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.In Sect. 3, we will share the topological aspect of a spinpolarized chiral p-wave superconducting state as a specific model having nontrivial w 2d ¼ Ch 1 . Topology subject to discrete symmetriesNaively, any one-dimensional closed loop S 1 cannot cover the target space S 2 . Thus, a generic 2  2 Hamiltonian in one dimension cannot provide a stable topological structure. However, discrete symmetries of H in Eq. (3) impose strong constraints on the spinorm, and thus nontrivial topological numbers can be introduced even in one dimension, as illustrated below.Particle-hole symmetry C 2 ¼ þ1 (class D)-Let us first suppose that the minimal Hamiltonian (13) holds the PHS C ¼ x K (C 2 ¼ þ1). The operation of PHS changes the spinor mðkÞ to ½Àm x ðÀkÞ; Àm y ðÀkÞ;m z ðÀkÞ. For the momentum space characterized by S 2 , there are two particle-hole invariant momenta, k ¼ 0 and jkj ¼ 1, where the infinite points are identical to a single point. At the particle-hole invariant momenta, the spinorm must point to the north or south pole on S 2 . Therefore, we have two different situations: One is that the spinorsm at k ¼ 0 and jkj ¼ 1 point in the same direction,m z ð0Þ ¼m z ð1Þ; ð26Þ Fig. 3. (Color online) Target spaces M subject to discrete symmetries T and C. Possible trajectories ofm on M with C 2 ¼ þ1 are also depicted.Fig. 24. (Color online) Low-lying quasiparticle spectra for the axisymmetric w-vortex (a) and v-vortex (b) with k z ¼ 0 and ...

The Mermin-Ho and Anderson-Toulouse coreless non-singular vortices are demonstrated to be thermodynamically stable in ferromagnetic spinor Bose-Einstein condensates with the hyperfine state F = 1. The phase diagram is established in a plane of the rotation drive vs the total magnetization by comparing the energies for other competing non-axis-symmetric or singular vortices. Their stability is also checked by evaluating collective modes.PACS numbers: 03.75. Fi, 67.57.Fg, 05.30.Jp Topological structure plays an important and decisive role in various research fields, ranging from condensed matter physics to high energy physics. They provide a common framework to connect diverse fields, enhancing mutual understanding [1].Recent advance of experimental techniques on BoseEinstein condensation (BEC) prompts us to closely and seriously look into theoretical possibilities which were mere imagination for theoreticians in this field. This is particularly true for spinor BEC where all hyperfine states of an atom Bose-condensed simultaneously, keeping these "spin" states degenerate and active. Recently, Barrett at al [3] have succeeded in cooling 87 Rb with the hyperfine state F = 1 by all optical methods without resorting to a usual magnetic trap in which the internal degrees of freedom is frozen. Since the spin interaction of the 87 Rb atomic system is ferromagnetic, based on the refined calculation of the atomic interaction parameters by Klausen at al[4], we now obtain concrete examples of the three component spinor BEC (F = 1, m F = 1, 0, −1) for both antiferromagnetic ( 23 Na) [5] and ferromagnetic interaction cases. In the present spinor BEC the degenerate internal degrees of freedom play an essential role to determine the fundamental physical properties. There is a rich variety of topological defect structures, which are already predicted in the earlier studies [6,7] on the spinor BEC. These are followed by others [8,9,10,11,12,13,14,15,16] who examine these topological structures more closely, such as skyrmion, monopole, meron or axis-symmetric or non axis-symmetric vortices both for antiferromagnetic and ferromagnetic cases.Superfluid 3 He is analogous to the spinor BEC where the neutral Cooper pair possesses the orbital and spin degrees of freedom, thus the order parameter is a multi-component [17]. Among various topological structures, the Mermin-Ho (MH) [18] and Anderson-Toulouse (AT) [19] vortices of a coreless and non-singular l-vector texture are proposed in 3 He-A phase. These are an extremely interesting object to study if they exist. The MH vortex is expected to spontaneously appear in a cylindrical vessel without any external rotation as an equilibrium state because the rigid wall boundary imposes the l-vector perpendicular to the vessel wall. The MH vortex is stable also under slow rotation because of their non-singular coreless structures [20].A similar topological structure, called skyrmion in general is proposed in the spinor BEC. Khawaja and Stoof[10] study a skyrmion in the ferromagnetic BEC with ...

We here demonstrate that the superfluid (3)He-B under a magnetic field in a particular direction stays topological due to a discrete symmetry, that is, in a symmetry protected topological order. Because of the symmetry protected topological order, helical surface Majorana fermions in the B phase remain gapless and their Ising spin character persists. We unveil that the competition between the Zeeman magnetic field and dipole interaction involves an anomalous quantum phase transition in which a topological phase transition takes place together with spontaneous symmetry breaking. Based on the quasiclassical theory, we illustrate that the phase transition is accompanied by anisotropic quantum criticality of spin susceptibilities on the surface, which is detectable in NMR experiments.

