size. The deterministic feedback stabilizes the macroscopic spectral variance, while a separate stochastic step restores the local fluctuations required for a GOE-like surrogate spacing distribution. The reconstruction loop therefore does not require a full eigendecomposition, while exact diagonalization is retained here only for finite-size validation.