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The coupling of spin angular momentum and mechanical rotation is utilized for generating a spin current of nuclear spins of the liquid helium-3. A hydrodynamic flow in the liquid helium-3 provides a vorticity gradient which can be regarded as a gradient of effective magnetic field by the spin-rotation coupling. The effective field gradient generates a nuclear spin current owing to the Stern–Gerlach effect. We quantitatively estimate the magnitude of the field gradient based on the microscopic Boltzmann equation. The effective field gradient is large enough to be detected by NMR measurements.

We theoretically investigate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state by using the microscopic quasi-classical Eilenberger equation. The Pauli paramagnetic effects and the orbital depairing effects due to vortices are treated in an equal footing for three dimensional spherical Fermi surface model and $s$-wave pairing. The field evolution of the LO state is studied in detail, such as the $H$-$T$ phase diagram, spatial structures of the order parameter, the paramagnetic moment, and the internal filed. Field-dependences of various thermodynamic quantities: the paramagnetic moment, entropy, and the zero-energy density of states are calculated. Those quantities are shown to start quickly growing upon entering the LO state. We also evaluate the wave length of the LO modulation, the flux line lattice form factors for small angle neutron scattering, and the NMR spectra to facilitate the identification of the LO state. Two cases of strong and intermediate Pauli paramagnetic effect are studied comparatively. The possibility of the LO phase in Sr$_2$RuO$_4$, CeCoIn$_5$, CeCu$_2$Si$_2$, and the organic superconductors is critically examined and crucial experiments to identify it are proposed., 33 pages, 7 figures

On the basis of the microscopic quasiclassical Eilenberger theory, we analyze the recent angle-resolved specific heat experiment carried out at low temperature for ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ to identify the superconducting gap symmetry, comprising either horizontal or vertical line nodes relative to the tetragonal crystal symmetry. Several characteristics, in particular, the landscape of the in-plane oscillation amplitude ${A}_{4}(B,T)$ with a definite sign for almost the entire $B\text{\ensuremath{-}}T$ plane are best explained by the horizontal line node symmetry, especially when the multiband and Pauli paramagnetic effects are taken into account. The present analysis of ${A}_{4}(B,T)$ with definite sign points to the presence of an anomalous field region at a lower temperature in the experimental data, whose origin is investigated. Our theory demonstrates the application and uniqueness of the field-rotating thermodynamic measurements in uncovering the precise gap structure for target materials.

In order to gain insight into the superconducting (SC) gap of UBe 13 , we studied its quasiparticle excitations by means of heat-capacity measurements. Quite unexpectedly, we found the isotropic C ( H ) ∝ H behavior in low fields at low temperatures, implying the absence of nodal quasiparticle excitations. This result indicates that the SC gap in UBe 13 is fully open over the Fermi surfaces. Furthermore, we observed a characteristic oscillation of heat capacity both in the SC and non-Fermi-liquid normal states above ∼2 T, and the angular variation of heat capacity possibly originates from anisotropic magnetic response of the heavy-electron state. Our result regarding the low-energy quasiparticle excitations in the SC and normal states will be a clue to understand the unusual nature of UBe 13 .

We have formulated the scattering theory on Majorana fermions emerging in the surface bound state of the superfluid ^{3}He B phase (^{3}He-B) by an impurity. By applying the theory to the electron bubble, which is regarded as the impurity, trapped below a free surface of ^{3}He-B, the observed mobility of the electron bubble [J. Phys. Soc. Jpn. 82, 124607 (2013)JUPSAU0031-901510.7566/JPSJ.82.124607] is quantitatively reproduced. The mobility is suppressed in low temperatures from the expected value in the bulk ^{3}He-B by the contribution from the surface Majorana fermions. By contrast, the mobility does not depend on the trapped depth of the electron bubble in spite of the spatial variation of the wave function of the surface Majorana fermions. Our formulated theory demonstrates the depth-independent mobility by considering intermediate states in the scattering process. Therefore, we conclude that the experiment has succeeded in observing Majorana fermions in the surface bound state.

In the superfluid 3He A-phase, the amplitude and direction of angular momentum by topological edge mass current coincide with the expected amplitude and direction of ground-state angular momentum by Cooper pairs. The topological mass current also flows at a domain wall in the A-phase, namely, an interface between the A-phases with the opposite direction of l-vector. We show that the amplitude of the topological mass current at the energetically favorable domain wall is almost equal to that at the specular edge; however, the topological mass current flows toward the opposite direction expected from angular momentum of Cooper pairs. Thus, the direction of topological mass current cannot be fixed only by the topological nature which is related to angular momentum of Cooper pairs.

