Symmetry Protected Topological Superfluids and Superconductors --- From the Basics to ${}^3$He ---

Abstract

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 ${}^3$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 ${}^3$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${}_3$ and superconducting doped topological insulators, in connection with the superfluid ${}^3$He.

82 pages, 33 figures

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Cited by in corpus (109)

• Topological superconductors: a review
• Superconductivity and spin-orbit coupling in non-centrosymmetric materials: a review
• Observation of time quasicrystal and its transition to superfluid time crystal
• Topology of nonsymmorphic crystalline insulators and superconductors
• Observation of half-quantum vortices in topological superfluid 3He
• Nematic superconductivity in doped Bi2Se3 topological superconductors
• Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments
• Pairing states of spin-3/2 fermions: Symmetry-enforced topological gap functions
• Evidence of nodes in the order parameter of the superconducting doped topological insulator Nb${}_x$Bi${}_2$Se${}_3$ via penetration depth measurements
• Vortex lattices and broken time reversal symmetry in the topological superconductor UPt3
• Three-Dimensional Majorana Fermions in Chiral Superconductors
• Realizing Quantum Materials with Helium: Helium films at ultralow temperatures, from strongly correlated atomically layered films to topological superfluidity
• Square-root topological semimetals
• Topological Crystalline Materials of $J=3/2$ Electrons: Antiperovskites, Dirac points, and High Winding Topological Superconductivity
• Symmetry-protected line nodes and Majorana flat bands in nodal crystalline superconductors
• Majorana surface modes of nodal topological pairings in spin-$\frac{3}{2}$ semi-metals
• Observation of unitary p-wave interactions between fermions in an optical lattice
• Theory of surface Andreev bound states and tunneling spectroscopy in three-dimensional chiral superconductors
• Theory of spin-polarized superconductors -- an analogue of superfluid ${}^3$He A-phase
• Disorder-enhanced topological protection and universal quantum criticality in a spin-3/2 topological superconductor
• Majorana multipole response of topological superconductors
• Topology and symmetry of surface Majorana arcs in cyclic superconductors
• ${}^3{P}_2$ Superfluids Are Topological
• Non-unitary triplet superconductivity tuned by field-controlled magnetization --URhGe, UCoGe and UTe${}_2$--
• Multifaceted properties of Andreev bound states: Interplay of symmetry and topology
• $AbInitio$ Simulation of Non-Abelian Braiding Statistics in Topological Superconductors
• Electron bubbles and Weyl Fermions in chiral superfluid ${}^3$He-A
• Spin triplet nematic pairing symmetry and superconducting double transition in U${}_{1-x}$Th${}_x$Be${}_{13}$
• Standard Model as the topological material
• Topological Crystalline Superconductivity in Dirac Semimetal Phase of Iron-based Superconductors
• Edge current and orbital angular momentum of chiral superfluids revisited
• Microscopic description of axisymmetric vortices in ${}^3{P}_2$ superfluids
• Andreev Bound States in Superconducting Films and Confined Superfluid He-3
• ${}^3$He Universe 2020
• Muon spin rotation study of the topological superconductor SrxBi2Se3
• Manipulation of Majorana Kramers Qubit and its Tolerance in Time-Reversal-Invariant Topological Superconductor
• Majorana multipole response: General theory and application to wallpaper groups
• Power-law Temperature Dependence of the Penetration Depth in a Topological Superconductor due to Surface States
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• Dirac and Weyl fermions: from Gor'kov equations to Standard Model (in memory of Lev Petrovich Gorkov)
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• Non-Abelian Half-Quantum Vortices in ${}^3{P}_2$ Topological Superfluids
• Spontaneous helical order of a chiral p-wave superfluid confined in nano-scale channels
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• Topological Anomalous Skin Effect in Weyl Superconductors
• Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors
• Robust and tunable coreless vortices and fractional vortices in chiral $d$-wave superconductors
• Scattering theory on surface Majorana fermions by an impurity in 3He-B
• Evidence of triplet superconductivity in Bi/Ni bilayers: Theoretical analysis of point contact Andreev reflection spectroscopy results
