Symmetry Protected Topological Superfluid $^3$He-B

TL;DR

This paper reviews the topological superfluidity of $^3$He-B, focusing on symmetry protection, Majorana fermions, and phase transitions under magnetic fields, highlighting the interplay of topology, symmetry, and emergent phenomena.

Contribution

It provides a comprehensive analysis of symmetry protected topological superfluidity in $^3$He-B, including microscopic calculations and the role of hidden symmetries.

Findings

Identification of symmetry protected Majorana fermions in $^3$He-B

Quantitative determination of critical magnetic field for phase transition

Observation of odd-frequency pairing in low-lying quasiparticles

Abstract

Owing to the richness of symmetry and well-established knowledge on the bulk superfluidity, the superfluid $^3$He has offered a prototypical system to study intertwining of topology and symmetry. This article reviews recent progress in understanding the topological superfluidity of $^3$He in a multifaceted manner, including symmetry consideration, the Jackiw-Rebbi's index theorem, and the quasiclassical theory. Special focus is placed on the symmetry protected topological superfuidity of the $^3$He-B confined in a slab geometry. The $^3$He-B under a magnetic field is separated to two different sub-phases: The symmetry protected topological phase and non-topological phase. The former phase is characterized by the existence of symmetry protected Majorana fermions. The topological phase transition between them is triggered off by the spontaneous breaking of a hidden discrete symmetry. The critical field is quantitatively determined from the microscopic calculation that takes account of magnetic dipole interaction of $^3$He nucleus. It is also demonstrated that odd-frequency even-parity Cooper pair amplitudes are emergent in low-lying quasiparticles. The key ingredients, symmetry protected Majorana fermions and odd-frequency pairing, bring an important consequence that the coupling of the surface states to an applied field is prohibited by the hidden discrete symmetry, while the topological phase transition with the spontaneous symmetry breaking is accompanied by anomalous enhancement and anisotropic quantum criticality of surface spin susceptibility. We also illustrate common topological features between topological crystalline superconductors and symmetry protected topological superfluids, taking UPt$_3$ and Rashba superconductors as examples.