We have formulated the scattering theory on Majorana fermions emerging in the surface bound state of the superfluid ^{3}He B phase (^{3}He-B) by an impurity. By applying the theory to the electron bubble, which is regarded as the impurity, trapped below a free surface of ^{3}He-B, the observed mobility of the electron bubble [J. Phys. Soc. Jpn. 82, 124607 (2013)JUPSAU0031-901510.7566/JPSJ.82.124607] is quantitatively reproduced. The mobility is suppressed in low temperatures from the expected value in the bulk ^{3}He-B by the contribution from the surface Majorana fermions. By contrast, the mobility does not depend on the trapped depth of the electron bubble in spite of the spatial variation of the wave function of the surface Majorana fermions. Our formulated theory demonstrates the depth-independent mobility by considering intermediate states in the scattering process. Therefore, we conclude that the experiment has succeeded in observing Majorana fermions in the surface bound state.

This paper presents the results of specific-heat and magnetization measurements, in particular their field-orientation dependence, on the first discovered heavy-fermion superconductor ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ (${T}_{\mathrm{c}}\ensuremath{\sim}0.6$ K). We discuss the superconducting gap structure and the origin of the anomalous pair-breaking phenomena, leading, e.g., to the suppression of the upper critical field ${H}_{\mathrm{c}2}$, found in the high-field region. The data show that the anomalous pair breaking becomes prominent below about 0.15 K in any field direction, but occurs closer to ${H}_{\mathrm{c}2}$ for $H\ensuremath{\parallel}c$. The presence of this anomaly is confirmed by the fact that the specific-heat and magnetization data satisfy standard thermodynamic relations. Concerning the gap structure, field-angle dependencies of the low-temperature specific heat within the $ab$ and $ac$ planes do not show any evidence for gap nodes. From microscopic calculations in the framework of a two-band full-gap model, the power-law-like temperature dependencies of $C$ and $1/{T}_{1}$, reminiscent of nodal superconductivity, have been reproduced reasonably. These facts further support multiband full-gap superconductivity in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$.

The measured heavy electron density of states in the correlated (unconventional) superconductor UPd${}_{2}$Al${}_{3}$ contains horizontal line nodes in the gap symmetry, a signal that the superconducting pairing mechanism could be mediated by magnetic fluctuations.

We propose a spectroscopic method to identify the nodal gap structure in unconventional superconductors. This method best suits for locating the horizontal line node and for pinpointing the isolated point nodes by measuring polar angle ($\theta$) resolved zero energy density of states $N(\theta)$. This is measured by specific heat or thermal conductivity at low temperatures under a magnetic field. We examine a variety of uniaxially symmetric nodal structure, including point and/or line nodes with linear and quadratic dispersions, by solving Eilenberger equation in vortex states. It is found that (A) the maxima of $N(\theta)$ continuously shift from the anti-nodal to the nodal direction ($\theta_{\rm n}$) as a field increases accompanying the oscillation pattern reversal at low and high fields. Furthermore, (B) local minima emerge next to $\theta_{\rm n}$ on both sides except for the case of linear point node. These features are robust and detectable experimentally. Experimental results of $N(\theta)$ performed on several superconductors, UPd$_2$Al$_3$, URu$_2$Si$_2$, Cu$_x$Bi$_2$Se$_3$, and UPt$_3$, are examined and commented in light of the present theory., Comment: 10 pages, 8 figures, full paper version

Quasiparticle excitations in UPd_{2}Al_{3} were studied by means of heat-capacity (C) measurements under rotating magnetic fields using a high-quality single crystal. The field dependence shows C(H)∝H^{1/2}-like behavior at low temperatures for both two hexagonal crystal axes, i.e., H∥[0001] (c axis) and H∥[112[over ¯]0] (a axis), suggesting the presence of nodal quasiparticle excitations from heavy bands. At low temperatures, the polar-angle (θ) dependence of C exhibits a maximum along H∥[0001] with a twofold symmetric oscillation below 0.5 T, and an unusual shoulder or hump anomaly has been found around 30°-60° from the c axis in C(θ) at intermediate fields (1≲μ_{0}H≲2 T). These behaviors in UPd_{2}Al_{3} purely come from the superconducting nodal quasiparticle excitations, and can be successfully reproduced by theoretical calculations assuming the gap symmetry with a horizontal linear line node. We demonstrate the whole angle-resolved heat-capacity measurements done here as a novel spectroscopic method for nodal gap determination, which can be applied to other exotic superconductors.

Motivated by a recent experiment on the superfluid 3He confined in a thin slab, we discuss the Majorana edge modes under the experimental situation. We solve the quasi-classical Eilenberger equation, which is quantitatively reliable, to evaluate several observables, such as local density of states, mass current for the A-phase, and spin current for the B-phase. On the basis of the quantitative calculation, we propose several experiments to check the existence of the Majorana modes.

