Ising criticality can drive vortex deconfinement in a spin-orbit coupled Bose gas

Abstract

Spin-orbit coupling in Bose gases is known to lead to an Ising-symmetry-broken phase where the bosons condense at one of two nonzero momenta. In two dimensions, the finite momentum of the order parameter allows vortex-antivortex pairs that are typically bound in the superfluid phase to freely separate along Ising domain walls. This non-trivial interaction between the superfluid and the Ising order suggests that the critical fluctuations near an Ising transition could drive a Berezinskii-Kosterlitz-Thouless transition of the superfluid. We present numerical evidence of this phenomenon using a Monte Carlo simulation that shows the disappearance of superfluid stiffness near an Ising transition. Additionally, we find numerical evidence that the Ising phase transition becomes first order and we justify this claim with a variational approximation.

Figures (6)

• Figure 1: (a) Cartoon schematic of a deconfined vortex on a kink in a vertical domain wall between vortex-confined states with different momenta $k^*$. As this kink is a local defect, its energy is independent of system size, and it does not experience a long-range force from other kinks. (b) Plot of the twisting of the phase $\theta$ above and below the kink. The dotted line shows the wrapping from $2\pi$ to $0$. (c) Schematic phase diagram of the model in Eq. \ref{['eq:lattice-action']} as a function of Ising coupling $J$ and XY coupling $K$ demonstrating four phases, denoted by ferromagnetic/paramagnetic (F/P) and vortex confined/deconfined (C/D). The Ising transition leads to a stiffness collapse in the XY model. The red line denotes a first order transition and the star denotes a purported tricritical point. The dashed lines represent decoupled ($k^*=0$) phase boundaries.
• Figure 2: Monte Carlo phase diagram demonstrating vortex deconfinement near the Ising transition and a fluctuation-driven first order phase transition. (a) Helicity modulus and (b) magnetization for XY and Ising couplings ($K$ and $J$ respectively), scaled by their uncoupled critical values $K_\text{c}$ and $J_\text{c}$, marked with black lines. The BKT transition occurs at $\Upsilon = 2 / \pi K$ which is shown in white. (c) Helicity modulus $\Upsilon$, (d) magnetization $M$ and (e) magnetic susceptibility $\chi$ over a vertical cut at $K/K_\text{c} = 3$, shown as yellow lines in (a). The dotted line shows the universal BKT jump $\Upsilon = 2/\pi K$. Parameters: $k^*=0.1 \pi$, $\delta_y=0$, $\delta_x$ determined by minimizing the free energy using the modified Newton's method Eq. \ref{['eq:newtons-method']}.
• Figure 3: Variational free energy density $f_\text{var}$ in units of $g$ from the variational method shown for ferromagnetic (F) and paramagnetic (P) states for two values of the vortex free energy prefactor $C$. The effective vortex coupling $K_\mathrm{eff}$ (Eq. \ref{['eq:Keff']}) is shown compared to the critical value $K_\text{c}$ differentiating confinement and deconfinement. The unrealized states are displayed in faded colors. Though the $C=0$ exhibits a small discontinuity in the derivative, this point does not satisfy the Ginzburg criterion (see Fig. \ref{['fig:ginzburg-criterion']}) and the variational approximation is unreliable. $K=3$, $k^*=1$, $g=2$, $\Lambda=1$.
• Figure A1: The order parameter $\bar{\phi}^2$ compared with the fluctuation strength $S$ for the variational state that minimizes the free energy in Fig. \ref{['fig:free-energy']}, demonstrating the approximate satisfaction of the Ginzburg criterion (Eq. \ref{['eq:ginzburg-criterion']}) when $C>0$ since the $\bar{\phi}^2$ is around 3 times larger than $S$.
• Figure S1: Near-critical scaling of (a) magnetization $M$ and (b) susceptibility $\chi$ as a function of reduced temperature $\tau \equiv J^* / (J - J^*)$ where $J^* \approx 0.722$ at $K = 3K_\text{c}$. The magnetization is fit to the exponent $\beta=1/5.6$ before the first order transition, which differs from the 2D Ising scaling exponent $\beta=1/8$.