Nonperturbative effects on $T_c$ of interacting Bose gases in power-law traps

The critical temperature $T_c$ of an interacting Bose gas trapped in a general power-law potential $V\left(\mathbf{x}\right)={\sum }_i{U}_i{\mid x_i\mid }^{p_i}$ is calculated with the help of variational perturbation theory. It is shown that the interaction-induced shift in $T_c$ fulfills the relation $(T_c-T_c^0)∕T_c^0=D_1\left(\eta \right)\widehat{a}+D^{\prime }\left(\eta \right){\widehat{a}}^{2\eta }+\mathcal{O}\left({\widehat{a}}^2\right)$ with $T_c^0$ the critical temperature of the trapped ideal gas, $\widehat{a}$ the $s$-wave scattering length divided by the thermal wavelength at $T_c$, and $\eta =1∕2+{\sum }_i{p}_i^{-1}$ the potential-shape parameter. The terms $D_1\left(\eta \right)\widehat{a}$ and $D^{\prime }\left(\eta \right){\widehat{a}}^{2\eta }$ describe the leading-order perturbative and nonperturbative contributions to the critical temperature, respectively. This result quantitatively shows how an increasingly inhomogeneous potential suppresses the influence of critical fluctuations. The appearance of the ${\widehat{a}}^{2\eta }$ contribution is qualitatively explained in terms of the Ginzburg criterion.

Figure 1

Schematic diagram of shift in $T_c$ for a fixed small scattering length $\widehat{a}$ as a function of the potential shape parameter $\eta$. The upper curve shows the full result below second order according to Eq. (16), the dashed lower curve only displays the perturbative contribution linear in $\widehat{a}$. The nonperturbative contribution decreases fast with growing inhomogeneity.

Figure 2

Diagrams contributing to $N^{\left(2\right)}$ up to order ${\widehat{a}}^4$. The crosses denote the joining of two Green functions.

Figure 3

Diagrams contributing to $N^{\left(1\right)}(T,\mu )$ up to order ${\widehat{a}}^2$. The sunset subdiagram is taken at the external momentum $p=0$.

Figure 4

Results for $C_2^{\prime }\left(\eta \right)$ from VPT: (bold curve) fixed $\omega =0.805$; (dash-dotted) self-consistent determination of ${\omega }^{\prime }$. Note the divergence as $\eta \to 1$.

Figure 5

Shift of critical temperature as a function of the potential shape parameter $\eta$ for fixed $a∕{\lambda }_T={10}^{-4}$: (full curve) Result including all terms in Eq. (45) and $C_2^{\prime }\left(\eta \right)$ calculated with fixed ${\omega }^{\prime }=0.805$; (dash-dotted) same as full curve but with ${\omega }^{\prime }$ determined self consistently; (dashed) $C_2^{\prime }\left(\eta \right)$ omitted (i.e., “mean-field result”); (dotted) $C_1^{\prime }\left(\eta \right)$ and $C_2^{\prime }\left(\eta \right)$ omitted (i.e., only perturbative term).