Exact solution of the Bose-Hubbard model on the Bethe lattice

The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546 (1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.

Figure 1

A portion of a Bethe lattice of connectivity $c=4$, approximation of a square lattice.

Figure 2

Example of an Ising tree model on 7 vertices. The definition of $Z_{2\to 1}$ and its recursive computation reads here: $Z_{2\to 1}\left({\sigma }_2\right)={\sum }_{{\sigma }_3,\dots ,{\sigma }_7}{e}^{\beta h\left({\sigma }_2+{\sigma }_3+\cdots +{\sigma }_7\right)+\beta J\left[{\sigma }_2\left({\sigma }_3+{\sigma }_4\right)+{\sigma }_4\left({\sigma }_5+{\sigma }_6+{\sigma }_7\right)\right]}=e^{\beta h{\sigma }_2}{\sum }_{{\sigma }_3,{\sigma }_4}{Z}_{3\to 2}\left({\sigma }_3\right)Z_{4\to 2}\left({\sigma }_4\right)e^{\beta J{\sigma }_2\left({\sigma }_3+{\sigma }_4\right)}$.

Figure 3

Pictorial representation of Eq. (14). The bubbles on the first panel represent subtrees of the graph; their effect on the spins ${\sigma }_1,\dots ,{\sigma }_{c-1}$ is summarized by ${\eta }_{\text{cav}}$, represented as a bold arrow in the second panel. Tracing over these $c-1$ spins leads to the third panel.

Figure 4

Illustration of Eq. (15); the local magnetization of one site is computed by taking into account all the $c$ neighbors.

Figure 5

Example of the sampling process from the law $P\left(\mathit{y}\mid {\mathit{y}}_1,\dots ,{\mathit{y}}_{c-1}\right)$ defined in Eq. (31) for $c=4$. The $c-1$ incoming hopping trajectories $\left({\mathit{y}}_1,{\mathit{y}}_2,{\mathit{y}}_3\right)$ impose jumps in ${\tau }_1^{\text{in}}<{\tau }_2^{\text{in}}<{\tau }_3^{\text{in}}<{\tau }_4^{\text{in}}$, a particle arriving on the central site at the first three imaginary times, leaving at the fourth one, corresponding to the symbols in the upper part of the rightmost figure. A trajectory $n\left(\tau \right)$ respecting these constraints is shown; it contains two other jumps at ${\tau }_1^{\text{out}}<{\tau }_2^{\text{out}}$, associated to hopping events in $\mathit{y}$. The symbols on the bottom are reversed: particles leaving the central site will arrive on the next level of the recursive equation.

Figure 6

(Color online) (Main panel) $⟨n⟩$ and $⟨a⟩$ as functions of $\mu /U$ for $J/U=0.02$, at $\beta J=1$ and $N_{\text{cut}}=6$ for Bethe lattices of connectivity $c=4$. Full lines are obtained by joining points measured every $\Delta \mu /U\sim 0.05$, except close to the critical points where $\Delta \mu /U\sim 0.001$. (Inset) Blow up of the region close to the first SF-MI transition that happens at ${\mu }_c=0.0648U$. Here $⟨a⟩$ is plotted as a function of $\delta =\left({\mu }_c-\mu \right)/U$ (circles). The full line is a fit to $⟨a⟩\sim A{\delta }^{0.5}$ with $A=2.9475$.

Figure 7

(Main panel) Total energy, $e=⟨H⟩/N$, as a function of $\mu /U$ for $J/U=0.02$, at $\beta J=1$ and $N_{\text{cut}}=6$ for Bethe lattices of connectivity $c=4$. (Inset) Kinetic energy $e_K$ as a function of the chemical potential for the same values of the parameters.

Figure 8

(Color online) “Zero-temperature” phase diagram of the Bose-Hubbard model on the Bethe lattice in the $\left(J/U,\mu /U\right)$ plane at connectivity $c=4$ (left panel) and 6 (right panel). The results obtained within the cavity method (black lines and filled circles) are compared to Monte Carlo simulation (blue lines and open diamonds) on the square (data from Ref. 17) and cubic (data from Ref. 18) lattices, and to the prediction of the mean-field approach (red dashed curves) and of B-DMFT (brown full curve in right panel, data from Ref. 24).

Figure 9

(Color online) Interaction dependence of the transition temperature at unit filling. The result obtained on the $c=6$ Bethe lattice (black lines and filled circles) is compared to Monte Carlo simulation (blue lines and open diamonds) on the cubic lattice (data from Ref. 18), and to the prediction of the mean-field approximation (red dashed curves).

Figure 10

(Color online) Green’s function $G\left(\tau \right)$ as a function of the imaginary time at $c=4$. (Left panel) Results in the MI phase, at $J/U=0.0523$, $\mu /U=0.39$, $N_{\text{cut}}=4$, and $\beta J=2$ (black dashed-dotted curve), 4 (red dashed curve), and 6 (blue continuous curve). Also reported is the mean-field result for the same parameters (black dotted curve). (Right panel) Results in the SF phase, at $J/U=0.0555$, $\mu /U=0.39$, $N_{\text{cut}}=6$, and $\beta J=2$ (black dashed curve) and 4 (blue curve). Also reported is the long-time limit of $G\left(\tau \right)$, which corresponds to the value of ${⟨a⟩}^2$ for these values of the parameters.

Figure 11

(Color online) (Main Panel) Green’s function $G\left(\tau \right)$ as a function of the rescaled imaginary time $\tau U$ for $c=4$ in the MI phase, close to the tip of the first lobe at $\mu /U=0.39$, $N_{\text{cut}}=4$, and $\beta J=6$, at different values of $J/U$ (approaching the transition to the superfluid, from bottom to top). Also reported is the mean-field result for the same parameters (black dashed curve). (Inset) Particle excitation gap $E_p/U$ is plotted as a function of $J/U$ (see text for details); the dashed black line is the mean-field result $E_p/U=1-\mu /U$, the vertical line marks the transition to the superfluid.

Figure 12

(Color online) (Main Panel) One-particle density matrix ${\rho }_d=⟨a_0^{†}{a}_d⟩$ at $c=4$, fixed $\mu /U=0.39$, $\beta J=2$, $N_{\text{cut}}=4$, and different values of $J/U$ crossing the tip of the first MI lobe in Fig. 8. In the MI phase ${\rho }_d$ decays to zero, while in the SF phase it decays to ${\left|⟨a⟩\right|}^2$. (Inset) Correlation length as a function of $J/U$ in the MI phase. The vertical line marks the MI-SF transition.

Figure 13

Computation of the one-particle density matrix at distance $d$, ${\rho }_d=⟨a_0^{†}{a}_d⟩$.