• Localized surfaces of three dimensional topological insulators
• Effects of the phase coherence on the local density of states in superconducting proximity structures
• Exhaustive derivation of static self-consistent multi-soliton solutions in the matrix Bogoliubov-de Gennes systems
• Spin-polarized phases of ${}^3{P}_2$ superfluids in neutron stars
• Non-thermodynamic nature of the orbital angular momentum in neutral fermionic superfluids
• Weyl Fermions and Broken Symmetry Phases of Laterally Confined ${}^3$He Films
• Spin-Nernst Effect in Time-Reversal-Invariant Topological Superconductors
• Vortices in polar and $\beta$ phases of ${}^3$He
• Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities
• Grand Challenges in Quantum Fluids and Solids
• Optical spin transport theory of spin-1/2 topological Fermi superfluids
• Design of a system for controlling a levitating sphere in superfluid ${}^3$He at extremely low temperatures
• Electric Multipoles of Double Majorana Kramers Pairs
• Topological Quantum Criticality in Superfluids and Superconductors: Surface criticality, Thermal properties, and Lifshitz Majorana fields
• Accurate projective two-band description of topological superfluidity in spin-orbit-coupled Fermi gases
• Nodal gap detection through polar angle-resolved density of states measurements in uniaxial superconductors
• Transport of bound quasiparticle states in a two-dimensional boundary superfluid
• Spin susceptibility of Andreev bound states
• Violation of the orbital depairing limit in a non-unitary state --on the high field phase in the heavy Fermion superconductor UTe${}_2$--
• Enhanced chiral edge currents and orbital magnetic moment in chiral $d$-wave superconductors from mesoscopic finite-size effects
• Anisotropic paramagnetic response of topological Majorana surface states in the superconductor ${\text{UTe}}_2$
• Theoretical studies on off-axis phase diagrams and Knight shifts in UTe${}_2$ -- Tetra-critical point, d-vector rotation, and multiple phases
• Andreev spectroscopy of the triplet superconductivity state in Bi/Ni bilayer system
• Magnetic Susceptibility of Andreev Bound States in Superfluid ${}^3$He-B
• On chiral magnetic effect in Weyl superfluid 3He-A
• Directly Detecting the Edge Current in a $p_x+i{p}_y$ Topological Superfluid
• Quantum Monte Carlo simulation of a two-dimensional Majorana lattice model
• Symmetry analysis of transport properties in helical superconductor junctions
• Spin Susceptibility of a J=3/2 Superconductor
• Odd-frequency pairs and anomalous proximity effect in nematic and chiral states of superconducting topological insulators
• Anomalous acoustoelectric effect induced by clapping modes in chiral superconductors
• Spin vortex lattice in the Landau vortex-free state of rotating superfluids
• Subgap in the surface bound states spectrum of Superfluid ${}^3$He-B with Rough Surface
• Fulde-Ferrell state in ferromagnetic chiral superconductor with magnetic domain wall
• Higher-order topological phases for time-reversal-symmetry breaking superconductivity in UTe${}_2$
• Helical spin texture in a thin film of superfluid ${}^3$He
• Dimensional crossover of superfluid ${}^3$He in a magnetic field
• Impurity bands, line-nodes, and anomalous thermal Hall effect in Weyl superconductors
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• Coherence Effects of Caroli-de Gennes-Matricon Modes in Nodal Topological Superconductors
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• Edge Modes and Teleportation in a Topologically Insulating Quantum Wire
• Two-parameter boundary condition applied to transverse acoustic impedance of a Fermi liquid
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• Designing edge currents using mesoscopic patterning in chiral d-wave superconductors
• Double-twisted surface spectrum from hybridized Majorana Kramers pairs and wallpaper fermions
• Chiral Current Inversion Induced by Flat-Band Andreev Bound States
• Observation of the crossover from quantum fluxoid to half-quantum fluxoid in a chiral superconducting device
• Condensation of fermion zero modes in the vortex in nodal superfluids/superconductors
• Influence of Magnetic Field on Surface Andreev Bound States in Superfluid ${}^3$He-B Studied by Mobility of Electron Bubble
• Superfluid ${}^3$He-B Surface States in a Confined Geometry Probed by a Microelectromechanical Oscillator
• Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems