Phase transition of interacting Bose gases in weak periodic potentials

We study the condensation of a homogeneous interacting Bose gas in the presence of a superimposed weak periodic potential. In contrast to, e.g., a harmonic trap, the periodic potential does not reduce the significance of long-wavelength fluctuations and thus influences the phase transition in a more profound way. We present general thermodynamic descriptions of the system above condensation in terms of mean-field as well as renormalization-group theory. Our approaches are based on describing the interplay between the effects of the interactions and the potential by means of a renormalization of the potential amplitude. These theories are then used to study the critical chemical potential and particle density and their dependence on the properties of the applied potential. A main result of our investigations concerns the characteristic nonperturbative momentum scale $k_c$ of the critical interacting Bose gas. It is shown that the existence of this momentum scale becomes macroscopically manifest in the behavior of the critical thermodynamic properties.

Figure 1

Function ${\overline{f}}^{\left(\mathrm{id}\right)}\left(x\right)$ characterizing the shift in the critical particle number for the ideal Bose gas, together with the approximations obtained from Eqs. (16) (dashed) and (17) (dotted). The asymptotic expansion (16) was calculated up to order $x^{-3}$.

Figure 2

Function $\overline{v}\left(x\right)$ characterizing the shift in the critical chemical potential for the interacting Bose gas. Full curve: $\overline{v}\left(x\right)$ for $\widehat{a}={10}^{-5}$ where MF and RG results coincide. Dashed (dotted) curve: $\overline{v}\left(x\right)$ for $\widehat{a}={10}^{-2}$ calculated with the RG (MF) approach. If potential renormalization is not taken into account, $\overline{v}\left(x\right)$ becomes equal to ${\overline{f}}^{\left(\mathrm{id}\right)}\left(x\right)$ (dash-dotted curve).

Figure 3

Function ${\overline{f}}^{\left(\mathrm{MF}\right)}\left(x\right)$ describing the shift in the critical density for the interacting Bose gas in the mean-field approximation. Full (dash-dotted) curve: ${\overline{f}}^{\left(\mathrm{MF}\right)}\left(x\right)$ for $\widehat{a}={10}^{-5}$ $(\widehat{a}={10}^{-2})$.

Figure 4

Function ${\overline{f}}^{\left(\mathrm{RG}\right)}\left(x\right)$ describing the critical-density shift in the RG calculation without renormalization of the potential amplitude. The curves show ${\overline{f}}^{\left(\mathrm{RG}\right)}\left(x\right)$ for $\widehat{a}={10}^{-5}$ (full), ${10}^{-4}$ (dashed), ${10}^{-3}$ (dotted), and ${10}^{-2}$ (dash-dotted).

Figure 5

Function ${\overline{f}}^{\left(\mathrm{RG}\right)}\left(x\right)$ characterizing the shift in the critical particle density due to the periodic potential in the presence of interactions. The curves show ${\overline{f}}^{\left(\mathrm{RG}\right)}\left(x\right)$ for $\widehat{a}={10}^{-5}$ (full), ${10}^{-4}$ (dashed), ${10}^{-3}$ (dotted), and ${10}^{-2}$ (dash-dotted), and are calculated following the RG approach of Sec. 4.