The gap structure of Sr$_2$RuO$_4$, which is a longstanding candidate for a chiral p-wave superconductor, has been investigated from the perspective of the dependence of its specific heat on magnetic field angles at temperatures as low as 0.06 K ($\sim 0.04T_{\rm c}$). Except near $H_{\rm c2}$, its fourfold specific-heat oscillation under an in-plane rotating magnetic field is unlikely to change its sign down to the lowest temperature of 0.06 K. This feature is qualitatively different from nodal quasiparticle excitations of a quasi-two-dimensional superconductor possessing vertical lines of gap minima. The overall specific-heat behavior of Sr$_2$RuO$_4$ can be explained by Doppler-shifted quasiparticles around horizontal line nodes on the Fermi surface, whose in-plane Fermi velocity is highly anisotropic, along with the occurrence of the Pauli-paramagnetic effect. These findings, in particular, the presence of horizontal line nodes in the gap, call for a reconsideration of the order parameter of Sr$_2$RuO$_4$., Comment: 5 pages, 3 figures (main text) + 3 pages, 3 figures (supplemental material), accepted for publication in J. Phys. Soc. Jpn

There has been intense interest in various Fermion superfluids in neutral atom liquids and gases, including chiral p-wave pairing in 3He-A phase and Feshbach-resonanced 6Li atom gases and d-wave pairing in atom gases. It is particularly interesting to find exotic vortices and associated low-lying Fermionic excitations under rotation. Here we report on our efforts of those topics: (1) Majorana Fermion in chiral superfluids near a p-wave Feshbach resonance. (2) Possible half-quantum vortices in p-wave superfluids of trapped Fermion atom gases. (3) Stability of a half-quantum vortex in rotating superfluid 3He-A between parallel plates. (4) Majorana bound state in rotating superfluid 3He-A between parallel plates. (5) Non-Abelian Fractional vortex in d-wave Feshbach resonance superfluids. We will summarize some of those works in a coherent manner in order to bridge the understanding between cold atom community and superfluid 3He community by stressing the importance of cross fertilization between them.

Motivated by recent experiments on heavy fermion materials ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ and ${\mathrm{UBe}}_{13}$, we develop a framework to capture generic properties of multiband superconductors with strong Pauli paramagnetic effect (PPE). In contrast to the single band case, the upper critical field ${H}_{\mathrm{c}2}$ can remain a second-order transition even for strong PPE cases. The expected first-order transition is hidden inside ${H}_{\mathrm{c}2}$ and becomes a crossover due to the interplay of multibandness. The present theory based on full self-consistent solutions of the microscopic Eilenberger theory explains several mysterious anomalies associated with the crossover and the ``empty'' vortex core state which is observed by recent STM experiment on ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$.

The superfluid $^{3}\mathrm{He}$ formed by spin-triplet $p$-wave Cooper pairs is a typical topological superfluid. In the superfluid $^{3}\mathrm{He}\text{\ensuremath{-}}B$ phase, several kinds of vortices classified by spatial symmetries ${P}_{1}$, ${P}_{2}$, and ${P}_{3}$ are produced, where ${P}_{1}$ is inversion symmetry, ${P}_{2}$ is magnetic reflection symmetry, and ${P}_{3}$ is magnetic $\ensuremath{\pi}$-rotation symmetry. We have calculated the vortex bound states by the Bogoliubov--de Gennes theory and the quasiclassical Eilenberger theory, and also clarified symmetry protection of the low-energy excitations by the spatial symmetries. On the symmetry protection, ${P}_{3}$ symmetry plays a key role which gives twofold-degenerate Majorana zero modes. Then, the bound states in the most symmetric $o$ vortex with ${P}_{1}$, ${P}_{2}$, and ${P}_{3}$ symmetries and in $w$ vortex with ${P}_{3}$ symmetry have the symmetry-protected degenerate Majorana zero modes. On the other hand, zero-energy modes in the $v$ vortex believed to be realized in the actual $B$ phase are not protected, and in consequence become gapped by breaking axial symmetry. The excitation gap may have been observed as the variation of critical velocity. We have also suggested an experimental setup to create the $o$ vortex with Majorana zero modes by a confinement and a magnetic field.

