Dynamical mean-field theory for the Bose-Hubbard model

The dynamical mean-field theory (DMFT), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [K. Byczuk and D. Vollhardt, Phys. Rev. B 77, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green’s function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-type regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\mu /U\text{−}\tilde{t}/U$ plane and on the $T/U\text{−}\tilde{t}/U$ plane are produced ($\tilde{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.

Figure 1

(Color online) The total boson occupation as functions of chemical potential $\mu$. [(a) and (b)]: $U=50.0$, $\tilde{t}=0$, and $T=1.0$; (c) and (d): $U=0$, $\tilde{t}=1.0$, and $T=1.0$.

Figure 2

(Color online) The self-energy of the noninteracting system as a function of Matsubara frequency. [(a) and (b)]: diagonal component ${\mathbf{\Sigma }}_1$; (c) and (d): off-diagonal component ${\mathbf{\Sigma }}_2$. They are calculated at $U=0$, $\tilde{t}=1.0$, $\mu =-2.1$, and $T=1.0$. Symbols are denoted in the figure.

Figure 3

(Color online) DMFT result (dots with eye-guiding lines) and the exact result (solid line) for the thermal excited boson number $N_e$ at $U=0$ and $\tilde{t}=1.0$. (a) as a function of $\mu$ at $T=1.0$; (b) as a function of $T$ at $\mu =-2\tilde{t}$. Inset: $N_e$ as a function of $N$ at $\mu =-2\tilde{t}$ and $T=1.5$.

Figure 4

(Color online) The total number of bosons as functions of the chemical potential $\mu$ in the atomic limit with $U=50.0$ for different temperatures. The lines are the exact results and symbols for B-DMFT results. Inset: the compressibility $\partial N_{tot}/\partial \mu$ as functions of $\mu$ obtained from B-DMFT.

Figure 5

(Color online) [(a) and (b)] The connected diagonal GF $G_{c1}$ and [(c) and (d)] off-diagonal GF $G_{c2}$ in the three phases: BEC phase (circle): $\tilde{t}/U=0.2$; normal phase (triangle): $\tilde{t}/U=0.11$; and MI phase (pentacle): $\tilde{t}/U=0.05$. All are calculated at $\tilde{t}=1.0$, $\mu /U=0.5$, and $T/U=0.05$.

Figure 6

(Color online) $N_{tot}$ and $N_0$ as functions of $\mu$. [(a) and (b)] $T/U=0.05$ for different $\tilde{t}/U$ $\left(\tilde{t}=1.0\right)$; (c) $\tilde{t}=1.0$, $\tilde{t}/U=0.05$ for different temperatures. Symbols with eye-guiding lines are denoted in the figure.

Figure 7

(Color online) The condensed boson number $N_0$ as functions of temperature $T$ for fixed total number $N_{tot}=1.5$ and different $\tilde{t}/U$ $\left(\tilde{t}=1.0\right)$. From top to bottom, $\tilde{t}/U=0.5,0.3,0.15,0.1,0.02$, respectively. Inset: $N_0$ changes with $\tilde{t}/U$ for $T=0.3$ (circle) and $T=2.0$ (pentacle), respectively. The dashed line is $N_0=1.5$. Solid lines are guiding lines.

Figure 8

(Color online) Phase diagrams. (a) in $\mu /U\text{−}\tilde{t}/U$ plane at $T/U=0.05$; (b) and (c) in $T/U\text{−}\tilde{t}/U$ plane at $\mu /U=0.5$ and $\mu /U=1.0$, respectively. Three phases, the BEC, normal, and MI phases are marked out in the figures. Lines are for eye guiding.