Double Helix Nodal Line Superconductor

Time-reversal invariant superconductors in three dimensions may contain nodal lines in the Brillouin zone, which behave as Wilson loops of 3D momentum-space Chern-Simons theory of the Berry connection. Here we study the conditions of realizing linked nodal lines (Wilson loops), which yield a topological contribution to the thermal magnetoelectric coefficient that is given by the Chern-Simons action. We find the essential conditions are the existence of torus or higher genus Fermi surfaces and spiral spin textures. We construct such a model with two torus Fermi surfaces, where a generic spin-dependent interaction leads to double-helix-like linked nodal lines as the superconductivity is developed.

Figure 1

Examples of Fermi surfaces compatible with TRI linked nodal lines. (a)–(c) shows how such linked nodal lines may occur on torus Fermi surfaces. (d) shows a possible configuration of linked nodal lines on a genus two Fermi surface.

Figure 2

The arrows show the spin textures of the lower electron band in the $k_z=0$ plane for $\lambda =-0.02t$, ${\lambda }_z=0.005t$, and the brightness is proportional to the SOC strength $|\mathbf{g}(\mathbf{k})|$ (brighter color for larger $|\mathbf{g}(\mathbf{k})|$). In particular, $|\mathbf{g}(\mathbf{k})|=0$ at $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^{\prime }$ points. The nodal lines occur at $A$, $B$ and $A^{\prime }$, $B^{\prime }$ points which are intersections between the Fermi surfaces as given by the white circles and the constant $|g(\mathbf{k})|=\psi /\xi$ contours (yellow circles).

Figure 3

(a) Phase diagram for fixed $V<0$, where the horizontal and vertical axes are $J$ and $V_0$, respectively. Both $\alpha$ and ${\alpha }^{\prime }$ phases are double-helix linked nodal line phases, except that their helicities are opposite. The $\beta$ phase contains two unlinked nodal lines on each Fermi surface. The $\gamma$ phase is the conventional fully gapped phase. Panels (b)–(d) show the nodal-line configuration on the Fermi surface at $k_x=\pi /2$ for points $A$, $B$, and $C$ in the phase diagram, respectively, where $\theta =\arg (\widetilde{k_x}+i{k}_y)$ is the poloidal angle of the torus Fermi surface. (e) The 3D plot of the spin texture (arrows) and nodal lines (blue and red lines) on the Fermi surfaces at point $A$ of the phase diagram.

Figure 4

(a) Sketch of a typical example of double-helix nodal lines in the BZ. (b)–(d) Illustration of the zero-energy surface Majorana flat bands (colored in green) on surface cuts along the $\mathrm{\Gamma }\text{−}Z$, $\mathrm{\Gamma }\text{−}X$, and $\mathrm{\Gamma }\text{−}M$ directions, which are bounded by the projection of nodal lines.