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., Comment: 82 pages, 33 figures

Low-energy quasiparticle (QP) excitations in the heavy-fermion superconductor URu$_2$Si$_2$ were investigated by specific-heat $C(T, H, \phi, \theta)$ measurements of a high-quality single crystal. The occurrence of QP excitations due to the Doppler-shift effect was detected regardless of the field direction in $C(H)$ of the present clean sample, which is in sharp contrast to a previous report. Furthermore, the polar-angle-dependent $C(\theta)$ measured under a rotating magnetic field within the ac plane exhibits a shoulder-like anomaly at $\theta \sim 45$ deg and a sharp dip at $\theta = 90$ deg ($H \parallel a$) in the moderate-field region. These features are supported by theoretical analyses based on microscopic calculations assuming the gap symmetry of $k_z(k_x+ik_y)$, whose gap structure is characterized by a combination of a horizontal line node at the equator and point nodes at the poles. The present results have settled the previous controversy over the gap structure of URu$_2$Si$_2$ and have authenticated its chiral $d$-wave superconductivity., Comment: 4 pages, 4 figures, published in J. Phys. Soc. Jpn

Coherence effects by the impurity scattering of Caroli–de Gennes–Matricon (CdGM) modes in a vortex for the E1u planar state and the E1u chiral state which are possible superconducting states in UPt3 have been studied. First, we analytically derived the eigenvalue and eigenfunction of the CdGM modes under magnetic fields producing a vortex without impurities. Then, we studied impurity effects on the CdGM modes by introducing the impurity self-energy, which are dominated by the coherence factor depending on the eigenfunction of the CdGM modes. The difference of the impurity effects for the E1u planar and E1u chiral states owing to the magnetic field is possibly utilized to identify the gap function in UPt3.

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., Invited Topical Review Paper: 99 pages, 24 figures

Low-energy quasiparticle excitations in the superconducting (SC) state of UBe$_{13}$ were studied by means of specific-heat ($C$) measurements in a rotating field. Quite unexpectedly, the magnetic-field dependence of $C(H)$ is linear in $H$ with no angular dependence at low fields in the SC state, implying that the gap is fully open over the Fermi surfaces, in stark contrast to the previous expectation. In addition, a characteristic cubic anisotropy of $C(H)$ was observed above 2~T with a maximum (minimum) for $H$ $||$ $[001]$ ($[111]$) within the $(1\bar{1}0)$ plane, both in the normal as well as in the SC states. This oscillation possibly originates from the anisotropic response of the heavy quasiparticle bands, and might be a key to understand the unusual properties of UBe$_{13}$., Comment: 5 pages, 4 figures

The temperature, field, and field-orientation dependences of the electronic specific heat Ce of the ironpnictide superconductor KFe2As2 have been investigated. Thermodynamic evidence of the presence of line nodes is obtained from the T and $\sqrt{H}$ linear dependences of Ce/T in the low-T and low-H region. Under a magnetic field rotated within the tetragonal ab plane, a fourfold oscillation is observed in Ce with a sign change at 0.08Tc. On the basis of the Doppler-shift analysis, the observed Ce minima in H // [100] at low T indicate the presence of line nodes somewhere on the Fermi surface where the Fermi velocity is parallel to the [100] direction; this is consistent with the octet-line-node scenario proposed recently by a photoemission experiment. In addition, the low-T Ce/T exhibits an unusual upturn on cooling at moderate fields only for H // ab, which is understood in terms of the strong Pauli paramagnetic effect on multiband superconductivity., 4 pages, 4 figures, published in J. Phys. Soc. Jpn

Quantum Handedness When a rotating object is placed in circulating fluid, an imbalance of pressures on either side of it causes a deflecting force called the Magnus force. The quantum analog of this effect has been predicted to appear in the low-temperature A phase of 3 He, where the Cooper pairs forming the superfluid have a specific handedness. An impurity traveling through such a superfluid would experience a deflecting force in the direction determined by the chirality of the pairs. Ikegami et al. (p. 59 ) trapped impurities beneath the free surface of 3 He, set them in motion, and demonstrated the existence of this deflecting force by measuring the differential transverse current. The sign of the deflection varied over different cooling runs, indicating that the system was choosing one or the other chirality upon entering the superfluid phase—a signature of spontaneous symmetry breaking.

On the basis of Eilenberger theory, surface bound states and spontaneous current are studied in cyclic d-wave superconductors as a broken time-reversal symmetry of superconductivity in cubic lattice symmetry. We discuss how the spontaneous current and the electronic states depend on the orientation of the surface relative to the symmetry of superconductivity. The condition for topological Fermi arcs of zero-energy surface bound states to appear is identified in the complex pairing function of the cyclic d-wave., 4 pages, 4 figures

