Takeshi Mizushima

Symmetry-Protected Topological Superfluids and Superconductors —From the Basics to 3He—

In this article, we give a comprehensive review of recent progress in research on symmetry-protected topological superfluids and topological crystalline superconductors, and their physical consequences such as helical and chiral Majorana fermions. We start this review article with the minimal model that captures the essence of such topological materials. The central part of this article is devoted to the superfluid

Symmetry protected topological superfluid 3He-B

^3

Dirac Fermion induced Parity Mixing in Superconducting Topological Insulators

He, which serves as a rich repository of novel topological quantum phenomena originating from the intertwining of symmetries and topologies. In particular, it is emphasized that the quantum fluid confined to nanofabricated geometries possesses multiple superfluid phases composed of the symmetry-protected topological superfluid B-phase, the A-phase as a Weyl superfluid, the nodal planar and polar phases, and the crystalline ordered stripe phase. All these phases generate noteworthy topological phenomena, including topological phase transitions concomitant with spontaneous symmetry breaking, Majorana fermions, Weyl superfluidity, emergent supersymmetry, spontaneous edge mass and spin currents, topological Fermi arcs, and exotic quasiparticles bound to topological defects. In relation to the mass current carried by gapless edge states, we also briefly review a longstanding issue on the intrinsic angular momentum paradox in

Topological phases of quasi-one-dimensional fermionic atoms with a synthetic gauge field

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Superconducting doped topological materials

He-A. Moreover, we share the current status of our knowledge on the topological aspects of unconventional superconductors, such as the heavy-fermion superconductor UPt

Fulde–Ferrell–Larkin–Ovchinnikov States in Two-Band Superconductors

_3

Odd-frequency pairing and Ising spin susceptibility in time-reversal-invariant superfluids and superconductors

and superconducting doped topological insulators, in connection with the superfluid

Mirror Majorana zero modes in spinful superconductors/superfluids Non-Abelian anyons in integer quantum vortices

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On Nambu's Fermion-Boson Relations for Superfluid

He.

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Owing to the richness of symmetry and well-established knowledge of 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 considerations, the Jackiw-Rebbis 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 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 the (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 UPt3 and Rashba superconductors as examples.