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

Abstract

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 $χ(T_c^{-}) = χ_N$ 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 $χ(0)$ which approaches $2χ_N/3$ in the strong SOC limit; combined fields create a Bogoliubov Fermi surface, resulting in a kink in $χ(0)$. 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 $χ_{zz}(0)$ 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.

Figures (19)

• Figure 1: $\chi_{\mu\mu}^{ph+}(T_c) / \chi_N = 1 - \chi_{\mu\mu}^{ph-}(T_c) / \chi_N$ plotted against the ratio $g k_F / \mu_B H$, assuming a continuous superconducting phase transition.
• Figure 2: $s$-wave pairing state: Spin susceptibility $\chi(T)$ under a sole Zeeman field $\mathbf{H}=H_z\hat{z}$, where $H_0=H_P=\Delta_0/\sqrt{2}\mu_B$. A second-order phase transition occurs for $T_c > T_0 = 0.556 T_{c0}$, while a first-order transition is seen for $T_c < T_0$, as noted by Maki and Tsuneto Maki1964.
• Figure 3: $s$-wave superconductor: Spin susceptibility $\chi(T)$ with Rashba SOC, in the absence of a Zeeman field.
• Figure 4: $s$-wave pairing state: Temperature-dependent spin susceptibility $\chi(T)$ with both Zeeman field $H_z$ and Rashba SOC $g$. (a) Rashba SOC fixed at $g k_F / k_B T_{c0} = 1$, with varying $H_z$; the transition becomes first-order for small $T_c / T_{c0}$, similar to $g = 0$Maki1964. (b) Zeeman field fixed at $H_z / H_0 = 0.8$, with varying Rashba SOC. Here the field scale $H_0 = H_P$ is provided at the end of Section \ref{['sec:s-wave-H']}.
• Figure 5: $s$-wave pairing state: Zero-temperature spin susceptibility $\chi_{zz}(T=0)$ as a function of (a) Zeeman field $H_z$ (fixed Rashba SOC) and (b) Rashba SOC $g$ (fixed Zeeman field). In (a), curves end at critical field $H_c^Z$, with a kink at $H_{c2}^Z$ (solid circles) for $g > g_c$, indicating Bogoliubov Fermi surface formation pang2025.