The total angular momentum associated with the edge mass current flowing at the boundary in the superfluid ${}^{3}$He $A$ phase confined in a disk is proved to be $L=N\ensuremath{\hbar}/2$, consisting of ${L}^{\mathit{MJ}}=N\ensuremath{\hbar}$ from the Majorana quasiparticles (QPs) and ${L}^{\mathrm{cont}}=\ensuremath{-}N\ensuremath{\hbar}/2$ from the continuum state. We show this based on an analytic solution of the chiral order parameter for the quasiclassical Eilenberger equation. Important analytic expressions are obtained for mass current, angular momentum, and density of states (DOS). Notably the DOS of the Majorana QPs is exactly ${N}_{0}/2$ (${N}_{0}$: normal state DOS) responsible for the factor 2 difference between ${L}^{\mathit{MJ}}$ and ${L}^{\mathrm{cont}}$. The current decreases as ${E}^{\ensuremath{-}3}$ against the energy $E$, and $L(T)\ensuremath{\propto}\ensuremath{-}{T}^{2}$. This analytic solution is fully backed up by numerically solving the Eilenberger equation. We touch on the so-called intrinsic angular momentum problem.

We propose a method utilizing edge current to observe Majorana fermions in the surface Andreev bound state for the superfluid $^3$He A- and B-phases. The proposal is based on self-consistent analytic solutions of quasi-classical Green's function with an edge. The local density of states and edge mass current in the A-phase or edge spin current in the B-phase can be obtained from these solutions. The edge current carried by the Majorana fermions is partially cancelled by quasiparticles (QPs) in the continuum state outside the superfluid gap. QPs contributing to the edge current in the continuum state are distributed in energy even away from the superfluid gap. The effect of Majorana fermions emerges in the depletion of the edge current by temperature within a low-temperature range. The observations that the reduction in the mass current is changed by $T^2$-power in the A-phase and the reduction in the spin current is changed by $T^3$-power in the B-phase establish the existence of Majorana fermions. We also point out another possibility for observing Majorana fermions by controlling surface roughness., Comment: 13 pages, 4 figures, published version

Stability of a half-quantum vortex (HQV) in superfluid $^3$He has been discussed recently by Kawakami, Tsutsumi and Machida in Phys. Rev. B {\bf 79}, 092506 (2009). We further extend this work here and consider the A$_2$ phase of superfluid $^3$He confined in thin slab geometry and analyze the HQV realized in this setting. Solutions of HQV and singly quantized singular vortex are evaluated numerically by solving the Ginzburg-Landau (GL) equation and respective first critical angular velocities are obtained by employing these solutions. We show that the HQV in the A$_2$ phase is stable near the boundary between the A$_2$ and A$_1$ phases. It is found that temperature and magnetic field must be fixed first in the stable region and subsequently the angular velocity of the system should be increased from zero to a sufficiently large value to create a HQV with sufficiently large probability. A HQV does not form if the system starts with a fixed angular velocity and subsequently the temperature is lowered down to the A$_2$ phase. It is estimated that the external magnetic field with strength on the order of 1 T is required to have a sufficiently large domain in the temperature-magnetic field phase diagram to have a stable HQV., Comment: 5 pages, 5 figures

In type-II superconductors (Fetter & Hohenberg, 1969), magnetic fields penetrate into superconductors as quantized flux lines with flux quanta φ0 = hc/2e, where h is Planck constant, c velocity of light, e electron’s charge. Around a flux line, pair potential ∆(r) of Cooper pair has a vortex structure, where ∆(r) has phase winding 2π reflecting screening super-current around a flux line. At the vortex core, amplitude |∆(r)| is suppressed, and low energy excitations appear within the superconducting gap of the electronic states. Vortex physics makes important roles in the study of unconventional superconductors, because unconventional characters hidden in the uniform superconducting state at a zero field appear around vortices. For example, in superconductors with anisotropic superconducting gap on Fermi surface in momentum space, electronic states around a vortex show real-space anisotropy in the local density of states (LDOS) N(E, r) around vortex core. The vortex core image is observed by scanning tunneling microscopy (STM) (Hess et al., 1990; Nishimori et al., 2004). Around vortex cores, local zero-energy electronic states at Fermi level are seen as star shape with tails extending toward node or weak-gap directions (Hayashi et al., 1996; 1997; Ichioka et al., 1996; Schopohl & Maki, 1995). From the spatial average of the zero-energy states, we can estimate the zero-energy density of states (DOS) N(E = 0), which determines low temperature (T) behaviors of physical quantities. Due to the differences of electronic states around the vortex core, N(E = 0) shows different magnetic field (H) dependences. These H-dependences are studied to identify the pairing symmetry in the experiments for vortex states, such as, electronic specific heat (Moler et al., 1994; Nohara et al., 1999), electronic thermal conductivity, and paramagnetic susceptibility (Zheng et al., 2002). For example, the H-dependence of low temperature specific heat C(H) is often used to distinguish the presence of nodes in the pairing potential. As for Sommerfeld coefficient γ(H) ≡ limT→0 C(H)/T, γ(H) ∝ H in s-wave pairing with full gap, and γ(H) ∝ √ H by the Volovik effect in d-wave pairing with line nodes (Ichioka et al., 1999a;b; Miranovic et al., 2003; Nakai et al., 2004; Volovik, 1993). The curves of γ(H) are expected to smoothly recover to the normal state value towards the upper critical field Hc2. However, in some heavy fermion superconductors, C(H) deviates from these curves. In CeCoIn5, C(H) shows convex curves, i.e., C(H) ∝ Hα (α > 1) at higher fields(Ikeda et al., 2001). This behavior is not understood only by effects of the pairing symmetry. A similar C(H) behavior is observed also in UBe13 (Ramirez et al., 1999). The experimental data of magnetization curve Mtotal(H) in CeCoIn5 show a convex curve at higher fields, instead of a conventional concave curve (Tayama et al., 2002). As an 10

