Interplay of Zeeman field, Rashba spin-orbit interaction, and superconductivity: spin susceptibility

We present a self-consistent theory to calculate the static and uniform spin susceptibility in superconductors under simultaneous Zeeman magnetic fields and Rashba-type spin-orbit coupling (SOC). Employing a single-band Bogoliubov-de Gennes Hamiltonian, we solve the gap equation for both conventional $s$-wave spin-singlet and six representative $p$-wave spin-triplet pairing states, categorized into opposite-spin-pairing (OSP) and equal-spin-pairing (ESP) classes. The Kubo formula, decomposed into intra- and interband particle-hole and particle-particle channels, provides two key constraints: at zero temperature, only particle-particle terms contribute, while at the critical temperature $T_c$, only particle-hole terms remain, ensuring $\chi(T_c^{-}) = \chi_N$ for continuous phase transitions. For $s$-wave pairing, a Zeeman field reduces $T_c$, whereas Rashba SOC preserves $T_c$ but yields a residual zero temperature spin susceptibility $\chi(0)$ which approaches $2\chi_N/3$ in the strong SOC limit; combined fields create a Bogoliubov Fermi surface, resulting in a kink in $\chi(0)$. In contrast, $p$-wave states exhibit strong anisotropy: OSP states mimic spin-singlet pairing behavior for parallel Zeeman fields and ESP for transverse ones, while ESP states show the opposite, with Rashba SOC potentially changing the quasiparticle nodal structure, lowering $T_c$, or causing $\chi_{zz}(0)$ divergences. This framework offers quantitative benchmarks for Knight-shift experiments in non-centrosymmetric superconductors like A$_2$Cr$_3$As$_3$ (A = Na, K, Rb, and Cs), enabling diagnostics to disentangle pairing symmetry, SOC strength, and Zeeman effects.