Pseudo-Ising superconductivity induced by $p$-wave magnetism

TL;DR

The work introduces pseudo-Ising superconductivity, a robust superconducting state induced in a conventional superconductor by proximity to a compensated $p$-wave magnet, and shows it can persist under strong magnetism without spin-orbit coupling. By developing a low-energy two-band model, deriving the Gor\'kov pairing correlations, and computing the Edelstein response, the authors demonstrate that $p$-wave magnetism generates out-of-plane spin-triplet pairing, yields a finite in-plane spin susceptibility at $T=0$, and enhances the in-plane upper critical field $B_{c2}$ with a second-order superconductor–metal transition. The analysis extends to odd-parity $f$-wave magnets and corroborates the key physics with four-band models, including a route to Majorana zero modes in a 1D wire on a $p$-wave magnet without spin-orbit coupling. Overall, the paper provides a new mechanism for unconventional superconductivity, with potential applications in superconducting spintronics and topological quantum computation through Majorana platforms.

Abstract

Unconventional magnetic orders usually interplay with superconductivity in intriguing ways. Here we propose that a conventional superconductor in proximity to a compensated $p$-wave magnet exhibits behaviors analogous to those of Ising superconductivity found in transition-metal dichalcogenides, which we refer to as pseudo-Ising superconductivity. The pseudo-Ising superconductivity is characterized by several distinctive features: (i) it stays much more robust under strong $p$-wave magnetism than usual ferromagnetism or $d$-wave altermagnetism, thanks to the apparent time-reversal symmetry in $p$-wave spin splitting; (ii) in the low-temperature regime, a second-order superconducting phase transition occurs at a significantly enhanced in-plane upper critical magnetic field $B_{c2}$; (iii) the supercurrent-carrying state establishes non-vanishing out-of-plane spin magnetization, which is forbidden by symmetry in Rahsba and Ising superconductors. We further propose a spin-orbit-free scheme to realize Majorana zero modes by placing superconducting quantum wires on a $p$-wave magnet. Our work establishes a new form of unconventional superconductivity generated by $p$-wave magnetism.

Figures (7)

• Figure 1: (a) Lattice model featuring a coplanar, non-collinear spin arrangement for compensated $p$-wave magnet, adapted from hellenes2023exchange. Nonmagnetic (magnetic) atoms are shown as orange (purple) sites. The green solid curve indicates spin-independent hopping, while the pink dashed line represents exchange-dependent hopping. (b) Upper panel: Fermi surfaces with $p$-wave spin splitting in momentum space, including the inter-pocket Cooper pairing in the superconducting state. Lower panel: A thin superconducting (SC) metal film placed atop a $p$-wave magnetic substrate to realize pseudo-Ising superconductivity via inverse proximity effect.
• Figure 2: (a) Superconducting critical temperature with increasing strength for several types of magnetism. We label the $x$-axis as $J=J_s=J_p=J_d$ because they cause the same maximal band spin splitting. (b) In-plane spin susceptibility normalized by the normal-state value ($\chi^s_{\|}/\chi_{0}$) in the low-field limit as a function of temperature $T$ with (red line) and without (blue line) $p$-wave magnetism. (c) $B_{c2}\text{--}T$ curves for $\boldsymbol{B}_{\parallel}=B_x \hat{x}$ with different strengths of $p$-wave magnetism. (d) Pairing amplitude $\Delta$ as a function of both $B_x$ and $T$, when $J=0.2$. Other parameters: $t=1,U=1.5,\mu=-1$.
• Figure 3: (a) The linear vertical Edelstein susceptibility $\alpha_{zx}$ as a function of the strength $J$ of $p$-wave magnetism at $T=0.2T_c$. (b) $\alpha_{zx}$ as a function of the temperature $T$. We also take a small Rashba SOC with $\lambda=0.03$. We adopt the standard BCS temperature dependence as $\Delta(T) = \Delta_0 \mathrm{tanh}(1.74\sqrt{T_c/T - 1})$ (for $p$-wave magnetism, this is verified numerically). Parameters: $(\Delta_0, \mu) = (0.06,-1)$. (c) Linear and nonlinear vertical Edelstein effects induced by $p$, $d$, $f$ magnetic orders.
• Figure 4: (a) A superconducting QW is placed on top of a $p$-wave magnetic substrate. MZMs (purple dots) appear at the ends of QW when an in-plane magnetic field $\boldsymbol{V}_{\parallel}$ is applied. (b) The energy spectrum of the setup in (a) as a function of the chemical potential of the wire, using the tight-binding model in Eq. \ref{['majorana']}. The red line highlights the topological regime with MZMs. Here we only plot the $\Delta^2+(2t_x-\mu)^2<V_y^2$ branch as a representative. Parameters: $J_p=-0.1$, $t_x=0.5$, $\Delta=0.1$, $V_y=0.2$.
• Figure S1: (a) Band structure from Eq. \ref{['eq:tbdmodel']} with parameters $(t,t_J, \mu,k_y) = (2, 0.4,6,0)$. (b) Edelstein effect in $p$-wave magnet with $(J_p,\Delta_0, \mu) = (0.3, 0.06,-1)$. (c) $B_{c2}\text{--}T$ curve of $f$-wave magnet with $(t, \mu) = (2, 2)$. (d) Edelstein effect in $f$-wave magnet with $t=1,\mu=0.3,\lambda=0.05,\Delta_0=0.05, T=0.3T_c$. We add the SOC term as $H_{soc}=\lambda(k_y \sigma_x-k_x \sigma_y)$.