Motivated by experiments on the superfluid 3He confined in a thin slab, we design a concrete experimental setup for observing the Majorana surface states. We solve the quasi-classical Eilenberger equation, which is quantitatively reliable, to evaluate several quantities, such as local density of states (LDOS), mass current for the A-phase, and spin current for the B-phase. In connection with realistic slab samples, we consider the upper and lower surfaces and the side edges including the corners with several thicknesses. Consequently the influence on the Majorana zero modes from the spatial variation of l-vector for the A-phase in thick slabs and the energy splitting of the zero-energy quasi-particles for the B-phase confined in thin slabs are demonstrated. The corner of slabs in the B-phase is accompanied by the unique zero-energy LDOS of corner modes. On the basis of the quantitative calculation, we propose several feasible and verifiable experiments to check the existence of the Majorana surface states, such as the measurement of specific heat, edge current, and anisotropic spin susceptibility.

In order to identify the gap structure of CeIrIn5, we measured field-angle-resolved specific heat C(phi) by conically rotating the magnetic field H around the c axis at low temperatures down to 80 mK. We revealed that C(phi) exhibits a fourfold angular oscillation, whose amplitude decreases monotonically by tilting H out of the ab plane. Detailed microscopic calculations based on the quasiclassical Eilenberger equation confirm that the observed features are uniquely explained by assuming the dx2-y2-wave gap. These results strongly indicate that CeIrIn5 is a dx2-y2-wave superconductor and suggest the universal pairing mechanism in CeMIn5 (M = Co, Rh, and Ir)., Comment: 4 pages, 4 figures, Selected for an Editors' Suggestion and a Synopsis in Physics

Motivated by a recent experiment on the superfluid 3He A-phase with a chiral p-wave pairing confined in a thin slab, we propose designing a concrete experimental setup for observing the Majorana edge modes that appear around the circumference edge region. We solve the quasi-classical Eilenberger equation, which is quantitatively reliable, to evaluate several observables. To derive the property inherent to the Majorana edge state, the full quantum mechanical Bogoliubov-de Gennes equation is solved in this setting. On the basis of the results obtained, we perform decisive experiments to check the Majorana nature., 5 pages, 5 figures

Motivated by the on-going rotating cryostat experiments in ISSP, Univ. of Tokyo, we explore the textures and vortices in superfluid 3He-A phase confined in narrow cylinders, whose radii are R=50mum and 115mum. The calculations are based on the Ginzburg-Landau (GL) framework, which fully takes into account the orbital (l-vector) and spin (d-vector) degrees of freedom for chiral p-wave pairing superfluid. The GL free energy functional is solved numerically by using best known GL parameters appropriate for the actual experimental situations at P=3.2MPa and H=21.6mT. We identify the ground state l-vector configuration as radial disgyration (RD) texture with the polar core both at rest and low rotations and associated d-vector textures for both narrow cylinder systems under high magnetic fields. The RD which has a singularity at center, changes into Mermin-Ho texture above the critical rotation speed which is determined precisely, providing an experimental check for own proposal., 22 pages, 12 figures

The stability conditions for the singular vortex which accompanies Majorana zero modes at the core are investigated for p-wave resonant superfluids of atomic Fermi gases. Within the Ginzburg-Landau framework we determine the stable conditions in the parameter space for the external rotation frequency and the harmonic trap frequency. There exists the narrow stable region in this parameter space for quasi-two-dimensional condensates. We also describe the detailed characterizations of the spatial structure of the order parameter in the chiral p-wave superfluids under rotation., Comment: 6 pages, 5 figures, replaced by the version to appear in J. Phys. Soc. Jpn.

