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 3He, 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, Maj...

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 $^3$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.

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

Topological Lifshitz transitions involve many types of topological structures in momentum and frequency–momentum spaces, such as Fermi surfaces, Dirac lines, Dirac and Weyl points, etc., each of which has its own stability-supporting topological invariant (N1, N2, N3, , etc.). The topology of the shape of Fermi surfaces and Dirac lines and the interconnection of objects of different dimensionalities produce a variety of Lifshitz transition classes. Lifshitz transitions have important implications for many areas of physics. To give examples, transition-related singularities can increase the superconducting transition temperature; Lifshitz transitions are the possible origin of the small masses of elementary particles in our Universe, and a black hole horizon serves as the surface of the Lifshitz transition between vacua with type-I and type-II Weyl points.

https://iopscience.iop.org/article/10.3367/UFNe.2017.01.038218/pdf

Exotic Lifshitz transitions in topological materials

Topological Lifshitz transitions involve many types of topological structures in momentum and frequency-momentum spaces: Fermi surfaces, Dirac lines, Dirac and Weyl points, etc. Each of these structures has their own topological invariant ($N_1$, $N_2$, $N_3$, $\tilde N_3$, etc.), which supports the stability of a given topological structure. The topology of the shape of Fermi surfaces and Dirac lines, as well as the interconnection of the objects of different dimensions, lead to numerous classes of Lifshitz transitions. The consequences of Lifshitz transitions are important in different areas of physics. The singularities emerging at the transition may enhance the transition temperature to superconductivity; the Lifshitz transition can be in the origin of the small masses of elementary particles in our Universe; the black hole horizon serves as the surface of Lifshitz transition between the vacua with type-I and type-II Weyl points; etc.

Symmetries of the physical world have guided formulation of fundamental laws, including relativistic quantum field theory and understanding of possible states of matter. Topological defects (TDs) often control the universal behavior of macroscopic quantum systems, while topology and broken symmetries determine allowed TDs. Taking advantage of the symmetry-breaking patterns in the phase diagram of nanoconfined superfluid 3He, we show that half-quantum vortices (HQVs)-linear topological defects carrying half quantum of circulation-survive transitions from the polar phase to other superfluid phases with polar distortion. In the polar-distorted A phase, HQV cores in 2D systems should harbor non-Abelian Majorana modes. In the polar-distorted B phase, HQVs form composite defects-walls bounded by strings hypothesized decades ago in cosmology. Our experiments establish the superfluid phases of 3He in nanostructured confinement as a promising topological media for further investigations ranging from topological quantum computing to cosmology and grand unification scenarios.

/pdf/half-quantum-vortices-and-walls-bounded-by-strings-in-the-kt6t0u3v0e.pdf

Half-quantum vortices and walls bounded by strings in the polar-distorted phases of topological superfluid 3He

One of the most sought-after objects in topological quantum-matter systems is a vortex carrying half a quantum of circulation. They were originally predicted to exist in superfluid ^{3}He-A but have never been resolved there. Here we report an observation of half-quantum vortices (HQVs) in the polar phase of superfluid ^{3}He. The vortices are created with rotation or by the Kibble-Zurek mechanism and identified based on their nuclear magnetic resonance signature. This discovery provides a pathway for studies of unpaired Majorana modes bound to the HQV cores in the polar-distorted A phase.

/pdf/observation-of-half-quantum-vortices-in-topological-4kshsitrs5.pdf

Observation of Half-Quantum Vortices in Topological Superfluid ^{3}He.

We report the first observation of the polar phase of superfluid $^{3}\mathrm{He}$. This phase appears in $^{3}\mathrm{He}$ confined in a new type of aerogel with a nearly parallel arrangement of strands which play the role of ordered impurities. Our experiments qualitatively agree with theoretical predictions and suggest that in other systems with unconventional Cooper pairing (e.g., in unconventional superconductors) similar phenomena may be found in the presence of anisotropic impurities.

Polar Phase of Superfluid (3)He in Anisotropic Aerogel.

We report results of experiments with superfluid ^{3}He confined in aerogels with parallel strands which lead to anisotropic scattering of ^{3}He quasiparticles. We vary boundary conditions for the scattering by covering the strands with different numbers of atomic ^{4}He layers and observe that the superfluid phase diagram and the nature of superfluid phases strongly depend on the coverage. We assume the main reason for these phenomena is a magnetic channel of the scattering which becomes important at low coverages and can be essential in other Fermi systems with triplet pairing.

Effect of Magnetic Boundary Conditions on Superfluid ^{3}He in Nematic Aerogel

The polar phase of $^{3}\mathrm{He}$, which is topological spin-triplet superfluid with the Dirac nodal line in the spectrum of Bogoliubov quasiparticles, has been recently stabilized in a nanoconfined geometry. We pump magnetic excitations (magnons) into the sample of polar phase and observe how they form a Bose-Einstein condensate, revealed by coherent precession of the magnetization of the sample. Spin superfluidity, which supports this coherence, is associated with the spontaneous breaking of U(1) symmetry by the phase of precession. We observe the corresponding Nambu-Goldstone boson and measure its mass emerging when applied rf field violates the U(1) symmetry explicitly. We suggest that the magnon BEC in the polar phase is a powerful probe for topological objects such as vortices and solitons and topological nodes in the fermionic spectrum.

/pdf/bose-einstein-condensation-of-magnons-and-spin-superfluidity-37m3ow1y9r.pdf

Bose-Einstein Condensation of Magnons and Spin Superfluidity in the Polar Phase of HE 3

The ground state of a fermionic condensate is well protected against perturbations in the presence of an isotropic gap. Regions of gap suppression, surfaces and vortex cores which host Andreev-bound states, seemingly lift that strict protection. Here we show that the role of bound states is more subtle: when a macroscopic object moves in superfluid $^3$He at velocities exceeding the Landau critical velocity, little to no bulk pair breaking takes place, while the damping observed originates from the bound states covering the moving object. We identify two separate timescales that govern the bound state dynamics, one of them much longer than theoretically anticipated, and show that the bound states do not interact with bulk excitations.

/pdf/fundamental-dissipation-due-to-bound-fermions-in-the-zero-1l9bb9cmbi.pdf

A. A. Soldatov

Papers

Observation of Half-Quantum Vortices in Topological Superfluid ^{3}He.

Polar Phase of Superfluid (3)He in Anisotropic Aerogel.

Effect of Magnetic Boundary Conditions on Superfluid ^{3}He in Nematic Aerogel

Bose-Einstein Condensation of Magnons and Spin Superfluidity in the Polar Phase of HE 3

Fundamental dissipation due to bound fermions in the zero-temperature limit.