Domain-area distribution anomaly in segregating multicomponent superfluids

The domain-area distribution in the phase transition dynamics of $Z_2$ symmetry breaking is studied theoretically and numerically for segregating binary Bose-Einstein condensates in quasi-two-dimensional systems. Due to the dynamic-scaling law of the phase ordering kinetics, the domain-area distribution is described by a universal function of the domain area, rescaled by the mean distance between domain walls. The scaling theory for general coarsening dynamics in two dimensions hypothesizes that the distribution during the coarsening dynamics has a hierarchy with the two scaling regimes, the microscopic and macroscopic regimes with distinct power-law exponents. The power law in the macroscopic regime, where the domain size is larger than the mean distance, is universally represented with the Fisher's exponent of the percolation theory in two dimensions. On the other hand, the power-law exponent in the microscopic regime is sensitive to the microscopic dynamics of the system. This conjecture is confirmed by large-scale numerical simulations of the coupled Gross-Pitaevskii equation for binary condensates. In the numerical experiments of the superfluid system, the exponent in the microscopic regime anomalously reaches to its theoretical upper limit of the general scaling theory. The anomaly comes from the quantum-fluid effect in the presence of circular vortex sheets, described by the hydrodynamic approximation neglecting the fluid compressibility. It is also found that the distribution of superfluid circulation along vortex sheets obeys a dynamic-scaling law with different power-law exponents in the two regimes. An analogy to quantum turbulence on the hierarchy of vorticity distribution and the applicability to chiral superfluid ${}^3\mathrm{He}$ in a slab are also discussed.

Figure 1

The expected asymptotic form of the rescaled domain-area distribution $\tilde{\rho }$.

Figure 2

A time evolution of domain coarsening dynamics in a numerical experiment of segregating binary BECs. Black regions show $\uparrow$ domains ($\downarrow$ domains) from $t/t_0=21.2$ to $t/t_0=1049$ in upper (lower) panels. The gradation represents spatial distribution of the phase ${\theta }_{\downarrow (\uparrow )}=\arg {\mathrm{\Psi }}_{\downarrow (\uparrow )}$ in $\downarrow (\uparrow )$ domains. A vortex is represented by a end point of branch cut (jump from ${\theta }_{\uparrow ,\downarrow }=-\pi$ to $\pi$). The phase ${\theta }_{\uparrow (\downarrow )}$ is nearly homogeneous in some of small $\uparrow (\downarrow )$ domains ($S<S_l$), at which branch cuts of ${\theta }_{\downarrow (\uparrow )}$ end, by forming vortex sheets along their domain walls.

Figure 3

Dynamic-scaling plot of the averaged domain-area distribution $\langle \rho \left(S\right)\rangle$. The graph legends represent the time from $t/t_0=21.1$ to 1049 with $t_0=\frac{\hslash }{gn}$. The table provides information on the corresponding values of $l/\xi$ and $L/l$. The inset shows the corresponding scaling plot for ${\tilde{l}}_{\mathrm{w}}$ from $t/t_0=$ 101 to 1049 with the solid line ${\tilde{l}}_{\mathrm{w}}=2\pi \tilde{S}$. Error bars correspond to the standard deviation $S_l^2\sqrt{{\delta }_X^2}\left[{\left(2\pi \right)}^{-1}\sqrt{{\delta }_X^2}\right]$ of the ensemble average $S_l^2\langle X\rangle \left[{\left(2\pi \right)}^{-1}\langle X\rangle \right]$ for the quantity $X\left(=\rho \mathrm{or}{\tilde{l}}_{\mathrm{w}}/\sqrt{\tilde{S}}\right)$, where the average $\langle X\rangle$ and its variance ${\delta }_X^2=\langle {\left(X-\langle X\rangle \right)}^2\rangle$ is calculated over the 64 samples of the numerical simulations.

Figure 4

Schematic of typical shapes of (a) a larger domain ($S\gg S_l$) and (b) a smaller domain ($S\ll S_l$) that appear in a domain structure in the SSB development. The black curves show domain walls whose thickness $l_{\min }$ is much smaller than the mean distance $l$ between domain walls in the late stage of the SSB development. (a) A domain in the macroscopic regime is like a snaky and branching trail of width $\sim l$ and length $\sim l_{\mathrm{w}}/2$; the area of a trail is given by $S\approx l{l}_{\mathrm{w}}/2$. This relation between $S$ and $l_{\mathrm{w}}$ is consistent with the dynamic-scaling relation ${\tilde{l}}_{\mathrm{w}}=2\pi \tilde{S}\Rightarrow l_{\mathrm{w}}/l=2\pi S/S_l\Rightarrow S=l{l}_{\mathrm{w}}/2$ of the macroscopic regime in the inset of Fig. 3. (b) The domain is almost circular by obeying the relation of Eq. (7) approximately; $S\approx l_{\mathrm{w}}^2/\left(4\pi \right)$. The $\uparrow$ domain (white region) contains a vortex with a circulation $\kappa n_{\mathrm{v}}$, whose core is occupied by a circular $\downarrow$ domain (black region) at rest. The hydrostatic pressure of the $\uparrow$ domain on the wall is smaller than that of the $\downarrow$ domain, $p_{\uparrow }^h<p_{\downarrow }^h$, in the presence of the circular superflow ($|{\mathit{v}}_{\uparrow }|=\frac{\kappa |n_{\mathrm{v}}|}{2\pi R}$).

Figure 5

Dynamic-scaling plot of the circulation quantum number $|n_{\mathrm{v}}\left(S\right)|$ from $t/t_0=101$ to 1049. The inset shows the ratio $P_{\mathrm{v}}\left(\tilde{S}\right)$ of the number of domains with $|n_{\mathrm{v}}|\ne 0$ from $t/t_0=$ 21.2 to 1049. Error bars correspond to the standard deviation for the quantities $\sqrt{\xi /l}\left|n_{\mathrm{v}}\right|$ and $P_{\mathrm{v}}$.