Discontinuities in the First and Second Sound Velocities at the Berezinskii-Kosterlitz-Thouless Transition

We calculate the temperature dependence of the first and second sound velocities in the superfluid phase of a 2D dilute Bose gas by solving Landau’s two fluid hydrodynamic equations. We predict the occurrence of a significant discontinuity in both velocities at the critical temperature, as a consequence of the jump of the superfluid density characterizing the Berezinskii-Kosterlitz-Thouless transition. The key role of the thermal expansion coefficient is discussed. We find that second sound in this dilute Bose gas can be easily excited through a density perturbation, thereby, making the perspective of the measurement of the superfluid density particularly favorable.

Figure 1

Phase space density $\mathcal{D}(x)$ (upper line) and superfluid density ${\mathcal{D}}_s(x)$ (lower line), calculated from the approximate analytical formula given in [9] for $g=0.1$. The numerical values given in the same reference are also plotted as dots. The transition $x_c\approx 0.16$ is marked as a vertical line. For large $x$, corresponding to $T/T_c\to 0$ limit, both functions approach $\sim 2\pi x/g$.

Figure 2

Value of ${\kappa }_T/{\kappa }_s={\overline{c}}_p/{\overline{c}}_v=2\mathcal{P}{\mathcal{D}}^{\prime }/{\mathcal{D}}^2$ as a function of $T/T_c$ for $g=0.1$. The line is calculated using the approximate analytical expressions on the universal functions, and dots are calculated using the numerical results from [9].

Figure 3

Normalized superfluid density $n_s/n$ for $g=0.1$. The line is calculated from the approximate analytical expression, and the dots are calculated using the numerical values from [9]. Two analytical expressions valid at low and high temperatures are connected to give the curve, resulting in an unphysical kink at $T/T_c\sim 0.7$.

Figure 4

First and second sound velocities in units of the zero temperature Bogoliubov sound velocity $c_0=\sqrt{gn}/m$ with $g=0.1$. The blue and green solid lines are the first and second sound velocities calculated from solving Eq. (1). The blue and green dashed lines are the $c_{10}$ and $c_{20}$ defined in Eq. (5). The dots are the first and second sound velocities calculated from the numerical values of [9].

Figure 5

Relative contributions of first and second sound to the compressibility sum rule for $g=0.1$. The bottom line corresponds to $W_1/(W_1+W_2)$ and the top line is $W_2/(W_1+W_2)$, which are the contributions from the first and the second sound, respectively. As before, the dots are calculated using the numerical values from [9].