A topological superconductor (TSC) is characterized by the topologically protected gapless surface state that is essentially an Andreev bound state consisting of Majorana fermions. While a TSC has not yet been discovered, the doped topological insulator Cu(x)Bi(2)Se(3), which superconducts below ∼3 K, has been predicted to possess a topological superconducting state. We report that the point-contact spectra on the cleaved surface of superconducting Cu(x)Bi(2)Se(3) present a zero-bias conductance peak (ZBCP) which signifies unconventional superconductivity. Theoretical considerations of all possible superconducting states help us conclude that this ZBCP is due to Majorana fermions and gives evidence for a topological superconductivity in Cu(x)Bi(2)Se(3). In addition, we found an unusual pseudogap that develops below ∼20 K and coexists with the topological superconducting state.

A theory of dispersionless Andreev bound states on surfaces of time-reversal invariant unconventional superconductors is presented. The generalized criterion for the dispersionless Andreev bound state is derived from the bulk-edge correspondence, and the chiral spin structure of the dispersionless Andreev bound states is argued from which the Andreev bound state is stabilized.Then we summarize the criterion in a form of index theorems. The index theorems are proved in a general framework to certify the bulk-edge correspondence. As concrete examples, we discuss (i) d xy -wave superconductor (ii) p x -wave superconductor, and (iii) noncentrosymmetric superconductors. In the last example, we find a peculiar time-reversal invariant Majorana fermion. The time-reversal invariant Majorana fermion shows an unusual response to the Zeeman magnetic field, which can be used to identify it experimentally.

We study edge states of noncentrosymmetric superconductors where spin-singlet d-wave pairing mixes with spin-triplet p (or f)-wave one by spin-orbit coupling. For d(xy)-wave pairing, the obtained Andreev bound state has an anomalous dispersion as compared to conventional helical edge modes. A unique topologically protected time-reversal invariant Majorana bound state appears at the edge. The charge conductance in the noncentrosymmetric superconductor junctions reflects the anomalous structures of the dispersions, particularly the time-reversal invariant Majorana bound state is manifested as a zero bias conductance peak.

We develop a theory of the tunneling spectroscopy for superconducting topological insulators (STIs), where the surface Andreev bound states (SABSs) appear as helical Majorana fermions. Based on the symmetry and topological nature of parent topological insulators, we find that the SABSs in the STIs have a profound structural transition in the energy dispersions. The transition results in a variety of Majorana fermions, by tuning the chemical potential and the effective mass of the energy band. We clarify that Majorana fermions in the vicinity of the transitions give rise to robust zero bias peaks in the tunneling conductance between normal metal/STI junctions.

It has been noted that certain surfaces of Weyl semimetals have bound states forming open Fermi arcs, which are never seen in typical metallic states. We show that the Fermi arcs enable them to support an even more exotic surface state with crossed flat bands in the superconducting state. We clarify the topological origin of the crossed flat bands and the relevant symmetry that stabilizes the cross point. We also discuss their possible experimental verification by tunneling spectroscopy.

Several samples of chrysotile asbestos from different localities, including a synthetic sample, were electron-microscopically observed by the lattice imaging method along two directions parallel and perpendicular to the fibre axis. The results are as follows: (a) The lattice fringes of 4.5 A corresponding to 020 were often tilted to the edge of the fibrils with an angular distribution ranging up to about 10 ° with a peak value at a few degrees, depending on the sample. (b) Most of the fibrils examined were hollow cylinders and their circumferential lattice layers form spiral or multi-spiral layers. The perfectly concentric cylindrical layers were also found with a frequency depending on the sample. (c) Unusual growth patterns which cannot be explained by Jagodzinski and Kunze's model were observed. (d) The lattice images of the cortical fibrils (cone-in-cone shape) were observed in the synthetic sample. (e) Most fibrils greater than about 350 A in diameter showed traces of discontinuous growth in two or three steps, depending on the growth conditions, and this gave rise to various distributions of the fibril diameters.

We analyze the normal electronic states of Na 0:35 CoO 2 based on the effective d-p model with a full dorbital freedom using the fluctuation-exchange (FLEX) approximation. The electronic states sensitively depend on the topology of Fermi surfaces, which changes with the crystalline electric splitting (CES) due to the trigonal deformation. We succeed in reproducing the weak pseudogap behaviors in the density of states (DOS) and in the uniform magnetic susceptibility below 300 K, assuming that six small hole pockets predicted by LDA band calculations are absent. If these pockets exist, on the contrary, ''antipseudogap behaviors'' should inevitably appear. Thus, the present study strongly supports the absence of the small hole pockets in Na 0:35 CoO 2 , as reported by recent ARPES measurements. A large Fermi surface around the À-point would account for the superconductivity in water-intercalated samples.

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