Energy level statistics of the two-dimensional Hubbard model at low filling

TL;DR

This study numerically analyzes the energy level statistics of the two-dimensional Hubbard model at low filling, revealing that the spectral properties align with Gaussian ensemble predictions regardless of lattice size or interaction strength.

Contribution

It provides a detailed symmetry-aware spectral analysis of the Hubbard model, demonstrating universal Gaussian ensemble behavior across various parameters.

Findings

Level spacing distribution matches Gaussian ensemble statistics.

Spectral rigidity and number variance are consistent with random matrix theory.

Universality persists even at very weak coupling, indicating nonperturbative effects.

Abstract

The energy level statistics of the Hubbard model for $L \times L$ square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of the coupling strength. All known symmetries of the model (space, spin and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance $\Sigma^2$, and the spectral rigidity $\Delta_3$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to the Gaussian ensemble statistics stressing the nonperturbative nature of the Hubbard model.