Numerical calculation of spectral functions of the Bose-Hubbard model using bosonic dynamical mean-field theory

We calculate the momentum dependent spectral function of the Bose-Hubbard model on a simple cubic lattice in three dimensions within the bosonic dynamical mean-field theory (B-DMFT). The continuous-time quantum Monte Carlo method is used to solve the self-consistent B-DMFT equations together with the maximum entropy method for the analytic continuation to real frequencies. Results for weak, intermediate, and strong interactions are presented. In the limit of weak and strong interactions very good agreement with results obtained by perturbation theory is found. By contrast, at intermediate interactions the results differ significantly, indicating that in this regime perturbative methods fail to describe the dynamics of interacting bosons.

Figure 1

Comparison of the error of the real part of the self-energy calculated with the conventional (red) and the improved (blue) method. Parameters are $U=20,\mu =0.23U$, and $T=1$. The improved method significantly reduces the error by making it linearly, rather than quadratically, dependent on frequency.

Figure 2

Phase diagram of the Bose-Hubbard model on the simple cubic lattice obtained with the static Fisher mean-field (MF) theory [41] and the B-DMFT, which is solved by CT-QMC and LCE. Only the first lobe corresponding to $\langle n\rangle \approx 1$ is plotted. Circles, diamonds, and triangles represent sets of parameters at which calculations for strong and intermediate interactions, respectively, were performed. Circles (orange): strong-coupling regime; triangles (violet) and diamonds (blue): intermediate interaction regime.

Figure 3

(a) Momentum resolved spectral functions at $T=0.5,U=0.25$, and $\mu =-2.875$ along the symmetry lines in the first Brillouin zone of a simple cubic lattice. (b) Results obtained by B-DMFT (red) and by the Bogoliubov approximation (black) for the dispersion relations along the $\mathrm{\Gamma }$-R line. The Brillouin zone of a simple cubic lattice with points of special symmetry is shown in the inset of panel (b).

Figure 4

Momentum resolved spectral function obtained from analytic continuation with MaxEnt of the CT-QMC data. Top panel: Mott insulating phase at $\mu =0.4U$ and $U=50$; bottom panel: superfluid phase at $\mu =0.9U$ and $U=50$. Spectral functions are plotted along the symmetry lines in the first Brillouin zone for a simple cubic lattice.

Figure 5

Dispersion relation as calculated within the B-DMFT and the static Fisher mean-field (MF) theory [41], respectively. Top panel: Mott insulating phase at $\mu =0.4U$ and $U=50$; bottom panel: superfluid phase at $\mu =0.9U$ and $U=50$. Only the three dominant bands are plotted.

Figure 6

Spectral function $A\left(\omega \right)$ obtained with LCE (dotted line) and CT-QMC (full lines) at $T=1$. Padé approximants were used for the data obtained with the LCE. Left panel: Calculations at $\mu =0.4U$ for different values of $U$; right panel: calculations at $U=15.5$ for different values of $\mu$. For $U=12.5,\mu =0.4U$ and $U=15.5,\mu =0.2U$ the system is in the superfluid phase; for reference see Fig. 2.

Figure 7

Momentum resolved spectral function $A(\mathbf{k},\omega )$ in the first Brillouin zone of a simple cubic lattice at $\mu =0.4U$ and $T=1$. Interaction strengths are the same as in Fig. 6, left panel.

Figure 8

Dispersion relation of correlated bosons for intermediate interaction strengths obtained within the B-DMFT. The momentum resolved spectral function $A(\mathbf{k},\omega )$ presented in Fig. 7 was employed together with Eq. (15). The results are compared to those of the static Fisher mean-field (MF) theory [41] except for the value $U=14$, where the B-DMFT finds the system to be in the Mott insulating phase, while it is in the superfluid phase according to the Fisher mean-field theory.

Figure 9

(a) Momentum resolved spectral function at $T=0.5,U=5$, and $\mu =-2.625$ along the symmetry lines in the first Brillouin zone of a simple cubic lattice. (b) Results obtained by the B-DMFT (red) and by the Bogoliubov approximation (black) for the dispersion relations along the $\mathrm{\Gamma }$-R line.