Dimensional crossover of superfluid $^{3}$He in a magnetic field

TL;DR

This paper develops a comprehensive Ginzburg--Landau framework for superfluid $^3$He in a slab geometry under a perpendicular magnetic field, revealing a dimensional crossover from 3D to quasi-2D as slab height $D$ decreases. By solving the GL equations with various boundary conditions, the authors obtain analytic control over order-parameter profiles using elliptic and Lamé function theory, and map the stability regions of A-, B-, P-, Pol-, and stripe phases, including the A$_1$/A$_2$/P$_2$ set in a field and the B$_2$-phase at finite $H$ and $D$. Key findings include the universal critical confinement $ar D_c$ (π for maximally pair-breaking and 0 for specular) that delineates the onset of superfluidity, the emergence of stripe order for certain γ and boundary types, and the dominance of A- or A$_2$-phases under confinement and magnetic field, with strong-coupling corrections crucially differentiating A$_2$ from P$_2$. The work provides detailed $P$-$T$-$D$ and $P$-$T$-$H$-$D$ phase diagrams, offering experimentally testable predictions to constrain GL coefficients and boundary specularity, and it points to potential applications in 2D topological superfluid physics and quantum computation. Overall, the study advances understanding of how geometry and magnetic fields sculpt complex order-parameter landscapes in strongly correlated superfluids.

Abstract

Motivated by recent experiments on superfluid $^3$He in nanoscale-confined geometries, we theoretically investigate the associated phase diagram in a slab geometry and perpendicular magnetic field as the size of confinement is varied. Our analysis is based on minimizing the Ginzburg--Landau free energy for the $3\times 3$ matrix superfluid order parameter for three different boundary conditions. We observe a smooth crossover from the phase diagram of the 3D system to the quasi-2D limit for slab heights of several hundred nanometres and magnetic fields of several kilogauss. We illuminate that, despite the apparent complexity of the underlying equations, many precise numerical and even analytical statements can be made about the phase structure for general values of the coefficients of the free energy functional, which can in turn be used to constrain or measure these parameters. To guide future experimental studies, we compute the phase diagram in dependence of pressure, temperature, slab height, and magnetic field.

Figures (18)

• Figure 1: In this work, we consider a scenario where superfluid $^3$He is confined to a slab geometry of height $D$ in a perpendicular magnetic field of magnitude $H$. We choose the coordinate system such that the magnetic field points in the z-direction and the superfluid is limited to the region $z\in[0,D]$, with idealized infinite extension in the x- and y-directions. Three different boundary conditions are chosen at the surfaces $z=0$ and $z=D$.
• Figure 2: Order parameter profiles in a slab geometry of height $D$ with maximally pair-breaking boundaries, for various ratios $\bar{D}=D/\xi$. The functions $f(\bar{z})$ are solutions to Eqs. (\ref{['real13']}), (\ref{['real13b']}) and parametrize a variety of superfluid orders. They are elliptic functions in the mathematical sense with real period $2\bar{D}$ and follow the analytical expression (\ref{['real16']}). Solutions only exist for $\bar{D}>\pi$. Note that the functions approach the bulk value $f= 1$ in regions sufficiently far from the boundaries.
• Figure 3: Properties of the elliptic function $f(\bar{z})$. Panel (a). The function $f(\bar{z})$ with $\bar{z}\in[0,\bar{D}]$ can be extended to a periodic function on $\bar{z}\in\mathbb{R}$. It then corresponds to a Duffing oscillator with softening spring and period $2\bar{D}$. Panel (b). Solutions to Eq. (\ref{['real13']}) are Jacobi elliptic functions parametrized by $f'(0)=\sigma$ or, equivalently, the modular parameter $k^2$. The condition $0\leq k^2< 1$ for proper oscillations translates to $0\leq \sigma<1/\sqrt{2}$ and $\pi \leq \bar{D}<\infty$, although solutions with $\bar{D}=\pi$ have vanishing amplitude.
• Figure 4: Phase diagram for real order parameters with maximally pair-breaking boundaries. For sufficiently large $\bar{D}=D/\xi$ and $\zeta_{12}<\frac{1}{3}$, the system is in the pdB-phase with $f_1=f_2>f_3$. The phase boundary $\bar{D}_{\rm P}(\gamma,\zeta_{12})$ is determined analytically by Eq. (\ref{['real41']}). For tight confinement, the system is in the P- or Pol-phase, depending on $\zeta_{12}$, whereas no superfluid order is possible for $\bar{D}\leq \pi$. Representative order parameter profiles are shown in panels A--D along the weak-coupling line $\zeta_{12}=\frac{1}{5}$, with $f_1$ ($f_3$) shown in blue (orange), corresponding to the stars in the phase diagram. The phase diagram is shown here for $\gamma=3$, but has the same shape for all $\gamma>1$. For $\gamma\to1$, $\bar{D}_{\rm P}\to \pi$, the P-phase vanishes, and the pdB-phase becomes the B-phase.
• Figure 5: Phase diagram for maximally pair-breaking boundary conditions. Panel (a). The time-reversal symmetry breaking A-phase competes with the real orders of Fig. \ref{['FigRealPD']}, resulting in first-order phase transitions, shown here for $\zeta_{245}=0.38$ and $\gamma=3$. For $\zeta_{245}>\frac{1}{3}$, the A-phase is energetically favorable over the P-phase in the region $\zeta_{12}<1-2\zeta_{245}$ (red arrow). Panel (b). The competition between the A- and pdB-phase has to be determined numerically as function of $\zeta_{12},\zeta_{245},\gamma$, and $\bar{D}$. We show the phase boundary between the A-phase and the real orders for several values of $\zeta_{245}$. For $\zeta_{245}<\frac{1}{3}$, the phase diagram consist of only two phases, the A- and Pol-phase, separated by a vertical line at $\zeta_{12}=\frac{1}{2}(1-\zeta_{245})$. The phase diagrams in the 3D and quasi-2D limits, shown in Fig. \ref{['FigBulkPD']}, can be read off from the behavior at $\bar{D}=\infty$ and $\bar{D}=\pi$, respectively.