To identify the superconducting gap structure in URu2Si2, we perform field-angle-dependent specific heat measurements for the two principal orientations in addition to field rotations, and a theoretical analysis based on microscopic calculations. The Sommerfeld coefficient gamma(H)'s in the mixed state exhibit a distinctly different field dependence. This comes from point nodes and the substantial Pauli paramagnetic effect of URu2Si2. These two features combined give rise to a consistent picture of superconducting properties, including a possible first order transition of Hc2 at low temperatures.

We report details of the observation of the intrinsic Magnus (IM) force acting on negative and positive ions trapped just below a free surface of the A phase of superfluid 3He (3He-A). From the transport measurements of the ions along the surface, we found that the IM force acts on both the negative and positive ions. We also demonstrate that the transport measurements could distinguish whether the surface is composed of a chiral monodomain or multiple chiral domains. For multiple chiral domains, the current of the ions was found to be irreproducible and unstable, which was reasonably explained by the formation of the chiral domain structure and the dynamics of the chiral domain walls. For chiral monodomains, the appearance ratio of chirality emerging upon cooling through the superfluid transition temperature was found to depend on the direction of the external magnetic field, which implies the existence of an unknown coupling between the chirality and the magnetic field.

Junctions and interfaces consisting of unconventional superconductors provide an excellent experimental playground to study exotic phenomena related to the phase of the order parameter. Not only does the complex structure of unconventional order parameters have an impact on the Josephson effects, but it also may profoundly alter the quasiparticle excitation spectrum near a junction. Here, by using spectroscopic-imaging scanning tunneling microscopy, we visualize the spatial evolution of the LDOS near twin boundaries (TBs) of the nodal superconductor FeSe. The π/2 rotation of the crystallographic orientation across the TB twists the structure of the unconventional order parameter, which may, in principle, bring about a zero-energy LDOS peak at the TB. The LDOS at the TB observed in our study, in contrast, does not exhibit any signature of a zero-energy peak, and an apparent gap amplitude remains finite all the way across the TB. The low-energy quasiparticle excitations associated with the gap nodes are affected by the TB over a distance more than an order of magnitude larger than the coherence length ξab. The modification of the low-energy states is even more prominent in the region between two neighboring TBs separated by a distance ≈7ξab. In this region, the spectral weight near the Fermi level (≈±0.2 meV) due to the nodal quasiparticle spectrum is almost completely removed. These behaviors suggest that the TB induces a fully gapped state, invoking a possible twist of the order parameter structure, which breaks time-reversal symmetry. ISSN:2160-3308

We represent our recent advances on the study of the gap symmetry in UPt3 by thermal conductivity tensors κij (i, j = x, y, z). The field-angle-resolved thermal conductivity κzz shows spontaneous twofold symmetry breaking in the gap function for the high-field C-phase, indicating that the pairing symmetry of UPt3 belongs to an E1u representation in the f-wave category. We also demonstrate that the proposed pairing symmetry is compatible with most of the experimental results reported until now.

The present study assessed in healthy nocturnal juveniles the presence or absence of the hypnotic action of exogenous melatonin and its effect on core temperature by oral administration of exogenous melatonin at a pharmacologically low dose 15 min before going to bed at the normal time. Both sleep latency and the number of instances of waking up after falling asleep tended to decrease, while the amount of Stage 2 sleep increased significantly. However, no hypothermic action regarding the rectal temperature for changes in the circadian phase were observed. According to those results, it is possible that exogenous melatonin has some direct hypnotic action on habitual night sleep, although the extent of this action is considered to be relatively weak.

We investigate the topological aspect of the spin-triplet $f$-wave superconductor UPt$_3$ through microscopic calculations of edge- and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov-de Gennes theories. It is shown that a gapless and linear dispersion exists at the edge of the $ab$-plane. This forms a Majorana valley, protected by the mirror chiral symmetry. We also demonstrate that, with increasing magnetic field, vortex-bound quasiparticles undergo a topological phase transition from topologically trivial states in the double-core vortex to zero-energy states in the normal-core vortex. As long as the $d$-vector is locked into the $ab$-plane, the mirror symmetry holds the Majorana property of the zero-energy states, and thus UPt$_3$ preserves topological crystalline superconductivity that is robust against the crystal field and spin-orbit interaction., Comment: 5 pages, 2 figures

Salival and serum melatonin levels after melatonin ingestion were measured by gas chromatography-mass spectrometry. Ingestion of 3 mg melatonin caused a marked increase in serum melatonin (3561 ± 1201 pg/mL) within 20 min, followed by a gradual decrease, but the level still remained higher than the basal level at 240 min after the ingestion. The saliva melatonin 60 min after the ingestion showed the highest level (1177 ± 403 pg/mL) which was one-third of the plasma level. The saliva melatonin level was highly correlated with the serum level throughout the experimental period (r= 0.82, P= 0.0001). These data indicate that the measurement of saliva melatonin level may be a suitable indicator for the melatonin secretion into general circulation.

