Semiclassical field method for the equilibrium Bose gas and application to thermal vortices in two dimensions

We develop a semiclassical field method for the study of the weakly interacting Bose gas at finite temperature which, contrary to the usual classical field model, does not suffer from an ultraviolet cutoff dependence. We apply the method to the study of thermal vortices in spatially homogeneous, two-dimensional systems. We present numerical results for the vortex density and the vortex pair distribution function. Insight in the physics of the system is obtained by comparing the numerical results with the predictions of simple analytical models. In particular, we calculate the activation energy required to form a vortex pair at low temperature.

Figure 1

In the Bogoliubov model, minimal value of the temperature $T_{\min }$ ensuring regularity and positivity of the Glauber-$P$ distribution in a mode $\mathbf{k}$, as a function of the kinetic energy coefficient $E_k$ of the mode.

Figure 2

(Color online) In the Bogoliubov model, mean energy in a mode as a function of the mode kinetic energy coefficient $E_k$ for different values of the temperature $k_{B}T∕\mu =0$, 2, 3, 5 (from bottom to top). Solid lines: quantum result; dashed lines: semiclassical theory; dotted lines: classical field approximation.

Figure 3

For a two-dimensional Bogoliubov gas in the thermodynamic limit, normal fraction $f_n$ as a function of the temperature $k_{B}T$. Solid line: quantum prediction; dashed line: semiclassical prediction. In order to have (within Bogoliubov theory) a universal function of $k_{B}T∕\mu$, we actually plot the product of $f_n$ times $n{\xi }^2$, the healing length $\xi$ being defined by ${\hslash }^2∕m{\xi }^2=\mu$.

Figure 4

(Color online) For a two-dimensional lattice Bogoliubov gas in the thermodynamic limit, pair distribution $g^{\left(2\right)}\left(\mathbf{r}\right)$ as a function of $r$ for different values of the temperature, $k_{B}T∕\mu =0$, 2, 3, 5 (from bottom to top). Solid line: quantum result; dashed line: semiclassical approximation. In the plot, the product of $g^{\left(2\right)}-1$ with $n{\xi }^2$ is actually plotted, where $n$ is the density, and $\xi$ the healing length such that ${\hslash }^2∕\left(m{\xi }^2\right)=\mu$. For the Bogoliubov gas, this product is indeed a universal function of $k_{B}T∕\mu$ and $r∕\xi$. Here the lattice spacing is $0.07\xi$.

Figure 5

(Color online) Normal fraction $f_n$ (black with error bars) and noncondensed fraction $f_{\mathrm{nc}}$ (red, no error bars) as functions of temperature for a two-dimensional Bose gas with $N=1000$ particles on a square box of size $L$ with periodic boundary conditions. (a) Ideal Bose gas. (b) Interacting gas with a coupling constant $g_0=0.1{\hslash }^2∕m$. (c) Interacting gas with $g_0=0.333{\hslash }^2∕m$. Symbols: results of semiclassical simulations on a 64$\times$64 grid with 2000 realizations. Solid lines in (a) exact result from the canonization procedure (see text); in (b) and (c), solid lines are a guide to the eye. Dashed lines in (a) are the grand canonical predictions. The temperature is in units of the degeneracy temperature $T_d$ such that $k_B{T}_d=2\pi {\hslash }^{2}n∕m$.

Figure 6

(Color online) Pair distribution function $g^{\left(2\right)}\left(0\right)$ as a function of temperature for the same parameters as in Fig. 5. Symbols: results of the semiclassical simulations. From top to bottom, the value of the coupling constant increases from $g_0=0$ (black stars) to $g_0=0.1{\hslash }^2∕m$ (red diamonds) and $0.333{\hslash }^2∕m$ (green circles). Solid lines: for $g_0=0$ the exact result from the canonization procedure, for $g_0>0$ a guide to the eye. Horizontal dashed line: grand canonical prediction $g^{\left(2\right)}\left(0\right)=2$ for the ideal gas. The temperature is in units of the degeneracy temperature $T_d$ such that $k_B{T}_d=2\pi {\hslash }^{2}n∕m$.

Figure 7

(Color online) Mean density of positive charge vortices as a function of temperature for various interaction strengths. The parameters have the same values as in Fig. 5. (a) Linear scale, (b) logarithmic scale for the vortex density, reciprocal scale for the temperature. Symbols: results of the semiclassical simulation, $g_0=0$ (black stars), $g_0=0.1{\hslash }^2∕m$ (red diamonds), $g_0=0.333{\hslash }^2∕m$ (green circles). Solid lines: the exact canonical result (46) for $g_0=0$; prediction of the activation law model of Sec. 3C for $g_0>0$, $n_{v,+}∕n=C{e}^{-\Delta \left(T\right)∕k_{B}T}$, with the prefactor $C$ taken as a constant and fitted to the data ($C=0.134$ for $g_0=0.1{\hslash }^2∕m$ and $C=0.3355$ for $g_0=0.333{\hslash }^2∕m$). Dashed line: grand canonical result for $g_0=0$. Dot-dashed line: Bogoliubov prediction for $g_0=0$ for $T∕T_d<0.15$, essentially indistinguishable from the solid line in (a). Note that the circle with the largest value of $T_d∕T$ corresponds to $k_{B}T∕n{g}_0\simeq 1.4$, which is on the limit of the validity of both the semiclassical field method and of the simple model of Sec. 3C calculating $\Delta$.

