Signatures of a universal jump in the superfluid density of a two-dimensional Bose gas with a finite number of particles

We study, within the classical fields approximation, a two-dimensional weakly interacting uniform Bose gas of a finite number of atoms. By using a grand canonical ensemble formalism we show that such systems exhibit, in addition to the Berezinskii-Kosterlitz-Thouless (BKT) and thermal phases, an intermediate region. This intermediate region is characterized by a decay of current-current correlations at low momenta and by an algebraic decay of the first-order correlations with an exponent being larger than the critical value predicted by the BKT theory. The density of the superfluid fraction at the temperature which separates the BKT phase from the intermediate region approaches the one found by Nelson and Kosterlitz for two-dimensional superfluids while the number of atoms is increased.

Figure 1

First-order correlation function, $g^{\left(1\right)}\left(x\right)$, representative for the BKT phase (upper frame), the intermediate region (middle frame), and the thermal phase (lower frame). Black solid lines are the outcome of our numerical procedure and blue dashed lines stand for the best algebraic fits, whereas the red dotted curves represent the best exponential fits. Evidently, for the BKT and intermediate phases the algebraic decay behavior works better. However, for the case of the intermediate region the exponent is larger than the critical value given by 0.25, which is characteristic of this region. Above the transition temperature the exponential decay of $g^{\left(1\right)}\left(x\right)$ becomes clear (lower frame). Insets (solid lines) show the behavior of the current-current correlations in the momentum space. The onset of the intermediate region is defined by the temperature at which the current-current correlations start to diminish at low momenta. The dashed lines represent the standard deviation of ${\left(j_{\mathbf{k}}\right)}_x{\left(j_{\mathbf{k}}\right)}_y^{★}$. For all frames $g=0.35{\hslash }^2/m$.

Figure 2

Main frame: exponent $\eta \left(T\right)$ (solid line) determined by fitting the first-order correlation function obtained numerically to the algebraic fall off formula as a function of temperature. The solid line shows the results for $\mu \approx 2500$ ($N\approx 6000$ and atomic density of $66.2\mu {\text{m}}^{-2}$) and $g=0.35$. The dashed line comes from the formula $\eta \left(T\right)=m^2{k}_{B}T/2\pi {\hslash }^2{\varrho }_s\left(T\right)$ [6], with the superfluid density obtained numerically. Vertical dashed line is the transition temperature, $T_{tr}$, to the intermediate region (marked by a shaded area), whereas the solid one shows the transition to the BKT phase in the thermodynamic limit [59]. Horizontal dashed line is the value of the critical exponent. Inset: exponent $\eta \left(T\right)$ at the transition temperature, $T=T_{tr}$, to the intermediate region as a function of the number of atoms. Errors in the main panel and in the inset are smaller than the size of the symbols used. Clearly, $\eta \left(T_{tr}\right)$ approaches the critical value of $0.25$ [6] with increasing number of atoms.

Figure 3

Superfluid fraction as a function of temperature for a two-dimensional weakly interacting Bose gas with the interaction strength $g=0.15{\hslash }^2/m$ (upper frame) and $g=0.35{\hslash }^2/m$ (lower frame). There are three sets of curves in each frame corresponding to different numbers of atoms in the system. Going from the right to the left (from top to bottom in the legend) the number of atoms increases as depicted in figures. Clearly, the intermediate region shrinks when the number of atoms grows sustaining the atomic density (here, equal to $66.2\mu {\text{m}}^{-2}$), i.e., the system moves towards the thermodynamic limit. The shaded areas depict the intermediate regions for the highest number of atom cases. The vertical solid black lines show the critical temperature for the infinite system calculated from the formula $T^{\infty }=2\pi {\hslash }^{2}n/m{k}_{B}ln(\xi {\hslash }^2/mg)$ [59], whereas the dashed color ones represent the transition, $T_{tr}$, temperatures found numerically and corresponding to the case of the maximal number of atoms considered. Note that both temperatures, obtained by different approaches, are already very close to each other. Also note that for larger number of atoms the curves become steeper.

Figure 4

Superfluid fraction, ${\rho }_s\left(T_{tr}\right)$, at the transition temperature $T_{tr}$ (symbols). The transition temperature is determined from the criterion based on the behavior of the current-current correlations at low momenta. For a given interaction strength ($g=0.7$—green squares, $g=0.35$—blue circles, and $g=0.15$—red diamonds), data for the system with different numbers of atoms (as in Fig. 3) are shown. The solid line is the linear relation between the superfluid density at the BKT transition and the BKT transition temperature, ${\rho }_s\left(T_c\right)=(m^2{k}_B/{\hslash }^2)2T_c/\pi$, as proved by Nelson and Kosterlitz [6] and experimentally verified in [9, 10, 11, 12]. As argued in the text, the transition temperature, $T_{tr}$, approaches the critical one when the number of atoms is increased. In the figure both temperatures are set equal.

Figure 5

First-order correlation function, $g^{\left(1\right)}\left(x\right)$, as in Fig. 1 but on a log-log plot.