We investigate the Majorana fermions bound at vortices and specular surface of superfluid 3He A- and B-phases. It is demonstrated that the statistics of the integer vortex which is competitive to the half-quantum vortex is sensitive to the orientation of the magnetic field. We also discuss the surface Andreev bound state in the B-phase, which is expected to behave as Majorana fermions at the zero field limit. Based on the quasiclassical and Bogoliubov-de Gennes theories, we clarify that in the presence of the magnetic field parallel to the surface, the stable configuration of the -vector, which reflects the spin-orbit symmetry breaking, traces the point at which the surface state has the maximum gap.

In the superfluid 3He A-phase, the edge mass current which accompanies surface Andreev bound states flows due to topological features. We discuss the relation between the angular momentum by the edge mass current and the macroscopic intrinsic angular momentum L = Nℏ/2 by the Cooper pairs with aligned angular momenta, based on the quasi-classical theory. At the low temperature limit, half of the edge mass current from the bound state is canceled by that from the continuum state. Then, the magnitude of the total angular momentum is L = Nℏ/2 which corresponds to the macroscopic intrinsic angular momentum of N particles. The net angular momentum decreases monotonously toward zero at the transition temperature. However, that is larger than the macroscopic intrinsic angular momentum expected from the same behavior of the superfluid density in middle temperatures.

Motivated by a recent angle-resolved thermal conductivity experiment that shows a twofold gap symmetry in the high-field and low-temperature C phase in the heavy-fermion superconductor UPt$_3$, we group-theoretically identify the pairing functions as $E_{1u}$ with the $f$-wave character for all the three phases. The pairing functions are consistent with the observation as well as with a variety of existing measurements. By using a microscopic quasi-classical Eilenberger equation with the identified triplet pairing function under applied fields, we performed detailed studies of the vortex structures for three phases, including the vortex lattice symmetry, the local density of states, and the internal field distribution. These quantities are directly measurable experimentally by SANS, STM/STS, and NMR, respectively. It is found that, in the B phase of low $H$ and low $T$, the double-core vortex is stabilized over a singular vortex. In the C phase, thermal conductivity data are analyzed to confirm the gap structure proposed. We also give detailed comparisons of various proposed pair functions, concluding that the present scenario of $E_{1u}$ with the $f$-wave, which is an analogue to the triplet planar state, is better than the $E_{2u}$ or $E_{1g}$ scenario. Finally, we discuss the surface topological aspects of Majorana modes associated with the $E_{1u}^f$ state of planar like features., 13 pages, 10 figures, published version, selected for an Editor's Choice

We investigated temperature, field, and field-orientation dependences of the specific heat of CeIrIn 5 . The upper critical field H c2 was determined down to 80 mK from the specific heat measurements and exhibits a slight suppression at low temperatures. We revealed that the specific heat at 80 mK increases proportional to the square root of H up to nearly H c2 for both H ∥ a b and H ∥ c , implying the presence of line nodes and a weak contribution of the Pauli paramagnetic pair-breaking effect. Furthermore, we revealed that the specific heat shows a fourfold oscillation under a rotating magnetic field in the a b -plane. This result indicates that CeIrIn 5 has fourfold symmetry around the c -axis in its gap amplitude.

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in the vortex lattice state is quantitatively studied using the selfconsistent Eilenberger theory in three-dimensional (3D) space. We estimate free energy to determine the FFLO phase diagram in the H-T plane and stable FFLO wave number in the isotropic system with the 3D Fermi sphere and s-wave pairing. To facilitate the experimental identification of the FFLO state, we investigate the field evolution of NMR spectra and flux line lattice form factors obtained in neutron scattering in the FFLO vortex states. Possible applications of our results to experimental data on CeCoIn5 are mentioned., Comment: 5pages, 4 figures. v2: accepted version for JPSJ

Fundamental properties of superfluids with d-wave pairing symmetry are investigated theoretically. We consider neutral atomic Fermi gases in a harmonic trap, the Cooper pairing being produced by a Feshbach resonance via a d-wave interaction channel. A Ginzburg-Landau (GL) functional is constructed which is symmetry constrained for five component order parameters (OPs). We find stable OP textures and vortices for all the three phases which are known to be the energy minimum of the GL functional; the ferromagnetic, polar and cyclic phases both at rest and under rotation. In particular, we touch upon the stability conditions for a non-Abelian fractional 1/3-vortex in the cyclic phase under rotation. It is proposed how to create the intriguing 1/3-vortex experimentally in atomic gases via optical means., Comment: 10 pages, 6 figures