Chiral $d$-wave superconductivity in SrPtAs

Recent muon spin-rotation ($\mu$SR) measurements on SrPtAs revealed time-reversal-symmetry breaking with the onset of superconductivity [Biswas et al., Phys. Rev. B 87, 180503(R) (2013)], suggesting an unconventional superconducting state. We investigate this possibility via the functional renormalization group and find a chiral $(d+id)$-wave order parameter favored by the multiband fermiology and hexagonal symmetry of SrPtAs. This $(d+id)$-wave state exhibits significant gap anisotropies as well as gap differences on the different bands, but only has point nodes on one of the bands at the Brillouin zone corners. We study the topological characteristics of this superconducting phase, which features Majorana-Weyl nodes in the bulk, protected surface states, and an associated thermal Hall response. The lack of extended nodes and the spontaneously broken time-reversal symmetry of the $(d+id)$-wave state are in agreement with the $\mu$SR experiments. Our theoretical findings, together with the experimental evidence, thus suggest that SrPtAs is an example of chiral $d$-wave pairing and a Weyl superconductor.

Figure 1

Fermi surface at (a) $k_z=0$ and (b) $k_z=\pi$ of SrPtAs. (c) The density of states for the full three-dimensional band structure (bold black), for a (2D) plane with $k_z=0$ (dashed black), and the three-dimensional band structure only looking at bands around the corners of the Brillouin zone (band 3, green line). Note the van Hove singularity very close to the Fermi energy.

Figure 2

Eigenvalue flow of several RG channels for $g_1=g_2=0.4$ eV and $g_3=g_4=1.5$ eV. The first diverging mode is the doubly degenerate $d$-wave superconducting (SC) channel, followed by the subleading doubly degenerate $d$-wave Pomeranchuk channel (PI). Due to imperfect nesting, neither the charge-density wave (CDW) nor the spin-density wave (SDW) play a significant role.

Figure 3

Representative $(d+id)$-gap structure plotted along the discretized Fermi pockets ($g_1=g_2=0.4$ eV and $g_3=g_4=1.5$ eV). There is significant gap anisotropy on the pockets around $K$ and $K^{\prime }$. The pockets centered around $\Gamma$ only show small, isotropic gaps mainly induced by the proximity coupling to the outer pockets. The inset visualizes the gap magnitude and shows the discretization of the Fermi surfaces.

Figure 4

Fermi surface structure and edge states of the $E_g$ superconducting state of SrPtAs. (a) Bulk Fermi surfaces with Majorana-Weyl nodes of the superconducting order parameter (red dots) and their projection to the surface BZ of the [010] surface. In the surface BZ, the Fermi surfaces and Fermi arcs of the chiral Majorana fermions are sketched, with the arrows indicating their group velocity. (b) Cut along $k_z=0$ of the spectrum of a 150 layer slab stacked along the [010] direction showing Majorana fermion surface states. (c) The cut along $k_x=0$ of the spectrum of a 400 layer slab stacked along the [010] direction shows the position of two Majorana-Weyl points as well as the chiral Majorana surface modes connecting them.