Topological crystalline superconductivity in locally noncentrosymmetric odd-parity superconductor CeRh$2$As$2$

The recent discovery of superconductivity in CeRh$_2$As$_2$ clarified an unusual $H$-$T$ phase diagram with two superconducting phases [1]. CeRh$_2$As$_2$ crystallizes in the centrosymmetric tetragonal CaBe$_2$Ge$_2$-type structure with stacking Ce layers and Rh$_2$As$_2$ layers. Importantly, Rh$_2$As$_2$ layers at the top and bottom of the Ce layer have different compositions. Therefore, the inversion symmetry is locally broken at the Ce sites, although the global inversion center exists in the middle of the two Ce sites in the unit cell. The space group of CeRh$_2$As$_2$ is P4/nmm (No.129) including the nonsymmorphic glide symmetry. Surprisingly, the phase diagram was predicted in theoretical work about locally noncentrosymmetric superconductors [2,3]. The similarity of these phase diagrams between the experiment [1] and theory [2,3] suggests that the local inversion symmetry breaking plays an essential role in CeRh$_2$As$_2$, and the superconducting phase in the high magnetic field region is the pair-density-wave (PDW) state. PDW state is odd parity superconducting state in which the superconducting gap function changes sign depending on Ce layers. Indeed, the experimental observation has been interpreted based on the even-odd parity transition. The most crucial property of this phase is the odd-parity superconductivity in spite of dominantly spin-singlet pairing. These character of PDW state suggests a possibility of topological superconductivity as spin-triplet superconductors are. Thus, the locally noncentrosymmetric crystal is a platform of odd-parity superconductivity without requiring rare spin-triplet pairing.
In this talk, using the group theory, we first clarified the algebra of symmetry operations in the Bloch representation and decomposed the Hilbert space on the glide-invariant planes $k_z = 0, \pi$ into the glide sectors. Supposing the PDW state in the high-field superconducting phase as proposed, based on these results, we derived the Fermi-surface formula of $\mathbb{Z}_2$ invariants specifying the topological crystalline superconductivity protected by the nonsymmorphic glide symmetry [4]. Second, we conducted the first-principles calculation for the electronic structure of CeRh$_2$As$_2$. Combining the results, we evaluate the $\mathbb{Z}_2$ invariants and found the topological crystalline superconductivity due to heavy-fermion bands of Ce 4$f$ electrons hybridizing with conduction electrons. The emergence of Majorana fermions at the surface preserving the glide symmetry was demonstrated based on the tight-binding model [5].

[1] S. Khim et al., arXiv:2102.02735 (2021).
[2] T. Yoshida, M. Sigrist, and Y. Yanase, Phys. Rev. B 86, 134514 (2012).
[3] D. Mockli, Y. Yanase, and M. Sigrist, Phys. Rev. B 97, 144508 (2018)
[4] M. Sato, Phys. Rev. B 81, 220504(R) (2010).
[5] K. Nogaki, A. Daido, J. Ishizuka, and Y. Yanase, arXiv:2102.08088 (2021).