Topological relation between bulk gap nodes and surface bound states: Application to iron-based superconductors

In the past few years materials with protected gapless surface (edge) states have risen to the central stage of condensed matter physics. Almost all discussions centered around topological insulators and superconductors, which possess full quasiparticle gaps in the bulk. In this paper we argue that systems with topological stable bulk nodes offer another class of materials with robust gapless surface states. Moreover the location of the bulk nodes determines the Miller index of the surfaces that show (or do not show) such states. Measuring the spectroscopic signature of these zero modes allows a phase-sensitive determination of the nodal structures of unconventional superconductors when other phase-sensitive techniques are not applicable. We apply this idea to gapless iron-based superconductors and show how to distinguish accidental from symmetry-dictated nodes. We shall argue that the same idea leads to a method for detecting a class of the elusive spin liquids.

Figure 1

Representations of the Fermi surfaces of LaFePO and four examples of nodal structures. The blue/red curves represent the hole/electron Fermi surfaces. Red dots with $\pm$ signs indicate nodes with $\pm$ sign of winding numbers. (a), (b) Two different $s$-wave gap nodes (red dots), (c) $d_{x^2-y^2}$ nodes, and (d) $p_x$ nodes. The black circle in part (a) is a closed loop enclosing a gap node.

Figure 2

The surface local density of states $N\left(\omega \right)$ for some of the pairing symmetries in Table . For each symmetry two surface orientations are shown: one with zero bias peak and the other without. The unit of the density of states is arbitrary. The ${\Delta }_0$ used in this calculation is $0.01$ eV. Thermal broadening with temperature $T=0.4$ meV is used.

Figure 3

(a) The spin nondegenerate Fermi surfaces (the black and gray curves) of the noncentrosymmetric superconductor defined in Eq. (3). The nodes are designated by the red dots. (b) The band structure of the $\left\{11\right\}$ edges. The edge states exist between the momenta marked by the yellow arrows. The parameters we used in producing (b) are $\lambda =0.3,\mu =0.45$, ${\Delta }_0=0.1$, and $\alpha =1,\beta =0$.

Figure 4

(a) The 2D nodal band structure and its projections on $\left(1\overline{1}\right)$ and $\left(01\right)$ surfaces for a model $d$-wave superconductor. The first BZ and one extended zone are drawn. The solid black curves denote the Fermi surface and the shaded region is filled in the normal state. Red arrows are the unit vectors in Eq. (A2). The blue dots with $\pm$ signs represent nodes with vorticity $\pm 2$ respectively. Top left: the slanted thin black line segment is the surface BZ of the $\left(1\overline{1}\right)$ edge. The black arrow with letter “P” indicates the direction of projection. Thick red line segments mark the surface momenta with zero-energy bound states. Top right: the thin black horizontal line segment is the BZ of the $\left(01\right)$ edge. The $\pm$ vorticity nodes overlap after projection. The vorticity of the node enclosed by the parallelogram is equal to the winding number difference along the two side blue segments because the windings along the top and bottom segments cancel due to the periodicity in momentum space. This can be explicitly seen by following the turning of the red arrows (the actual winding number is twice that of the winding shown by the arrows, due to the spin degeneracy). (b) The (11) edge band structure. The surface flat bands are marked red. In constructing this figure we have used $\epsilon \left(\mathbf{k}\right)=-\cos k_x-\cos k_y-\mu$ and $\Delta \left(\mathbf{k}\right)={\Delta }_0(\cos k_x-\cos k_y)$ in Eq. (A1). Here $\mu =0.45,{\Delta }_0=0.1$.