Spin response and topology of a staggered-Rashba superconductor

Inversion symmetry is a key symmetry in unconventional superconductors, and even its local breaking can have profound implications. For inversion-symmetric systems, there is a competition on a microscopic level between the spin-orbit coupling associated with the local lack of inversion and hybridizing terms that “restore” inversion. Investigating a layered system with alternating mirror-symmetry breaking, we study this competition considering the spin response of different superconducting order parameters for the case of strong spin-orbit coupling. We find that signatures of the local noncentrosymmetry, such as an increased spin susceptibility in spin-singlet superconductors for $T\to 0$, persist even into the quasi-three-dimensional regime. This leads to a direction-dependent spin response that allows us to distinguish different superconducting order parameters. Furthermore, we identify several regimes with possible topological superconducting phases within a symmetry-indicator analysis. Our results may have direct relevance for the recently reported Ce-based superconductor ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ and beyond.

Figure 1

Schematic of the system with the triangles indicating the mirror symmetry breaking. Note that, for simplicity, we choose the layers to be spaced equidistantly in the $z$ direction with unit-cell size $c$.

Figure 2

Ratio of the van Vleck contribution to the total susceptibility in the normal state for fields along the $z$ direction (solid lines) and in-plane (dashed lines). For the case of $\delta t_z/t_z=0.2$, four different regimes are indicated, which are connected to the Fermi surface topology. The Fermi surfaces are illustrated by the insets, which show the Fermi surfaces in the $k_x-k_z$ plane of the Brillouin zone. For numerical reasons, we set $T=0.0025t$.

Figure 3

Out-of-plane (solid lines) and in-plane (dashed lines) susceptibility for all three order parameters for $T\to 0$ compared to the normal-state susceptibility as a function of $z$ axis hopping for $\delta t_z=0$. The pairing gaps are defined as $\mathrm{\Delta }\left(\mathbf{k}\right)=\psi$ for the spin-singlet, ${\vec{d}}_{\mathbf{k}}^{\parallel }=\mathrm{\Delta }(-sin{k}_y,sin{k}_x,0)$ for intralayer pairing, and $d_{\mathbf{k}}^z=\mathrm{\Delta }sin{k}_z/2$ for interlayer pairing. For the numerical evaluation, we used $\mathrm{\Delta }=0.025t$ and $T=0.0025t$.

Figure 4

Projection of the Fermi surface onto the $k_x-k_z$ plane for different values of interlayer hopping parameters $t_z$ and $\delta t_z$. Two colors correspond to two different momentum sectors with inversion eigenvalues $+1$ and $-1$, respectively.

Figure 5

The normal state susceptibility, and the Pauli and the van Vleck contributions to it for fields along the $z$ direction (solid lines) and in-plane (dashed lines). Four different regimes of the Fermi surface topology are indicated for the case of $\delta t_z/t_z=0.2$. For numerical reasons, we set $T=0.0025t$.

Figure 6

The Pauli (solid lines) and the van Vleck (dashed lines) contributions to the normal state susceptibility for fields along the $z$ direction for $\delta t_z/t_z=0.2$. Colored regions indicate regimes with different Fermi surface topology for three values of the electron density $n_{\text{tot}}$.