Phase transition of trapped interacting Bose gases and the renormalization group

We apply perturbative renormalization-group theory to the symmetric phase of a dilute interacting Bose gas which is trapped in a three-dimensional harmonic potential. Using Wilson’s energy-shell renormalization and the $\epsilon$ expansion, we derive the flow equations for the system. We relate these equations to the flow for the homogeneous Bose gas. In the thermodynamic limit, we apply our approach to examine the transition temperature of the harmonically trapped Bose gas as a function of the scattering length. The results are compared to previous studies of the problem.

Figure 1

Schematic for integrating out a high-energy shell. Shown is a one-dimensional cut through the potential along the coordinate axis $x_j$. The high-energy eigenfunctions are located around $E_{\Lambda }⪢E_0$. In Eq. (11), the trap potential $V_j$ is thus negligible compared to $E_{\Lambda }$ around the trap center. The classical turning points of the high-energy eigenfunctions are denoted by ${\xi }_{n_j}^{<}$ and ${\xi }_{n_j}^{>}$, whereas the harmonic-oscillator length $L$ indicates the extension of the trap center.

Figure 2

Flow trajectories for finite-sized systems (full curves) and their thermodynamic limit (dashed). All trajectories have been determined from the RG equations (18). They have the same initial values of $b\left(0\right)=\mathrm{10\hspace{0.17em}657}$ and $\tilde{G}\left(0\right)=1.18\times {10}^{-3}$, but different values of $M\left(0\right)$. For the finite-sized system in (a), we use $\Omega \left(0\right)\equiv 1∕{\left(L\Lambda \right)}^2={10}^{-9}$ and $M\left(0\right)$ between $2.1195\times {10}^{-6}$ and $2.128\times {10}^{-6}$. In (b), we have $\Omega \left(0\right)={10}^{-6}$ and $M\left(0\right)$ between $1.4\times {10}^{-6}$ and $2.1\times {10}^{-6}$. In the thermodynamic limit, $\Omega \left(0\right)=0$, and $M\left(0\right)$ varies between $2.1387\times {10}^{-6}$ and $2.1391\times {10}^{-6}$. The corresponding value for $a{\tilde{N}}^{1∕6}∕L$ is $6.12\times {10}^{-3}$. The quantity $\overline{G}$ is related to $\tilde{G}$ by $\overline{G}=\tilde{G}d\left(l\right)$.

Figure 3

Scaled critical temperature $T_c∕T_c^0$ as a function of the interaction parameter $a{\tilde{N}}^{1∕6}∕L$ for the harmonically trapped Bose gas. (a) and (b) display different ranges of values for $a{\tilde{N}}^{1∕6}∕L$. Bold curves: RG result after $\epsilon$ expansion according to Eqs. (23), dashed: RG result without $\epsilon$ expansion according to Eqs. (18), dotted: mean-field approximation [36]. In (a), further predictions from the literature are shown, viz., Eq. (5.1) of Ref. 37 (dot-dashed curve), Fig. 1 of Ref. 38 (dot-dot-dashed curve), and DQS-Popov and DQS-G2 (full and open circles) according to Fig. 6(b) of Ref. 39.

Figure 4

Scaled critical temperature $T_c∕T_c^0$ as a function of the interaction parameter $a{n}_p^{1∕3}$ for the homogeneous Bose gas. Bold curve: RG result after $\epsilon$ expansion according to Eqs. (23), dashed: RG result without $\epsilon$ expansion according to Eqs. (18), dotted: RG result for symmetry-broken phase [9].

Figure 5

Numerical calculation of $\rho \left(n\right)$ of Eq. (14) (full curve) and approximations $2\pi \sqrt{n}$ (dashed) and $2\pi \sqrt{n}-5.701$ (dotted).