Bose-Einstein condensation temperature of a homogenous weakly interacting Bose gas in variational perturbation theory through seven loops

The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length $a$ is computed for given particle density $n$. Variational perturbation theory is used to resum the corresponding perturbative series for $\Delta ⟨{\varphi }^2⟩∕Nu$ in a classical three-dimensional scalar field theory with coupling $u$ and where the physical case of $N=2$ field components is generalized to arbitrary $N$. Our results for $N=1,2,4$ are in agreement with recent Monte Carlo simulations; for $N=2$, we obtain $\Delta T_c∕T_0=\left(1.27\pm 0.11\right)a{n}^{1∕3}$. We use seven-loop perturbative coefficients, extending earlier work by one loop order.

Figure 1

Results for $c_1\left(u_r\right)$ in VPT with fixed ${\omega }^{\prime }=0.805$ for $N=2$ for four through seven loops and with [1,1] and [2,2] Padé approximants (dashed lines; in the range of the plot, the [2,2] approximant is almost indistinguishable from the five-loop VPT curve; the [1,1] and [2,2] approximants use results through four and six loops, respectively). For $u_r\to \infty$ one obtains the flxed-${\omega }^{\prime }$results for $c_1$, indicated by the line segments at the upper right. Also indicated are the $u_r\to \infty$ limits of the Padé approximants. The loop order rises from bottom to top for both the curves at the right end of the plot and their limiting values. Also shown are the unresummed perturbative results from three through seven loops.

Figure 2

$c_1$ from VPT as a function of the number of loops $L$ for $N=1$ (solid), $N=2$ (long dashes), and $N=4$ (short dashes). In each case, the rising lines are obtained using a fixed ${\omega }^{\prime }$, while for the falling lines, ${\omega }^{\prime }$ is determined self-consistently. For comparison, the three-dimensional MC results from [13, 14] for $N=2$ and from [33] for $N=1,4$ have been included.

Figure 3

Comparison of $c_1$ from VPT as a function of the number of loops $L$ for $N=2$ with most results from other sources. Upper dots, self-consistent (sc) ${\omega }^{\prime }$. Lower dots, ${\omega }^{\prime }=0.805$. The label ${\text{LDE}}^{*}$ indicates use of a resummation method for accelerated convergence. The label ${\text{VPT}}^{*}$ indicates the inclusion of a nonzero one-loop term in Ref. 23, which is absent in the present treatment, labeled by VPT. Not shown are other approximate results of [18], the renormalization group results $c_1=4.7$ [2] and $c_1=3.42$ [11], and the canonical ensemble result $c_1=-0.93$ [9].