Leading-order auxiliary-field theory of the Bose-Hubbard model

We discuss the phase diagram of the Bose-Hubbard (BH) model in the leading-order auxiliary field (LOAF) theory. LOAF is a conserving nonperturbative approximation that treats on equal footing the normal and anomalous density condensates. The mean-field solutions in LOAF correspond to first-order and second-order phase transition solutions with two critical temperatures corresponding to a vanishing Bose-Einstein condensate $T_c$ and a vanishing diatom condensate $T^{☆}$. The second-order phase transition solution predicts the correct order of the transition in continuum Bose gases. For either solution, the superfluid state is tied to the presence of the diatom condensate related to the anomalous density in the system. In ultracold Bose atomic gases confined on a three-dimensional lattice, the critical temperature $T_c$ exhibits a quantum phase transition, where $T_c$ goes to zero at a finite coupling. The BH phase diagram in LOAF features a line of first-order transitions ending in a critical point beyond which the transition is second order while approaching the quantum phase transition. We identify a region where a diatom condensate is expected for temperatures higher than $T_c$ and less than $T_0$, the critical temperature of the noninteracting system. The LOAF phase diagram for the BH model compares qualitatively well with existing experimental data and results of ab initio Monte Carlo simulations.

Figure 1

Temperature dependence of the condensate fraction $\nu n_0$, scaled auxiliary fields ${\chi }^{\prime }$ and $A$, scaled chemical potential $\mu$, and effective potential $V_{\text{eff}}$ for a Hubbard interaction parameter value $U/J=10$, at unity filling $\nu =1$. Here the temperature $T$ is scaled by $T_0$, the critical temperature of the ideal (noninteracting) Bose-Hubbard model. Solutions II(i) and II(ii) corresponds to first- and second-order phase transitions, respectively.

Figure 2

LOAF scaled critical temperatures $T_c$ and $T^{☆}$ and the corresponding scaled normal-density auxiliary fields ${\chi }_c^{\prime }={\chi }_{c\left(\text{ii}\right)}^{\prime }$, ${\chi }^{\prime }{}^{☆}$, and ${\chi }_{c\left(\text{i}\right)}^{\prime }$, and effective potentials, as a function of the Bose-Hubbard model coupling constant ($U/J$) at unity filling $\nu =1$. We note that only the critical temperature $T_c$ features a downturn with the interaction strength and leads to a quantum phase transition (QPT) for ${(U/J)}_c\approxeq 56.07$, where $T_c$ goes to zero. In addition, the normal-density auxiliary field ${\chi }_{c\left(\text{i}\right)}^{\prime }$ for solution II(i) is not defined for $T_c<T_0$, where $T_0\approxeq 5.59$ is the critical temperature of the noninteracting Bose system. Therefore, LOAF predicts a critical point (CP) at coordinates $T_{\mathrm{CP}}=T_0$ and ${(U/J)}_{\mathrm{CP}}=46.02$. For coupling values ${(U/J)}_{\mathrm{CP}}<(U/J)\le {(U/J)}_c$, the transition is second order and LOAF predicts a diatom condensate $A\ne 0$ in the absence of the usual Bose-Einstein condensate fraction for temperatures $T_c<T<T_0$. Because $T^{☆}>T_0$, the system is in the normal phase for $T>T_0$ in this region. For coupling constants $(U/J)<{(U/J)}_{\mathrm{CP}}$ the transition is first order because $V_{\mathrm{eff},\mathrm{c}\left(\mathrm{i}\right)}<V_{\mathrm{eff},\mathrm{c}\left(\mathrm{ii}\right)}$ in that region.

Figure 3

Interaction strength dependence of the zero-temperature condensate fraction $\nu n_0$ and the corresponding scaled normal-density auxiliary field ${\chi }_0^{\prime }$ at unity filling $\nu =1$.

Figure 4

Comparison of the coupling constant dependence of the LOAF critical temperature $T_c$ at unity filling $\nu =1$ with experimental [6] and quantum Monte Carlo (QMC) results [20]. The LOAF value of the critical Hubbard parameter value ${(U/J)}_c=56.076$ should be compared to the QMC critical value ${(U/J)}_c=29.34\left(2\right)$ reported in Ref. 20. LOAF also predicts a critical point at ${(U/J)}_{\mathrm{CP}}=46.02$. The solid and dashed lines indicate first- and second-order phase transitions predicted by LOAF theory, respectively. The shaded area depicts the region where a diatom condensate without the usual Bose-Einstein condensate is expected.

Figure 5

Weak-coupling limit behavior of the critical temperature $T_c$ and corresponding critical value of the scaled normal-density auxiliary field ${\chi }_c^{\prime }$ as a function of the Hubbard interaction parameter value $U/J$ at unity filling $\nu =1$.

Figure 6

Ideal gas critical temperature $T_0$, critical value of the scaled Hubbard parameter ${(U/J)}_c$, and the corresponding critical value of the scaled normal-density auxiliary field ${\chi }_c^{\prime }$ as a function of the filling factor $\nu$.