Topological invariants for the Fermi surface of a time-reversal-invariant superconductor

A time-reversal-invariant (TRI) topological superconductor has a full pairing gap in the bulk and topologically protected gapless states on the surface or at the edge. In this paper, we show that in the weak pairing limit, the topological quantum number of a TRI superconductor can be completely determined by the Fermi-surface properties and is independent of the electronic structure away from the Fermi surface. In three dimensions, the integer topological quantum number in a TRI superconductor is determined by the sign of the pairing order parameter and the first Chern number of the Berry phase gauge field on the Fermi surfaces. In two dimensions and one dimension, the $Z_2$ topological quantum number of a TRI superconductor is determined simply by the sign of the pairing order parameter on the Fermi surfaces. Our results could directly aid the search for topological superconductors in real materials.

Figure 1

(Color online) (a) The path of ${ϵ}_{n\mathbf{k}}+i{\delta }_{n\mathbf{k}}$ in the complex plane for positive (red dashed) and negative (blue) ${\delta }_{n\mathbf{k}}$ around the Fermi surface. (b) ${\theta }_{n\mathbf{k}}$ and ${ϵ}_{n\mathbf{k}}$ vs momentum $\mathbf{k}$. The change in ${\theta }_{n\mathbf{k}}$ across $k_F$ is $-\pi$ $\left(+\pi \right)$ when ${\delta }_{n\mathbf{k}}$ is positive (negative), as shown by the red dashed (blue) curve.

Figure 2

(Color online) Dimensional reduction from a 3D TRI superconductor to a 2D TRI superconductor. The 2D TRI superconductor corresponds to the $\theta =\pi$ section of a 3D superconductor. The Fermi surfaces with blue (red dashed) color are those with positive (negative) pairing amplitude ${\delta }_s$.

Figure 3

(Color online) Simple examples of (a) nontrivial and (b) trivial pairing in a 1D system. The red (open) and blue (closed) dots are Fermi points with negative and positive pairing, respectively. The phase $\theta ={\theta }_{n\mathbf{k}}$ is taken to be the same at $k=0$ for the two bands. At $k=\pi$, $\theta$ of the two bands are differ by (a) $2\pi$ for nontrivial pairing and (b) by 0 for trivial pairing.

Figure 4

(Color online) (a) Schematic picture of a Fermi surface enclosing one TRI point (0,0,0). The Fermi surface is separated to two parts ${\text{FS}}_{+}$ and ${\text{FS}}_{-}$ by $k_z=0$ plane. The interface between the two parts is further split into curves $L_1$ (red dashed curve) and $L_2$ (blue curve) which are time-reversal partners of each other. The interface of $L_1$ and $L_2$ are given by points A and B. (b) Schematic picture of a Fermi surface enclosing two TRI points (0,0,0) and $\left(0,0,\pi \right)$. Similar to (a), the Fermi surface is separated to ${\text{FS}}_{+}$ and ${\text{FS}}_{-}$, and the interface between ${\text{FS}}_{+}$ and ${\text{FS}}_{-}$ is split into $L_1$ (red dashed curve) and $L_2$ (blue curve), which intersect at two pairs of points ${\text{A}}_1,{\text{B}}_1$ and ${\text{A}}_2,{\text{B}}_2$.