Spectral moment sum rules for the retarded Green's function and self-energy of the inhomogeneous Bose-Hubbard model in equilibrium and nonequilibrium

TL;DR

This paper derives spectral moment sum rules for the retarded Green's function and self-energy in the inhomogeneous Bose-Hubbard model, providing tools for benchmarking spectral function approximations in equilibrium and nonequilibrium scenarios.

Contribution

It introduces new spectral moment sum rules for the Bose-Hubbard model, including nonequilibrium cases, and discusses how to evaluate and use these moments for benchmarking approximations.

Findings

Derived spectral moment sum rules for Green's function and self-energy.

Showed how to evaluate moments in the Mott-insulating phase using strong-coupling expansion.

Compared exact moments with approximate spectral functions to benchmark accuracy.

Abstract

We derive expressions for the zeroth and the first three spectral moment sum rules for the retarded Green's function and for the zeroth and the first spectral moment sum rules for the retarded self-energy of the inhomogeneous Bose-Hubbard model in nonequilibrium, when the local on-site repulsion and the chemical potential are time-dependent, and in the presence of an external time-dependent electromagnetic field. We also evaluate these moments for equilibrium, where all time dependence and external fields vanish. Unlike similar sum rules for the Fermi-Hubbard model, in the Bose-Hubbard model case, the sum rules often depend on expectation values that cannot be determined simply from parameters in the Hamiltonian like the interaction strength and chemical potential, but require knowledge of equal time many-body expectation values from some other source. We show how one can approximately evaluate these expectation values for the Mott-insulating phase in a systematic strong-coupling expansion in powers of the hopping divided by the interaction. We compare the exact moments to moments of spectral functions calculated from a variety of different approximations and use them to benchmark their accuracy.