Focusing on a quantum-limit behavior, we study a single vortex in a clean s-wave type-II superconductor by selfconsistently solving the Bogoliubov-de Gennes equation. The discrete energy levels of the vortex bound states in the quantum limit is discussed. The vortex core radius shrinks monotonically up to an atomic-scale length on lowering the temperature T , and the shrinkage stops to saturate at a lower T . The pair potential, supercurrent, and local density of states around the vortex exhibit Friedel-like oscillations. The local density of states has particle-hole asymmetry induced by the vortex. These are potentially observed directly by STM. PACS number(s): 74.60. Ec, 61.16.Ch, Growing interest has been focused on vortices both in conventional and unconventional superconductors from fundamental and applied physics points of view. This is particularly true for high-T c cuprates, since it is essential that one understands fundamental physical properties of the vortices in the compounds to better control various superconducting characteristics of some technological importance. Owing to the experimental developments, it is not difficult to reach low temperatures of interest where distinctive quantum effects associated with the discretized energy levels of the vortex bound states are expected to emerge. The quantum limit is realized at the temperature where the thermal smearing is narrower than the discrete bound state levels [1]: T /T c ≤ 1/(k F ξ 0 ) with ξ 0 =v F /∆ 0 the coherence length (∆ 0 the gap at T = 0) and k F (v F ) the Fermi wave number (velocity). For example, in a typical layered type-II superconductor NbSe 2 with T c = 7.2 K and k F ξ 0 ∼ 70, the quantum limit is reached below T < 50 mK. As for the high-T c cuprates, the corresponding temperature is rather high: T < 10 K for YBa 2 Cu 3 O 7−δ (YBCO).Important microscopic works to theoretically investigate the quasiparticle spectral structure around a vortex in a clean limit are put forth by Caroli et al. The purposes of the present paper are to reveal the quantum-limit aspects of the single vortex in s-wave superconductors and to discuss a possibility for the observation of them. The present study is motivated by the following recent experimental and theoretical situations: (1) The socalled Kramer and Pesch (KP) effect [1,3,7,8]; a shrinkage of the core radius upon lowering T (to be exact, an anomalous increase in the slope of the pair potential at the vortex center at low T ) is now supported by some experiments [9]. The T dependence of the core size is studied by µSR on NbSe 2 and YBCO [10], which is discussed later. The KP effect, if confirmed, forces us radically alter the traditional picture [11] for the vortex line such as a rigid normal cylindrical rod with the radius ξ 0 . (2) The scanning tunneling microscopy (STM) experiment on YBCO by Maggio-Aprile et al. [12], which enables us to directly see the spatial structure of the low-lying quasiparticle excitations around the vortex, arouses much interest. They claim that surprisingly enough, ...

We study the vortex structure and its field dependence within the framework of the quasi-classical Eilenberger theory to find the difference between the d x 2 −y 2 -and s-wave pairings. We clarify the effect of the d x 2 −y 2 -wave nature and the vortex lattice effect on the vortex structure of the pair potential, the internal field and the local density of states. The d x 2 −y 2 -wave pairing introduces a fourfold-symmetric structure around each vortex core. With increasing field, their contribution becomes significant to the whole structure of the vortex lattice state, depending on the vortex lattice's configuration. It is reflected in the form factor of the internal field, which may be detected by small angle neutron scattering, or the resonance line shape of µSR and NMR experiments. We also study the induced s-and dxy-wave components around the vortex in d x 2 −y 2 -wave superconductors.

The vortex structure of pure d x 2 Ϫy 2-wave superconductors is microscopically analyzed in the framework of the quasiclassical Eilenberger equations. A self-consistent solution for the d-wave pair potential is obtained in the case of an isolated vortex. The vortex core structure, i.e., the pair potential, the supercurrent, and the magnetic field, is found to be fourfold symmetric even in the case that the mixing of the s-wave component is absent. The detailed temperature dependences of these quantities are calculated. The fourfold symmetry becomes clear when the temperature is decreased. The local density of states is calculated for the self-consistently obtained pair potential. From the results, we discuss the flow trajectory of the quasiparticles around a vortex, which is characteristic in d x 2 Ϫy 2-wave superconductors. The experimental relevance of our results to hightemperature superconductors is also given.

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 ...

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