Lattice model of three-dimensional topological singlet superconductor with time-reversal symmetry

We study topological phases of time-reversal invariant singlet superconductors in three spatial dimensions. In these particle-hole symmetric systems the topological phases are characterized by an even-numbered winding number $\nu$. At a two-dimensional (2D) surface the topological properties of this quantum state manifest themselves through the presence of $\nu$ flavors of gapless Dirac fermion surface states, which are robust against localization from random impurities. We construct a tight-binding model on the diamond lattice that realizes a topologically nontrivial phase, in which the winding number takes the value $\nu =\pm 2$. Disorder corresponds to a (non-localizing) random SU(2) gauge potential for the surface Dirac fermions, leading to a power-law density of states $\rho(\epsilon) \sim \epsilon^{1/7}$. The bulk effective field theory is proposed to be the (3+1) dimensional SU(2) Yang-Mills theory with a theta-term at $\theta=\pi$.Comment: 5 pages, 3 figure

Momentum space metric, non-local operator, and topological insulators

Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In particular, we introduce a non-local operator which represents distance square in real space and show that this corresponds to the Laplacian in curved momentum space, and also derive its path integral representation in momentum space. The quantum metric itself measures the second cumulant of the position operator in real space, much like the Berry gauge potential measures the first cumulant or the electric polarization in real space. By using the non-local operator and the metric, we study some aspects of topological phases such as topological invariants, the cumulants and topological phase transitions. The effect of interactions to the momentum space geometry is also discussed.Comment: 13 pages, 4 figure

Fermion zero modes at the boundary of superfluid 3He-B

Superfluid 3He-B belongs to the important special class of time-reversal invariant topological superfluids. It has Majorana fermions as edge states on the surface of bulk 3He-B. On the rough wall these fermion zero modes have finite density of states at E=0. It is possible that Lancaster experiments with a wire vibrating in 3He-B have already probed Majorana fermions living on the surface of the wire.Comment: 4 pages, no Figures, JETP Letters style, version to be published in JETP Letter

Quasiparticle interference from different impurities on the surface of pyrochlore iridates: signatures of the Weyl phase

Weyl semimetals are gapless three-dimensional topological materials where two bands touch at an even number of points in the bulk Brillouin zone. These semimetals exhibit topologically protected surface Fermi arcs, which pairwise connect the projected bulk band touchings in the surface Brillouin zone. Here, we analyze the quasiparticle interference patterns of the Weyl phase when time-reversal symmetry is explicitly broken. We use a multi-band $d$-electron Hubbard Hamiltonian on a pyrochlore lattice, relevant for the pyrochlore iridate R$_2$Ir$_2$O$_7$ (where R is a rare earth). Using exact diagonalization, we compute the surface spectrum and quasiparticle interference (QPI) patterns for various surface terminations and impurities. We show that the spin and orbital texture of the surface states can be inferred from the absence of certain backscattering processes and from the symmetries of the QPI features for non-magnetic and magnetic impurities. Furthermore, we show that the QPI patterns of the Weyl phase in pyrochlore iridates may exhibit additional interesting features that go beyond those found previously in TaAs.Comment: 15 pages, 16 figure

Screening in (d+s)-wave superconductors: Application to Raman scattering

We study the polarization-dependent electronic Raman response of untwinned YBa$_2$Cu$_3$O$_{7-\delta}$ superconductors employing a tight-binding band structure with anisotropic hopping matrix parameters and a superconducting gap with a mixing of $d$- and s-wave symmetry. Using general arguments we find screening terms in the B^{\}_{1g} scattering channel which are required by gauge invariance. As a result, we obtain a small but measurable softening of the pair-breaking peak, whose position has been attributed for a long time to twice the superconducting gap maximum. Furthermore, we predict superconductivity-induced changes in the phonon line shapes that could provide a way to detect the isotropic s-wave admixture to the superconducting gap.Comment: typos corrected, 6 pages, 3 figure

Influence of higher d-wave gap harmonics on the dynamical magnetic susceptibility of high-temperature superconductors

Using a fermiology approach to the computation of the magnetic susceptibility measured by neutron scattering in hole-doped high-Tc superconductors, we estimate the effects on the incommensurate peaks caused by higher d-wave harmonics of the superconducting order parameter induced by underdoping. The input parameters for the Fermi surface and d-wave gap are taken directly from angle resolved photoemission (ARPES) experiments on Bi{2}Sr{2}CaCu{2}O{8+x} (Bi2212). We find that higher d-wave harmonics lower the momentum dependent spin gap at the incommensurate peaks as measured by the lowest spectral edge of the imaginary part in the frequency dependence of the magnetic susceptibility of Bi2212. This effect is robust whenever the fermiology approach captures the physics of high-Tc superconductors. At energies above the resonance we observe diagonal incommensurate peaks. We show that the crossover from parallel incommensuration below the resonance energy to diagonal incommensuration above it is connected to the values and the degeneracies of the minima of the 2-particle energy continuum.Comment: 13 pages, 7 figure