Critical dynamics of the superfluid phase transition in Model F

Abstract

We describe numerical simulations of the critical dynamics near the superfluid phase transition. The calculations are based on an implementation of a stochastic hydrodynamic theory known as model F in the classification of Hohenberg and Halperin. This theory is expected to describe dynamic scaling near the lambda transition in liquid $^4$He, Bose-Einstein condensation in ultracold atomic gases, and the superfluid transition in the unitary Fermi gas. Our simulation is based on a Metropolis algorithm previously applied to the critical endpoint of the liquid-gas phase transition in ordinary fluids. In the model E truncation of model F we obtain the expected dynamical exponent $z\simeq 3/2$. We observe the emergence of a propagating second sound mode at the phase transition. The second sound diffusivity $D_s$ is consistent with the scaling relation $D_s\sim ξ^{x_κ}$, where $ξ$ is the correlation length and $x_κ=1/2$.

Figures (10)

• Figure 1: Left panel: The binder cumulant $U_4$ as a function of temperature $m^2$ for several values of the lattice size $L$. The universal value, $U_4^*$, is shown by the dashed black line. Right panel: The value of $m^2$ where $U_4$ crosses the critical value as a function of the scaled inverse lattice size $L$. The star indicates the infinite volume extrapolation, $m_c^2 = -3.1046(3)$.
• Figure 2: The magnetic equation of state and susceptibility as a function of the scaling variable $y$. The dots depict the values of $y$ at which we have simulated the dynamics of the system. Here, $y_{\rm pc} \approx 1.671$ is the position of the susceptibility peak.
• Figure 3: Order parameter correlation function $G_\sigma(t,0)$ at the critical point $m^2=m_c^2$ and $H=0$ for two different values of the lattice size, $L=32$ and $L=40$. The correlation function is plotted as a function of the scaling variable $t/L^z$ for two different values of the dynamic exponent, $z=2$ (left panel) and $z=3/2$ (right panel). The best fit value of $z$, based on the scaling behavior at $G_\sigma(t)/\chi_\sigma = 0.3$ is $z = 1.51 \pm 0.14$.
• Figure 4: Conserved density correlation function $G^\psi(t,k = 2\pi /L)$ at the critical point $m^2=m_c^2$ and $H=0$ for two different values of the lattice size, $L=32$ and $L=40$. The correlation function is plotted as a function of the scaling variable $t/L^z$ for two different values of the dynamic exponent, $z_\psi=2$ (left panel) and $z_\psi=3/2$ (right panel). The best fit value of $z_\psi$, based on the scaling behavior at $G^\psi(t)/\chi_\psi = 0.3$, is $z_\psi = 1.715 \pm 0.026$.
• Figure 5: Goldstone boson correlation function $G_\pi(t,0)$ at the critical temperature $m^2=m_c^2$ for different values of the external field $H$. We plot the correlation function as a function of the scaling variable $th^{z\nu_c}$ for two different values of the dynamic exponent, $z=2$ (left panel) and $z=3/2$ (right panel). Here, $h=H/H_{\it ref}$ with $H_{ref} = 0.004$ and the lattice size is $L=40$.