Symmetry Protected Topological Superfluid ${}^3$He-B

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.

arXiv commentInvited Topical Review Paper: 99 pages, 24 figures

References in corpus (185)

• Topological insulators and superconductors
• Fault-tolerant quantum computation by anyons
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Non-Abelian Anyons and Topological Quantum Computation
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Topological Insulators with Inversion Symmetry
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Helical liquids and Majorana bound states in quantum wires
• Topological Crystalline Insulators
• A spin-orbit coupled Bose-Einstein condensate
• A generic new platform for topological quantum computation using semiconductor heterostructures
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• Periodic table for topological insulators and superconductors
• Atomic Scattering in Presence of an External Confinement and a Gas of Impenetrable Bosons
• Topological Insulator Materials
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Spin-orbit coupling in quantum gases
• Spin-Orbit Coupled Degenerate Fermi Gases
• Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Odd-Parity Topological Superconductors: Theory and Application to Cu${}_x$Bi${}_2$Se${}_3$
• Long-range effects in superconductor-ferromagnet structures
• Non-Abelian Topological Order in S-Wave Superfluids of Ultracold Fermionic Atoms
• Topological Superconductivity and Superfluidity
• Topological Superconductivity in CuxBi2Se3
• Antiferromagnetic topological insulators
• The space group classification of topological band insulators
• Degenerate Quantum Gases with Spin-Orbit Coupling
• Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian Statistics
• Symmetry and Topology in Superconductors - Odd-frequency pairing and edge states -
• Using photoemission spectroscopy to probe a strongly interacting Fermi gas
• Spin-Imbalance in a One-Dimensional Fermi Gas
• Topology of crystalline insulators and superconductors
• Majorana Fermions in Semiconductor Nanowires
• Resonantly-paired fermionic superfluids
• Classification of topological insulators and superconductors in the presence of reflection symmetry
• Multichannel Generalization of Kitaev's Majorana End States and a Practical Route to Realize Them in Thin Films
• Topology of Andreev Bound States with Flat Dispersion
• The spin quantum Hall effect in unconventional superconductors
• Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
• Emergent Space-time Supersymmetry at the Boundary of a Topological Phase
• Flat bands in topological media
• Non-Abelian Topological Orders and Majorana Fermions in Spin-Singlet Superconductors
• Topological invariants for spin-orbit coupled superconductor nanowires
• Quantum Computation Using Vortices and Majorana Zero Modes of a $p_x$ + $i{p}_y$ Superfluid of Fermionic Cold Atoms
• Edge modes, edge currents, and gauge invariance in $p_x+i{p}_y$ superfluids and superconductors
• Topological classification with additional symmetries from Clifford algebras
• Theory of the proximity effect in junctions with unconventional superconductors
• Topological odd-parity superconductors
• Search for Majorana fermions in multiband semiconducting nanowires
• Spin current in ferromagnet/insulator/superconductor junctions
• Spin-Orbit Coupled Quantum Gases
• Realistic Rashba and Dressehaus spin-orbit coupling for neutral atoms
• Local Measurements of the Superconducting Pairing Symmetry in CuxBi2Se3
• Topological Mirror Superconductivity
• Distribution functions in non-equilibrium theory of superconductivity and Andreev spectroscopy in unconventional superconductors
• Synthetic 3D Spin-Orbit Coupling
• Classification of Two Dimensional Topological Crystalline Superconductors and Majorana Bound States at Disclinations
• Magnetically generated spin-orbit coupling for ultracold atoms
• Odd-frequency pairing in normal metal/superconductor junctions
• Inhomogeneous Condensates in the Thermodynamics of the Chiral NJL_2 model
• Splitting of Majorana modes due to intervortex tunneling in a p + ip superconductor
• Majorana Fermions and Disclinations in Topological Crystalline Superconductors
• Detecting the Majorana fermion surface state of ${}^3$He-B through spin relaxation
• Topological properties of spin-triplet superconductors and the Fermi surface topology in the normal state
• Majorana Fermions and Exotic Surface Andreev Bound States in Topological Superconductors: Application to CuxBi2Se3
• Topological non-symmorphic crystalline insulators
• Direct Imaging of Spatially Modulated Superfluid Phases in Atomic Fermion Systems
• Generalized Rashba spin-orbit coupling for cold atoms
• Spontaneous Hall effect in chiral p-wave superconductor
• Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems
• From relativistic quantum fields to condensed matter and back again: Updating the Gross-Neveu phase diagram
• Symmetry-Protected Majorana Fermions in Topological Crystalline Superconductors: Theory and Application to Sr2RuO4
• Majorana Fermions and Odd-frequency Cooper Pairs in a Nano Wire
• Topological Classification of Crystalline Insulators with Point Group Symmetry
• Lattice Models with N=2 Supersymmetry
• Spinless Topological Insulators without Time-Reversal Symmetry
• A Twisted Kink Crystal in the Chiral Gross-Neveu model
• Surface States, Edge Currents and the Angular Momentum of Chiral p-wave Superfluids
• Topological field theory and thermal responses of interacting topological superconductors
• Majorana Fermions in Nodal Superconductors: Gapless Topological Phase
• Anomalous features of the proximity effect in triplet superconductors
• Symmetry protected topological phases in non-interacting fermion systems
• Tunneling Conductance and Surface States Transition in Superconducting Topological Insulators
• Vortices and Other Topological Solitons in Dense Quark Matter
• Cross-Correlated Responses of Topological Superconductors and Superfluids
• Atomic spin-orbit coupling synthesized with magnetic-field-gradient pulses
• Fulde-Ferrell-Larkin-Ovchinnikov states in one-dimensional spin-polarized ultracold atomic Fermi gases
• Thermodynamic properties of thin films of superfluid 3He-A
• Tunneling of anyonic Majorana excitations in topological superconductors
• Zero modes of two-dimensional chiral p-wave superconductors
• Quantum Thermal Hall Effect in a Time-Reversal-Symmetry-Broken Topological Superconductor in Two Dimensions : Approach From Bulk Calculations
• An Index Theorem for the Majorana Zero Modes in Chiral P-Wave Superconductors
• Pairing symmetry conversion by spin-active interfaces in superconducting junctions
• Unconventional superconductivity in CuxBi2Se3
