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

We derive exact 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 expressions for the homogeneous case in 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 moment relations to the calculated moments of spectral functions determined from a variety of different numerical approximations and use them to benchmark their accuracy.

Figure 1

Local density of states for the one-dimensional Bose Hubbard model in the Mott-insulating phase with $n=1$. The parameters are $t/U=0.05$ and $\mu /U=0.5$. The energies are measured in units of $U$. We compare the variational cluster approach with two different broadenings (red and blue) to the quantum Monte Carlo plus maximum entropy approach (purple) and to the density matrix renormalization group approach (orange). The inset zooms in on the region just above $2U$, where the variational cluster approach has some structure which is needed to get high precision to the different moment sum rules. The data for the other two methods is cutoff before this frequency.

Figure 2

Spectral moments of the retarded Green's function as a function of momentum for the one-dimensional Bose Hubbard model in the Mott-insulating phase with $n=1$. The parameters are $t/U=0.05$ and $\mu /U=0.5$. We compare the exact result evaluated with the expectation values from Table (black) to the variational cluster approach (red dashed line), to the quantum Monte Carlo plus maximum entropy approach (purple), and to the density matrix renormalization group approach (orange). Panel (a) is the zeroth moment, panel (b) is the first moment, panel (c) is the second moment, and panel (d) is the third moment. The accuracy of the VCA is so good that one cannot see any deviation from the exact result in panels (a) and (b). The results of quantum Monte Carlo and density matrix renormalization group have higher errors.

Figure 3

Zeroth (a) and first (b) spectral moments of the retarded self-energy as a function of momentum for the one-dimensional Bose Hubbard model in the Mott-insulating phase with $n=1$. The parameters are $t/U=0.05$ and $\mu /U=0.5$, with energies measured in units of $U$. We compare the exact result (black) to the variational cluster approach (red dashed line), to the quantum Monte Carlo plus maximum entropy approach (purple), and to the density matrix renormalization group approach (orange). Deviations are visible for all approximations.

Figure 4

Spectral moments of the retarded Green's function as a function of momentum for the two-dimensional Bose Hubbard model on a square lattice in the Mott-insulating phase with $n=2$. The parameters are $t/U=0.03$ and $\mu /U=1.45$. We compare the exact moment values evaluated through second order in the strong-coupling expansion (black) to the variational cluster approach (red dashed line) and to the quantum Monte Carlo plus maximum entropy approach (purple). Panel (a) is the zeroth moment, panel (b) is the first moment, panel (c) is the second moment, and panel (d) is the third moment. The accuracy of the VCA is so good that one cannot see any deviation from the exact result in panels (a) and (b). The quantum Monte Carlo results have higher errors. The symmetry lines over which the data is plotted are shown with the schematic triangle.

Figure 5

Zeroth (a) and first (b) spectral moments of the retarded self-energy as a function of momentum for the two-dimensional Bose Hubbard model on a square lattice in the Mott-insulating phase with $n=2$. The parameters are $t/U=0.03$ and $\mu /U=1.45$, with energies measured in units of $U$. We compare the exact moment values evaluated in strong coupling (black) to the variational cluster approach (red line), and to the quantum Monte Carlo plus maximum entropy approach (purple). Deviations are visible for all approximations, indicating the self-energy is not determined as accurately here.