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

Nonsymmorphic symmetry-required band crossings in topological semimetals

We show that for two-band systems nonsymmorphic symmetries may enforce the existence of band crossings in the bulk, which realize Fermi surfaces of reduced dimensionality. We find that these unavoidable crossings originate from the momentum dependence of the nonsymmorphic symmetry, which puts strong restrictions on the global structure of the band configurations. Three different types of nonsymmorphic symmetries are considered: (i) a unitary nonsymmorphic symmetry, (ii) a nonsymmorphic magnetic symmetry, and (iii) a nonsymmorphic symmetry combined with inversion. For nonsymmorphic symmetries of the latter two types, the band crossings are located at high-symmetry points of the Brillouin zone, with their exact positions being determined by the algebra of the symmetry operators. To characterize these band degeneracies we introduce a \emph{global} topological charge and show that it is of $\mathbb{Z}_2$ type, which is in contrast to the \emph{local} topological charge of Fermi points in, say, Weyl semimetals. To illustrate these concepts, we discuss the $\pi$-flux state as well as the SSH model at its critical point and show that these two models fit nicely into our general framework of nonsymmorphic two-band systems.Comment: 6.5 pages, 4 figure