• Proximity induced superconductivity by Bi in topological $B{i}_{2}T{e}_{2}Se$ and $B{i}_{2}S{e}_3$ films: Evidence for a robust zero energy bound state possibly due to Majorana Fermions
• Anomalous Surface Impedance due to Odd-frequency Cooper Pairs
• Fermion zero modes at the boundary of superfluid 3He-B
• Quantizing Majorana Fermions in a Superconductor
• Topological Phase Transition without Gap Closing
• A phenomenological theory of zero-energy Andreev resonant states
• Symmetry Protected Topological Order and Spin Susceptibility in Superfluid 3He-B
• Surface spectral function in the superconducting state of a topological insulator
• Topological superfluid in one-dimensional spin-orbit coupled atomic Fermi gases
• Simulating Wess-Zumino Supersymmetry Model in Optical Lattices
• All about the Static Fermion Bags in the Gross-Neveu Model
• New class of topological superconductors protected by magnetic group symmetries
• Crystalline order in superfluid 3He films
• Flat band in the core of topological defects: bulk-vortex correspondence in topological superfluids with Fermi points
• Topological invariant for superfluid 3He-B and quantum phase transitions
• Strong Anisotropy in Spin Suceptibility of Superfluid 3He-B Film Caused by Surface Bound States
• Role of the Majorana Fermion and the Edge Mode in Chiral Superfluidity near a p-Wave Feshbach Resonance
• Twofold spontaneous symmetry breaking in a heavy fermion superconductor UPt${}_3$
• Signature of odd-frequency pairing correlations induced by a magnetic interface
• Non-Abelian Statistics of Axion Strings
• UPt${}_3$ as a Topological Crystalline Superconductor
• Surface Majorana Cone of the Superfluid ${}^3$He B Phase
• Electromagnetic Response of a Vortex in Layered Superconductors
• Topological superfluid 3He-B in magnetic field and Ising variable
• Charge frustration and quantum criticality for strongly correlated fermions
• Absence of zero-energy surface bound states in CuxBi2Se3 via a study of Andreev reflection spectroscopy
• Topological minigap in quasi-one-dimensional spin-orbit-coupled semiconductor Majorana wires
• Flat band in topological matter: possible route to room-temperature superconductivity
• Dirac-Fermion-Induced Parity Mixing in Superconducting Topological Insulators
• The Bogoliubov/de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry
• Topological Structure of a Vortex in Fulde-Ferrell-Larkin-Ovchinnikov State
• Discovery of the Acoustic Faraday Effect in Superfluid 3He-B
• Majorana zero modes in dislocations of Sr${}_2$RuO${}_4$
• Majorana surface states of superfluid 3He A- and B-phases in a slab
• Majorana fermions in one-dimensional spin-orbit coupled Fermi gases
• Quantum impurity spin in Majorana edge fermions
• Low-energy sub-gap states in multi-channel Majorana wires
• A Spin Triplet Superconductor UPt${}_3$
• Theory of pairing symmetry inside the Abrikosov vortex core
• Vortex structures and zero energy states in the BCS-to-BEC evolution of p-wave resonant Fermi gases
• Surface Specific Heat of ${}^3$He and Andreev Bound States
• Classification of engineered topological superconductors
• Topological superfluid ${}^3$He-B: fermion zero modes on interfaces and in the vortex core
• Splitting and oscillation of Majorana zero modes in the p-wave BCS-BEC evolution with plural vortices
• Majorana Excitations, Spin- and Mass Currents on the Surface of Topological Superfluid ${}^3$He-B
• CPT theorem and classification of topological insulators and superconductors
• Self-consistent multiple complex-kink solutions in Bogoliubov-de Gennes and chiral Gross-Neveu systems
• Topological phases of quasi-one-dimensional fermionic atoms with a synthetic gauge field
• Majorana bound state in rotating superfluid 3He-A between parallel plates
• Topological Fermi arcs in superfluid ${}^3$He
• A supersymmetric multicritical point in a model of lattice fermions
• Anomalous Attenuation of Transverse Sound in 3He
• Abelian Chern-Simons Term in Superfluid 3He-A
• Andreev-Majorana bound states in superfluids
• Edge Current due to Majorana Fermions in Superfluid ${}^3$He A- and B-Phases
• Superfluid 3He in a restricted geometry with a perpendicular magnetic field
• Odd-frequency Cooper pairs and zero-energy surface bound states in superfluid 3He
• Zero energy bound states in tunneling conductance spectra at the interface of an s-wave superconductor and a topological insulator in NbN-$B{i}_{2}S{e}_3$-Au thin film junctions
• Demonstrating Lattice-Symmetry-Protection in Topological Crystalline Superconductors
• Electromagnetic and thermal responses of Z topological insulators and superconductors in odd spatial dimensions
• Majorana meets Coxeter: Non-Abelian Majorana Fermions and Non-Abelian Statistics
• Mirror Majorana zero modes in spinful superconductors/superfluids -Non-Abelian braiding of integer quantum vortices-
• Discovery of an Excited Pair State in Superfluid 3He
• Magnetically robust topological edge states and flat bands
• Non-Abelian statistics of vortices with multiple Majorana fermions
• Sub-gap in the Edge States of 2-D Chiral Superconductor with Rough Surface
• Odd-frequency pairing effect on the superfluid density and the Pauli spin susceptibility in spatially nonuniform spin-singlet superconductors
• Majorana edge modes of superfluid 3He A-phase in a slab
• Manipulating Majorana fermions in one-dimensional spin-orbit coupled atomic Fermi gases
• Theory of pairing symmetry in Fulde-Ferrell-Larkin-Ovchinnikov vortex state and vortex lattice
• Topology of the planar phase of superfluid ${}^3$He and bulk-boundary correspondence for three dimensional topological superconductors
• Spin imbalance effect on Larkin-Ovchinnikov-Fulde-Ferrel state
• Bell's inequality and universal quantum gates in a cold atom chiral fermionic p-wave superfluid
• Majorana states and longitudinal NMR absorption in ${}^3$He-B film
• Zero Energy Modes and Statistics of Vortices in Spinful Chiral p-Wave Superfluids
• Odd-frequency Cooper pair amplitude around a vortex core in a chiral p-wave superconductor in the quantum limit
• Odd-frequency pairing and Ising spin susceptibility in time-reversal invariant superfluids and superconductors
• Marginally Stable Topologically Non-Trivial Solitons in the Gross-Neveu Model
• Momentum-resolved radio-frequency spectroscopy of a spin-orbit coupled atomic Fermi gas near a Feshbach resonance in harmonic traps
• Stripe order in superfluid 3He confined in narrow cylinders
• Magneto-Acoustic Spectroscopy in Superfluid 3He-B
• Evolution of edge states in topological superfluids during the quantum phase transition
• High frequency sound in superfluid 3He-B
• Topological Quantum Computing with p-Wave Superfluid Vortices
• Nonlinear field-dependence and f-wave interactions in superfluid 3He

