Weyl Fermions and Broken Symmetry Phases of Laterally Confined $^3$He Films

TL;DR

This paper investigates the effects of lateral confinement on the fermionic spectrum, edge states, and phase transitions in superfluid $^3$He films, revealing a sequence of symmetry-breaking phases including a pair density wave and polar phase.

Contribution

It introduces a detailed analysis of phase transitions and edge state hybridization in confined superfluid $^3$He, highlighting the emergence of a pair density wave phase under strong confinement.

Findings

Identification of a critical confinement width $D_{c2} \\sim 16 \\xi_0$ for the transition to a pair density wave phase.

Observation of a second-order transition to a polar phase at $D_{c1} \\sim 9 \\xi_0$.

Analysis of edge current suppression and angular momentum behavior with temperature.

Abstract

Broken symmetries in topological condensed matter systems have implications for the spectrum of Fermionic excitations confined on surfaces or topological defects. The Fermionic spectrum of confined (quasi-2D) $^3$He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum, $L_z = (N/2) \hbar$, for $N/2$ Cooper pairs. The power law suppression of the angular momentum, $L_z(T) \simeq (N/2)\,\hbar\,[1 - \frac{2}{3}(\pi T/\Delta)^2 ]$ for $0 \le T \ll T_c$, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We discuss the effects of wave function overlap, and hybridization between edge states confined near opposing edge boundaries on the edge currents, ground-state angular momentum and ground-state order parameter of superfluid $^3$He thin films. Under strong lateral confinement, the chiral A phase undergoes a sequence of phase transitions, first to a pair density wave (PDW) phase with broken translational symmetry at $D_{c2} \sim 16 \xi_0$. The PDW phase is described by a periodic array of chiral domains with alternating chirality, separated by domain walls. The period of PDW phase diverges as the confinement length $D\rightarrow D_{c_2}$. The PDW phase breaks time-reversal symmetry, translation invariance, but is invariant under the combination of time-reversal and translation by a one-half period of the PDW. The mass current distribution of the PDW phase reflects this combined symmetry, and originates from the spectra of edge Fermions and the chiral branches bound to the domain walls. Under sufficiently strong confinement a second-order transition occurs to the non-chiral ``polar phase'' at $D_{c1} \sim 9\xi_0$, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.