Benchmarkings for a semiclassical impurity solver for dynamical-mean-field theory: Self-energies and magnetic transitions of the single-orbital Hubbard model

An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to correlated-electron models. The method is based on performing an exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descent treatment of the charge part. We test this method by applying it to one- or two-site approximations to the single-band Hubbard model with different band structures, and comparing the results to quantum Monte Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states, and magnetic transition temperatures show reasonable agreement with the quantum Monte Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.