Spontaneous symmetry breaking of vortex number in binary alternating current countersuperflow

In binary superfluid counterflow systems, vortex nucleation arises as a consequence of hydrodynamic instabilities when the coupling coefficient and counterflow velocity exceed the critical value. When dealing with two identical components, one might naturally anticipate that the number of vortices generated would remain equal. However, through the numerical experiments of the holographic model and the Gross-Pitaevskii equation, our investigation has unveiled a remarkable phenomenon: in alternating current counterflow systems, once the coupling coefficient and frequency exceed certain critical values, a surprising symmetry-breaking phenomenon occurs. This results in an asymmetry in the number of vortices in the two components. We establish that this phenomenon represents a continuous phase transition, which, as indicated by the phase diagram, is exclusively observable in alternating current counterflow. We provide an explanation for this intriguing phenomenon through soliton structures, thereby uncovering the complex and unique characteristics of quantum fluid instabilities and their rich phenomena.

Figure 1

(a) The vortex evolution with coupling constant $\eta =4.5$; vortex number exhibits completely symmetrical behavior. (b) The vortex evolution with coupling constant $\eta =6$; symmetry breaking occurs in a number of vortices. (c) The dynamic evolution diagram of the densities corresponding to (b). The upper and lower rows refer to components ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$.

Figure 2

Order parameter $\mathrm{\Psi }$ as a function of $\eta$ for $\mathrm{\Omega }=0.1{\mu }_c, \mathrm{\Psi }=(1.542\pm 0.11){(\eta /5.445-1)}^{0.507\pm 0.027}$. The black line is the fit curve, and the inset magnifies the critical region. The spacing of the selected points in the main figure is $\mathrm{\Delta }\eta =0.1$, while in the inset it is $\mathrm{\Delta }\eta =0.01$.

Figure 3

Comprehensive phase diagram, delineated by oscillation frequency $\mathrm{\Omega }$ and coupling coefficient $\eta$, where the sampling points are spaced 0.1, partitioned into three distinct regions: the left phase represents the absence of vortex, the lower-right phase signifies a phase with symmetric vortex numbers, and the upper-right phase designates a phase with broken vortex symmetry. The dotted line on ${\eta }_{ps}=7.05$ is the critical boundary between miscible and immiscible phase, above which the two components are phase separated.

Figure 4

The evolution of vortex numbers in Gross-Pitaevskii equation. (a) The vortex number maintains symmetry. (b) The vortex number exhibits symmetry breaking. (c) Order parameter $\mathrm{\Psi }$ as a function of coupling coefficient $g_{12}$ for $\mathrm{\Omega }=0.4, \mathrm{\Psi }=(2.957\pm 0.383){(g/0.855-1)}^{0.475\pm 0.031}$. The black line is the fit curve, and the small graph at the bottom right magnifies the critical region.

Figure 5

In the vortex number unequal phase with $\eta =6, \mathrm{\Omega }=0.1{\mu }_c$. In both figures (a, b) the red line symbolizes the averaged velocity in the $y$ direction, whereas the blue line represents the averaged velocity in the $x$ direction. In Component 1 of (a), the peak value or amplitude of $\langle v_x\rangle$ increases with the increase in vortex number, while in Component 2 of (b), the peak value or amplitude of $\langle v_x\rangle$ decreases with the decrease in vortex number.

Figure 6

At $t{\mu }_c=8000$ where vortex number symmetry is completely broken. Panels (a) and (b) show the density profile and velocity field of Component 1 and Component 2, respectively. The color bar shows the superfluid density corresponding to the colors. In both panels, the left figure shows the complete diagram and the right figure shows the local enlarged diagram.

Figure 7

Evolution of vortex number as observed in beyond-mean-field simulations: (a) The vortex number maintains symmetry. (b) The vortex number exhibits symmetry breaking.

Figure 8

Nonperiodic system simulation. (a) The vortex evolution with coupling constant $\eta =4.5$; vortex number exhibits completely symmetrical behavior. (b) The vortex evolution with coupling constant $\eta =6$; symmetry breaking occurs in a number of vortices. (c) The dynamic evolution diagram of the densities corresponding to (b). The upper and lower rows refer to components ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$.

Figure 9

(a) Density and velocity field of soliton structure. (b) The phase of the soliton.

Figure 10

Variation of the characteristic time for soliton splitting as a function of the coupling constant.

Figure 11

(a) Global density distribution diagram and local enlarged velocity field diagram of soliton and vortex in DC counterflow. (b) Left panel shows the vortex number in counterflow with equal velocity of two components. Right panel shows the vortex number of two components with different velocity.

Figure 12

Total vortex number vs coupling coefficient. Black diamond dot is the numerical result; red line is the fitting curve as $N=(114\pm 1.6)\eta -402.4\pm 7.8$.

Figure 13

(a) The phase diagram as detailed in the main text. Altered phase diagram with counterflow velocity $U=2$.

Figure 14

Comparison of vortex number before and after time-averaging treatment, with the left image showing before treatment and the right image showing after. The inset shows a magnified view of the vortex changes in the treatment interval $\delta t$.