Symmetry protected topological order and spin susceptibility in superfluid 3He-B

Continuous Phase Transitions between Fractional Quantum Hall States and Symmetry-Protected Topological States

We study quantum phase transitions in Bose-Fermi mixtures driven by interspecies interaction in the quantum Hall regime. In the absence of such an interaction, the bosons and fermions form their respective fractional quantum Hall (FQH) states at certain filling factors. A symmetry-protected topological (SPT) state is identified as the ground state for strong interspecies interaction. The phase transitions between them are proposed to be described by Chern-Simons-Higgs field theories. For a simple microscopic Hamiltonian, we present numerical evidence for the existence of the SPT state and a continuous transition to the FQH state. It is also found that the entanglement entropy between the bosons and fermions exhibits scaling behavior in the vicinity of this transition.

Weyl Fermions and broken symmetry phases of laterally confined3He films

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)3He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum,Lz=(N/2)ℏ, forN/2Cooper pairs. The power law suppression of the angular momentum,Lz(T)≃(N/2)ℏ[1-23(πT/Δ)2]for0⩽T≪Tc, 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 superfluid3He 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 atDc2∼16ξ0. 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 lengthD→Dc2. 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' atDc1∼9ξ0, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.

Magnetic Susceptibility of Andreev Bound States in Superfluid ^{3}He-B

Nuclear magnetic resonance measurements of the magnetic susceptibility of superfluid ^{3}He imbibed in anisotropic aerogel reveal anomalous behavior at low temperatures. Although the frequency shift clearly identifies a low-temperature phase as the B phase, the magnetic susceptibility does not display the expected decrease associated with the formation of the opposite-spin Cooper pairs. This susceptibility anomaly appears to be the predicted high-field behavior corresponding to the Ising-like magnetic character of surface Andreev bound states within the planar aerogel structures.

Chiral Ising Gross-Neveu Criticality of a Single Dirac Cone: A Quantum Monte Carlo Study

We perform large-scale quantum Monte Carlo simulations of SLAC fermions on a two-dimensional square lattice at half filling with a single Dirac cone with N=2 spinor components and repulsive on-site interactions. Despite the presence of a sign problem, we accurately identify the critical interaction strength U_{c}=7.28±0.02 in units of the hopping amplitude, for a continuous quantum phase transition between a paramagnetic Dirac semimetal and a ferromagnetic insulator. Using finite-size scaling, we extract the critical exponents for the corresponding N=2 chiral Ising Gross-Neveu universality class: the inverse correlation length exponent ν^{-1}=1.19±0.03, the order parameter anomalous dimension η_{ϕ}=0.31±0.01, and the fermion anomalous dimension η_{ψ}=0.136±0.005.

New types of topological superconductors under local magnetic symmetries

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries, in which the time-reversal symmetry [Formula: see text] is broken, but its combinations with certain crystalline symmetries, such as [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the [Formula: see text] or [Formula: see text] superconducting wire, which is characterized by a chiral Z c invariant. More interestingly, we also find two types of totally new TSC phases in the [Formula: see text] and [Formula: see text] superinducting wires, which are beyond the known AZ class, and are characterized by a helical Z h invariant and Z h ⊕Z c invariants, respectively. In the Z h TSC phase, Z pairs of Majorana zero modes (MZMs) are protected at each end. In the [Formula: see text] case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving [Formula: see text] or [Formula: see text] symmetry are presented to illustrate their novel TSC properties and MZMs.

Majorana Multipole Response of Topological Superconductors

In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for helical MFs on time-reversal-invariant superconductors. The results show that the multipole response is governed by crystal symmetry, and that a one-to-one correspondence exists between the symmetry of Cooper pairs and the representation of magnetic multipoles under crystal symmetry. The latter property provides a way to identify unconventional pairing symmetry via the magnetic response of helical MFs. We also find that most helical MFs exhibit a magnetic-dipole response, but those on superconductors with spin-3/2 electrons may display a magnetic-octupole response in leading order, which uniquely characterizes high-spin superconductors. Detection of such an octupole response provides direct evidence of high-spin superconductivity, such as in half-Heusler superconductors.

Multifaceted properties of Andreev bound states: interplay of symmetry and topology

Andreev bound states (ABSs) ubiquitously emerge as a consequence of non-trivial topological structures of the order parameter of superfluids and superconductors and significantly contribute to thermodynamics and low-energy quantum transport phenomena. We here share the current status of our knowledge on their multifaceted properties such as Majorana fermions and odd-frequency pairing. A unified concept behind ABSs originates from a soliton state in the one-dimensional Dirac equation with mass domain wall and interplay of ABSs with symmetry and topology enrich their physical characteristics. We make an overview of ABSs with a special focus on superfluid 3He. The quantum liquid confined to restricted geometries serves as a rich repository of noteworthy quantum phenomena, such as the mass acquisition of Majorana fermions driven by spontaneous symmetry breaking, topological quantum criticality, Weyl superfluidity and the anomalous magnetic response. The marriage of the superfluid 3He and nano-fabrication techniques will take one to a new horizon of topological quantum phenomena associated with ABSs.This article is part of the theme issue 'Andreev bound states'.

Topological superconductors: a review

This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

Symmetry protected topological superfluid (3)He-B

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-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 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.

Surface-induced order parameter distortion in superfluid ³He-B measured by nonlinear NMR

The B phase of superfluid 3He is a three-dimensional time-reversal invariant topological superfluid, predicted to support gapless Majorana surface states. We confine superfluid 3He into a thin nanofluidic slab geometry. In the presence of a weak symmetry-breaking magnetic field, we have observed two possible states of the confined 3He-B phase manifold, through the small tipping angle NMR response. Large tipping angle nonlinear NMR has allowed the identification of the order parameter of these states and enabled a measurement of the surface-induced gap distortion. The results for two different quasiparticle surface scattering boundary conditions are compared with the predictions of weak-coupling quasiclassical theory. We identify a textural domain wall between the two B phase states, the edge of which at the cavity surface is predicted to host gapless states, protected in the magnetic field.

Topological phase transition without gap closing

Topological phase transition is accompanied with a change of topological numbers. According to the bulk-edge correspondence, the gap closing and the breakdown of the adiabaticity are necessary at the phase transition point to make the topological number ill-defined. However, the gap closing is not always needed. In this paper, we show that two topological distinct phases can be continuously connected without gap closing, provided the symmetry of the system changes during the process. Here we propose the generic principles how this is possible by demonstrating various examples such as 1D polyacetylene with the charge-density-wave order, 2D silicene with the antiferromagnetic order, 2D silicene or quantum well made of HgTe with superconducting proximity effects and 3D superconductor Cu doped Bi₂Se₃. It is argued that such an unusual phenomenon can occur when we detour around the gap closing point provided the connection of the topological numbers is lost along the detour path.

Symmetry-protected Majorana fermions in topological crystalline superconductors: theory and application to Sr2RuO4

Crystal point group symmetry is shown to protect Majorana fermions (MFs) in spinfull superconductors (SCs). We elucidate the condition necessary to obtain MFs protected by the point group symmetry. We argue that superconductivity in Sr2RuO4 hosts a topological phase transition to a topological crystalline SC, which accompanies a d-vector rotation under a magnetic field along the c axis. Taking all three bands and spin-orbit interactions into account, symmetry-protected MFs in the topological crystalline SC are identified. Detection of such MFs provides evidence of the d-vector rotation in Sr2RuO4 expected from Knight shift measurements but not yet verified.