Equality of phase-quenched and physically reweighted gap-ratio averages

Determine whether the ensemble average of the low-lying singular-value adjacent gap ratio in the phase-quenched reference ensemble coincides with its physically reweighted ensemble average for the complex-weight Haldane-Hubbard model studied with determinant quantum Monte Carlo.

Background

The Haldane-Hubbard model has generally complex determinant weights, so the determinant quantum Monte Carlo calculation samples the phase-quenched reference measure defined by the magnitude of the configuration weight, while physical averages require reweighting by the determinant phase. The paper finds that the low-lying singular-value adjacent gap ratio exhibits a transition-sensitive crossover in the phase-quenched ensemble, near previously reported estimates of the Chern-insulator-to-Mott-insulator transition.

Whether this phase-quenched statistic faithfully represents the physically reweighted statistic depends on the covariance between the gap ratio and the determinant phase relative to the average phase. The authors derive an exact covariance identity for the reweighting difference and report that the correction is small within the system sizes and interaction regimes investigated, but they do not establish equality or asymptotic equivalence in general.