Figure 8

(Color online) Results of the semiclassical simulations for the angular average $G_{v,+-}^{\left(2\right)}\left(r\right)$ of the pair distribution function for opposite charge vortices as a function of the distance $r$ between the two vortices. The parameters have the same values as in Fig. 5. (a) High-temperature, non Bose condensed regime, temperature $T∕T_d=2.5∕\left(2\pi \right)\simeq 0.398$, for $m{g}_0∕{\hslash }^2=0$ (black stars), 0.1 (red diamonds), 0.333 (green circles). The solid lines are a guide to the eye. Horizontal dashed lines: square of the mean vortex density $n_{v,+}^2$, showing the decorrelation at long distances. (b) Low temperature, Bose-condensed regime. The temperatures are adjusted to have similar vortex densities for the various values of $g_0=0$ [black stars, $T∕T_d=0.35∕\left(2\pi \right)\simeq 0.056$, leading to $n_{v,+}\simeq 0.28∕L^2$], $g_0=0.1{\hslash }^2∕m$ [red diamonds, $T∕T_d=0.5∕\left(2\pi \right)\simeq 0.08$, leading to $n_{v,+}\simeq 0.23∕L^2$], $g_0=0.333{\hslash }^2∕m$ [green circles, $T∕T_d=0.625∕\left(2\pi \right)\simeq 0.1$, leading to $n_{v,+}\simeq 0.23∕L^2$]. The solid lines are a guide to the eye. In both panels (a) and (b), the cross at $r=0$ gives the exact value of $G_{v,+-}^{\left(2\right)}$ for the ideal gas, obtained with the canonization procedure. The distance $r$ is in units of $L$ and $G_{v,+-}^{\left(2\right)}$ is in units of the squared particle density $n^2$.

Figure 9

(Color online) For arbitrary Monte Carlo realizations of the field with vortices, locations of the positive charge vortices (red plus symbols) and negative charge vortices (black minus symbols) in the field. Parameters as in some curves of Fig. 8: (a) $T∕T_d=2.5∕\left(2\pi \right)\simeq 0.398$ for $g_0=0$. (b) $T∕T_d=2.5∕\left(2\pi \right)\simeq 0.398$ for $g_0=0.333{\hslash }^2∕m$. (c) $T∕T_d=0.35∕\left(2\pi \right)\simeq 0.056$ for $g_0=0$. (d) $T∕T_d=0.625∕\left(2\pi \right)\simeq 0.1$ for $g_0=0.333{\hslash }^2∕m$. Note that the realizations shown in (c) and (d) are not fully typical since they contain several pairs.

Figure 10

(Color online) (a) Cut along the $x$ axis of the field ${\psi }_0$ minimizing the energy functional $U\left[\psi \right]$ over the fields with a node at the origin. Black solid line (the broadest hole): $g_0=0$, $T∕T_d=0.35∕\left(2\pi \right)\simeq 0.056$; red solid line: $g_0=0.1{\hslash }^2∕m$, $T∕T_d=0.5∕\left(2\pi \right)\simeq 0.08$; blue solid line (the narrowest hole): $g_0=0.333{\hslash }^2∕m$, $T∕T_d=0.625∕\left(2\pi \right)\simeq 0.1$. The total number of particles is $N=1000$. The dashed lines for $g_0>0$ correspond to a field value $(\mu ∕g_0{)}^{1∕2}$, where $\mu$ is the Lagrange multiplier defined in Eq. (74). (b) For a semiclassical Monte Carlo realization of the field with a single vortex pair with a small radius, comparison of the density profile of the field (green solid line) with the one of the minimizer ${\psi }_0$ of $U\left[\psi \right]$ with a node (black solid line). Here $g_0=0.333{\hslash }^2∕m$, $T∕T_d=0.5∕\left(2\pi \right)\simeq 0.08$, the vortex pair diameter is $\simeq 0.03L$ and the origin of the coordinates was redefined to match the location of the vortex pair.

Figure 11

(Color online) (a) Activation energy $\Delta \left(T\right)$ as a function of $(g_{0}n∕k_{B}T{)}^{1∕2}$ at a fixed particle density $n$ for increasing system size $L∕{\lambda }_{\mathrm{th}}=6$, 12, 24, 48 (thin solid lines, respectively, black, red, green, blue, from top to bottom; the crosses are the actually calculated values and the lines are a guide to the eye). The dashed line is the upper bound equation (78) for an infinite system size. The thick solid line is the improved upper bound discussed around Eq. (79), plotted for $(n{g}_0∕k_{B}T{)}^{1∕2}\ge 0.01$. (b) Vortex density as a function of the total particle number (for increasing system sizes) for fixed values of the density $n$ and the temperature $T=0.5T_d∕\left(2\pi \right)\simeq 0.08T_d$, and a coupling constant $g_0=0.333{\hslash }^2∕m$. Circles: semiclassical simulations. Solid line: prediction of the activation law $0.44e^{-\Delta ∕k_{B}T}$ where the numerical factor 0.44 was fitted to the data.