Instability toward Superconducting Stripe Phase in Altermagnets with Strong Rashba Spin-Orbit Coupling

TL;DR

This work addresses the existence and nature of finite-momentum superconductivity in two-dimensional altermagnets with strong Rashba spin-orbit coupling and $d$-wave spin-splitting. Using a quasiclassical framework in the RSOC basis, the authors first establish a helical phase with a single center-of-mass momentum and then perturbatively analyze an instability toward a stripe phase with multiple momenta by solving a linearized gap equation and examining a 2x2 coupling matrix. They map out phase diagrams in the $(T, \Delta_{AM})$ plane for various density-of-states asymmetries, showing a stripe phase that can reenter as altermagnetic splitting grows and that can be LO-like with two dominant momenta at large splitting; the stripe onset is governed by a zero-eigenvalue condition of the perturbation matrix. The stripe mechanism is traced to anisotropic deformation of the Fermi surfaces induced by altermagnetic splitting, which differentially weights contributions from inner and outer Fermi-surface sheets and leads to distinct pairing channels. These results highlight a robust route to multi-$q$ superconductivity in altermagnets and reveal the intricate interplay between spin-orbit coupling and altermagnetic spin-splitting that shapes finite-momentum pairing.

Abstract

We numerically investigate finite-momentum superconductivity in noncentrosymmetric metallic altermagnets with $d$-wave spin-splitting and strong Rashba-type spin-orbit coupling. Focusing on a stripe phase in which Cooper pairs acquire multiple center-of-mass momenta, we construct phase diagrams that reveal phase boundaries between the stripe phase and a helical phase characterized by a single center-of-mass momentum. Our results show that the stripe phase emerges at low temperatures and exhibits a reentrant behavior as a function of the strength of the altermagnetic splitting. We further analyze the stripe phase within a linearized gap equation, and uncover the mechanism of the pairing formation unique to the stripe phase. This mechanism originates from the anisotropic deformation of the Fermi surfaces induced by the altermagnetic splitting, highlighting the intriguing interplay between the spin-orbit coupling and the altermagnets.

Figures (3)

• Figure 1: Phase diagrams in the $(T,\Delta_{\mathrm{AM}})$ plane for (a) $\delta N=0.0$, (b) $\delta N=0.01$, (c) $\delta N=0.05$. The blue lines show second-order transitions between superconducting and normal states. The dashed red lines indicate first-order transitions of $\bm{Q}$; the red solid line in (a) shows its second-order transition. The red crosses in (b) and (c) mark the crossover onset. Scatter plots show the phase boundaries between the stripe phase and the helical phase. Their colors indicate the absolute value of $|\Delta^*_{2\bm{Q}-\bm{q}_{\mathrm{st}}}/\Delta_{\bm{q}_{\mathrm{st}}}|$ calculated from the gap equation Eq. \ref{['eq:linearized_gap_eq']}.
• Figure 2: $\Delta_{\mathrm{AM}}$ dependence of $\bm{Q}$ and $\bm{q}_{\mathrm{st}}$ for three $\delta N$ values. The solid, dashed, and dash-dotted lines represent $\bm{Q}$, $-\bm{q}_{\mathrm{st}}$, and $|\delta \bm{q}|=||\bm{Q}_{\mathrm{heli}}|-|\bm{q}_{\mathrm{st}}||$, respectively. The temperature is fixed at $T=0.1T_{\mathrm{c}}$. The dotted lines indicate the first-order transition of $\bm{Q}$. The inset shows an enlarged view in the region $0\le\Delta_{\mathrm{AM}}/T_{\mathrm{c}}\le1.8$ for visibility. The shaded region indicates where the ground state is helical.
• Figure 3: (a) Schematic illustration of the FSs in our model. The dashed lines represent FS with RSOC only, while the solid lines include both the RSOC and the altermagnetic splitting. The arrows on FSs indicate the spin orientation of the electrons. The open red (blue) arrows indicate that the dashed FSs around them are shifted toward the positive (negative) $k_x$ direction due to the altermagnetic splitting. The dotted lines indicate directions along which the altermagnetic splitting vanishes. (b,c) $\Delta_{\mathrm{AM}}$ dependence of $\epsilon_1(\bm{q}_{\mathrm{st}})$, $1 - M_{11}(\bm{q}_{\mathrm{st}})$, and $M^{\pm}_{11}(\bm{q}_{\mathrm{st}})$. $M^{\pm}_{11}$ is the contribution from $\lambda=\pm$ band and $M_{11}=M^{+}_{11}+M^{-}_{11}$. (d, e) $\phi_{\bm{k}}$ dependence of the integrand of $M^{\pm}_{11}$ for two $\Delta_{\mathrm{AM}}$ values at $\delta N=0.05$ and $T=0.1T_{\mathrm{c}}$. (f,g) $\delta q$ dependence of $M_{11}^{\pm}$. The dotted green (purple) lines indicate the position of $\bm{q}_{\mathrm{st}}$ ($-\bm{Q}$).