Auxiliary-field approach to dilute Bose gases with tunable interactions

We rewrite the Lagrangian for a dilute Bose gas in terms of auxiliary fields related to the normal and anomalous condensate densities. We derive the loop expansion of the effective action in the composite-field propagators. The lowest-order auxiliary field (LOAF) theory is a conserving mean-field approximation consistent with the Goldstone theorem without some of the difficulties plaguing approximations such as the Hartree and Popov approximations. LOAF predicts a second-order phase transition. We give a set of Feynman rules for improving results to any order in the loop expansion in terms of composite-field propagators. We compare results of the LOAF approximation with those derived using the Popov approximation. LOAF allows us to explore the critical regime for all values of the coupling constant, and we determine various parameters in the unitarity limit.

Figure 1

Feynman diagrams for $\varphi$ and $\chi$ to first order in $\epsilon$. Solid and wavy lines correspond to the propagators of $\varphi$ and $\chi$. Dashed lines denote the zeroth-order ${\varphi }^{(0)}$.

Figure 2

Comparison of the predictions of the Popov and LOAF approximations regarding the zero-temperature values of the densities ${\chi }^{\prime }=A$, condensate fraction ${\rho }_0/\rho$, chemical potential $\mu$, and effective potential $V_{\mathrm{eff}}$ as a function of dimensionless parameter ${\rho }^{1/3}a$. In the case of the LOAF approximation, the normal and anomalous densities are equal, ${\chi }^{\prime }=A$, whereas in the Popov approximation we have $A=\lambda {\rho }_0$. Note that there is no effective potential in the Popov approximation because this approximation is not derivable from an action.

Figure 3

Normal density ${\chi }^{\prime }$ and anomalous density $A$ from the LOAF and Popov approximations. The comparison between the LOAF and Popov approximations is carried out for ${\rho }^{1/3}a=1$, ${\rho }^{1/3}a=0.4$, and ${\rho }^{1/3}a=0.05$. $T_c$ and $T^{\star }$ indicate vanishing condensate density ${\rho }_0$ and anomalous density $A$, respectively. The Popov approximation leads to a first-order phase transition, whereas LOAF predicts a second-order phase transition. We have that $T_c=T^{\star }$ in the Popov approximation but not in LOAF. In LOAF, ${\chi }^{\prime }$ and $A$ are equal for $T⩽T_c$.

Figure 4

Condensate fraction ${\rho }_0/\rho$ from the LOAF and Popov approximations. Similar to Fig. 3. Because at $T_c$ the Popov approximation and noninteracting dispersion relations are the same, Popov approximation does not change $T_c$ relative to the noninteracting case. LOAF increases $T_c$.

Figure 5

Chemical potential $\mu$ from the LOAF and Popov approximations. Similar to Fig. 3.

Figure 6

LOAF predictions for the critical regime, $T=T_c$, as a function of ${\rho }^{1/3}a$. (top) Relative change in $T_c$ with respect to the noninteracting critical temperature $T_0$. (bottom) Critical value of the normal and anomalous densities, ${\chi }_c^{\prime }=A_c$. The insets illustrate the ${\rho }^{1/3}a$ dependence of $\Delta T_c/T_0=(T_c-T_0)/T_0$ and ${\chi }_c^{\prime }$ in the weak-coupling regime.

Figure 7

Feynman diagrams for the second derivatives of $\Gamma$. Solid and wavy lines correspond to the propagators of $\varphi$ and $\chi$. Dashed lines denote $\varphi$.