Phase transition of interacting Bose gases in general power-law potentials

We investigate the phase transition of interacting Bose gases in general power-law traps in the thermodynamic limit. Using energy-shell renormalization and the $\epsilon$ expansion, we evaluate the partition function for the uncondensed gas phase within a renormalization-group framework. This approach allows a unified description of homogeneous as well as inhomogeneous and anisotropic systems. Results for the critical temperature are compared to mean-field theory as well as to a local-density approximation based on renormalization-group theory for the homogeneous Bose gas. This comparison indicates the consistency of our approach. We also make suggestions for an optimized trap design in experiments that measure the transition temperature.

Figure 1

Relative shift $t$ of critical temperature as a function of $\eta$ at fixed $q=a∕ {\lambda }_T={10}^{-5}$ (a) and ${10}^{-2}$ (b). Bold curves: RG results with $\epsilon$ expansion. Dashed curves: mean-field LDA. Dotted curves: linear mean-field approximation $D_1\left(\eta \right)q$ [cf. Eqs. (30, 31)].

Figure 2

Relative shift $t$ of the critical temperature as a function of $a∕{\lambda }_T^0$ for $\eta =0.5,0.6,0.75,1.0,1.5$, and 2.0 (from top to bottom). Shown is the RG calculation with $\epsilon$ expansion.

Figure 3

Comparison between the RG method of Sec. 3 and RG-LDA. (a)–(c) display the relative shift $t$ of the critical temperature as a function of $a∕{\lambda }_T^0$ for $\eta =0.75$ (a), $\eta =1.2$ (b), and $\eta =2.0$ (c). (d) shows $t\left(\eta \right)$ for $q={10}^{-3}$. Bold curves: RG method of Sec. 3. Dotted curves: RG-LDA. Dashed curves: MFT. Dot-dashed curves: modified MFT according to [13]. All RG calculations use the $\epsilon$ expansion.