It is proposed that the spatially modulated superfluid phase, or the Fulde-Ferrell-Larkin-Ovchinnikov state could be observed in resonant fermion atomic condensates which are realized recently. We examine optimal experimental setups to achieve it by solving the Bogoliubov-de Gennes equation for both idealized one-dimensional and realistic three-dimensional cases. The spontaneous modulation of this superfluid is shown to be directly imaged as the density profiles either by optical absorption or by Stern-Gerlach experiments.

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 evenparity 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.

Plasma synthesis of ammonia was studied at atmospheric pressure using a dielectric-barrier-discharge-plasma reactor equipped with a metal-loaded membrane-like alumina tube as a catalyst between the electrodes. Introducing the pure alumina into N 2 -H 2 plasma resulted in an increase in the ammonia yield and the further improvement was achieved by loading the alumina with Ru, Pt, Ni, and Fe. These results clearly demonstrate the catalytic effects of the alumina and the metals in the plasma reaction. Temperature-programmed desorption and isotope exchange reaction of nitrogen revealed that plasma-excited N 2 molecules were subjected to dissociative adsorptions mainly on the alumina to form atomic N(a) (The suffix ''(a)'' denotes adsorbed species) species, which were converted into ammonia by H 2 plasma. A role of the metals is considered to be acceleration of ammonia formation by the reaction of the alumina-adsorbed N(a) atoms with plasma-activated hydrogen species.

We demonstrate that the three-dimensional Skyrmion, which has remained elusive so far, spontaneously appears as the ground state of SU(2) symmetric Bose-Einstein condensates coupled with a non-Abelian gauge field. The gauge field is a three-dimensional analogue of the Rashba spin-orbit coupling. Upon squashing the SO(3) symmetric gauge field to one- or two-dimensional shapes, we find that the ground state continuously undergoes a change from a three-dimensional to a one- or two-dimensional Skyrmion, which is identified by estimating winding numbers and helicity. All of the emerged Skyrmions are physically understandable with the concept of the helical modulation in a unified way. These topological objects might potentially be realizable in two-component Bose-Einstein condensates experimentally.

The stability of doubly quantized vortices in dilute Bose-Einstein condensates of 23 Na is examined at zero temperature. The eigenmode spectrum of the Bogoliubov equations for a harmonically trapped cigar-shaped condensate is computed and it is found that the doubly quantized vortex is spectrally unstable towards dissection into two singly quantized vortices. By numerically solving the full three-dimensional time-dependent Gross-Pitaevskii equation, it is found that the two singly quantized vortices intertwine before decaying. This work provides an interpretation of recent experiments [A. E. Leanhardt et al. Phys. Rev. Lett. 89, 190403 (2002) The stability of single-quantum vortices has been studied extensively after such vortices were first observed in dilute alkali atom Bose-Einstein condensates (BECs) [1]. Since a single-valued complex order parameter describes the state of the condensate, its phase must undergo a 2πn change along a loop encircling a vortex, where n is the quantum number of the vortex. However, the creation of multiquantum vortices is impossible just by rotating the harmonic trapping potential at a high frequency, since the existence of many singly quantized vortices is energetically more favorable than a single multiquantum vortex [2]. Indeed, vortex lattices composed of single-quantum vortices were observed in such experiments [3].Verifying the proposal of topological phase engineering by Nakahara et al. [4], Leanhardt et al. [5] have recently succeeded in creating vortices simply by reversing the bias magnetic field used to trap the condensate. The vortices created display winding numbers with n = 2 (4π phase winding) or with n = 4 (8π phase winding), depending on the hyperfine spin states used for 23 Na condensates. It was confirmed that the axial angular momentum per particle is 2h (4h) for the doubly (quartically) quantized vortex.In this experiment, after creating a vortex it is held for ∼ 20 ms, watching the vortex core to split into singlequantum vortices. However, no splitting was observed in this time span. Since a doubly quantized vortex is expected to decay spontaneously into two singly quantized vortices owing to energetics, this observation seems to be puzzling. This motivates our investigation of the detailed dynamics of the decay process of multiply quantized vortices in view of the present experimental situation. In dilute BECs, these were the first experimentally realized double-quantum vortices which are known to exist in other superfluids, such as superfluid 3 He-A [6]. Therefore, we have a unique opportunity to examine the physics of multiply quantized vortices.There are several theoretical investigations on the instability of the doubly quantized vortex: The instability due to the bound state in the vortex core was pointed out by Rokhsar [7], who claimed that the decay of vortex states requires the presence of thermal atoms. On the other hand, Pu et al. [8] found that even in the absence of thermal atoms, i.e. in the non-dissipative system, appearance of the modes with...