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• Symmetry Protected Topological Superfluids and Superconductors --- From the Basics to ${}^3$He ---
• Observation of half-quantum vortices in topological superfluid 3He
• Half-quantum vortices and walls bounded by strings in the polar-distorted phases of topological superfluid ${}^3$He
• From Standard Model of particle physics to room-temperature superconductivity
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• Non-unitary triplet superconductivity tuned by field-controlled magnetization --URhGe, UCoGe and UTe${}_2$--
• Topology and symmetry of surface Majorana arcs in cyclic superconductors
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• Current fluctuations in unconventional superconductor junctions with impurity scattering
• Multifaceted properties of Andreev bound states: Interplay of symmetry and topology
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• Bogoliubov-de Gennes Soliton Dynamics in Unconventional Fermi Superfluids
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• Spin-polarized phases of ${}^3{P}_2$ superfluids in neutron stars
• Effects of the phase coherence on the local density of states in superconducting proximity structures
• Vortex-bound solitons in topological superfluid ${}^3$He
• Low-temperature environments for quantum computation and quantum simulation
• Nanofluidic platform for studying the first-order phase transitions in superfluid helium-3
• NMR-like effect on Anisotropic Magnetic Moment of Surface Bound States in Topological Superfluid ${}^3$He-B
• Dimensional crossover of superfluid ${}^3$He in a magnetic field
• Chiral superfluid helium-3 in the quasi-two-dimensional limit
• Magnetic edge fields in UTe${}_2$ near zero background fields
• Quasiclassical theory of spin dynamics in superfluid ${}^3$He: kinetic equations in the bulk and spin response of surface Majorana states
• Coherence Effects of Caroli-de Gennes-Matricon Modes in Nodal Topological Superconductors
• Quantum Quenches of an SO(5) Pseudospin Reveal Higgs Bosons
• Unconventional superconductivity from electronic dipole fluctuations
• Influence of Magnetic Field on Surface Andreev Bound States in Superfluid ${}^3$He-B Studied by Mobility of Electron Bubble