Spin-Orbital Intertwined Topological Superconductivity in a Class of Correlated Noncentrosymmetric Materials

TL;DR

The paper proposes an intrinsic route to topological superconductivity in correlated noncentrosymmetric materials by exploiting a two-orbital Rashba-Hubbard framework. Using spin-fluctuation-mediated pairing within a multi-orbital RPA and linearized Eliashberg formalism, it shows that the leading instabilities can realize an A1(Spm) state that is fully gapped and Z2 topological, as well as B2 and B2(dpm) states with nodal or parity-mixed features. A key finding is that the A1(Spm) TSC is often spin-singlet-dominated and parity-mixed, enabling robust topological behavior without relying on triplet-pairing near van Hove singularities. The work suggests experimentally accessible platforms where orbital degrees of freedom and spin textures enable bulk TSC, with potential implications for topological quantum computation.

Abstract

In this study, we propose an alternative route to achieving topological superconductivity (TSC). Our approach applies to a new class of correlated noncentrosymmetric materials that host two spin-split Fermi surfaces with identical spin textures due to a spin-orbital intertwined effect. Incorporating multi-orbital repulsive Hubbard interactions, we calculate the superconducting pairings of a minimal two-orbital effective model within a spin-fluctuation-mediated superconductivity framework. We find that, depending on the effective Rashba spin-orbit coupling (RSOC) strength and filling level, the Hubbard interaction can drive the leading pairing symmetry into the $A_1(S_{\pm})$, $B_1$, $B_2$ or $B_2(d_{\pm})$ irreducible representations (IRs) of the $C_{4v}$ point group. Notably, the $A_1(S_{\pm})$ pairing gives rise to a fully gapped TSC characterized by a $Z_2$ invariant, while the $B_2(d_{\pm})$ pairing results in a nodal TSC. Our analysis reveals that the fully gapped TSC is predominated by spin-singlet regardless of the presence of the spin-triplet components. This distinguishes our model from noncentrosymmetric materials with conventional Rashba-split band structures, where TSC typically emerges near the van Hove singularity and is primarily driven by $p$-wave or $f$-wave spin-triplet pairing. These features enhances its experimental accessibility, and we discuss potential experimental systems for its realization.

Figures (11)

• Figure 1: (a) The two-orbital ($d_{x^2-y^2},d_{xy}$) unconventional Rashba model defined on a square lattice with $C_{4v}$ symmetry. The polar axis is defined as the axis perpendicular to the square lattice plane, indicating that along the two opposite directions of this axis the system experiences different polarization fields $V(z)\neq V(-z)$. This asymmetry can in turn generate Rashba-type SOC. (b) The transition from single-orbital limit with conventional spin texture (left part in (b)) to two-orbital limit with unconventional spin texture (right part in (b)) controlled by $\epsilon_{0}^{onsite}$ or $\lambda_{so}$. The circles with cyan color denote the Fermi surfaces and the red arrows indicate the spin texture. (c) The band structures under different values of parameters $\epsilon_{0}^{onsite}$ or $\lambda_{so}$. From left to right, the parameters are set to be $\lambda_{so}=0$, $\lambda_{so}=1.5$ and $\lambda_{so}=3$ respectively. $\lambda_{so}=3+\epsilon_{0}^{onsite}$ is also fixed in order to prevent significant changes in the band structure. Other parameters are set as follows: $t^{nn}_2=-0.1$, $\lambda_R=0.5$, $\epsilon=1$, $t^{nnn}_1=0.0125$, $t^{nnn}_2=0.0375$, the occupied particle number per site is set to be $n=0.5$. Here, the coordinates of high-symmetry points are defined as follows: $\Gamma=(0,0)$; $X=(\pm\pi,0)$, $(0,\pm\pi)$; $M=(\pm\pi,\pm\pi)$.
• Figure 2: (a), (f), (k) and (p) The plottings illustrate Fermi surfaces under various parameters: for (a),$\lambda_{so}=0$, $n=0.5$, $U=3.5$ and $\lambda_{R}=0.35$; for (f), $\lambda_{so}=3$, $n=0.5$, $U=3.5$and $\lambda_{R}=0.725$; for (k), $\lambda_{so}=0$, $n=0.28$, $U=7.5$and $\lambda_{R}=0.225$; for (p), $\lambda_{so}=3$, $n=0.26$, $U=7.5$and $\lambda_{R}=0.35$. The red and blue colors of the Fermi surfaces indicates the superconducting gap functions have opposite signs. Additionally, black arrows denote the spin texture of the relevant Fermi surface on the $k_x-k_y$ plane, while red arrows indicates a $\bm{q}$-vector corresponding to the peaks of $\chi^{RPA}_{zz}(\bm{q})$. The points on the Fermi surface are labeled from 1 to 64 for the outer Fermi surface and from 65 to 128 for the inner Fermi surface in (a), and the settings are the same in (f), (k) and (p). Figures (b), (g), (l) and (q) display $\chi^{RPA}_{zz}(\bm{q})$ defined in Eq. \ref{['sus_sca']} corresponding to (a), (f), (k) and (p) respectively, whereas figures (c), (h), (m) and (r) show the relevant $\chi^{RPA}_{+-}(\bm{q})$ defined in Eq. \ref{['sus_sca']}. Figures (d), (i), (n) and (s) present the intra-pocket effective interaction in $\tau^0\sigma^0$ channel of the outer Fermi surface corresponding to figures (a), (f), (k) and (p), while figures (e), (j), (o) and (t) depict the inter-pocket effective interaction in $\tau^0\sigma^0$ channel between different Fermi surfaces. The effective intra-pocket interaction of the inner Fermi surface can be neglected, because it is weaker than the other two types of interactions, as indicated by the absence of peaks connecting the inner Fermi surface to itself in $\chi^{RPA}_{zz}(\bm{q})$ in (l) and (q). On the right, for figures (d), (e), (i) and (j) correspond to the upper color bar, while for figures (n), (o), (s) and (t) correspond to the lower color bar.
• Figure 3: (a), (b), (c) and (d) present the phase diagram under varuios parameters. In (a), $\lambda_{so}=0$ and $U=3.5$; in (b), $\lambda_{so}=3$ and $U=3.5$; in (c),$\lambda_{so}=0$ and $U=7.5$; in (d), $\lambda_{so}=3$ and $U=7.5$. The x-axis represents particle number, varying from 0.2 to 0.5 for $U=3.5$ and from 0.2 to 0.34 for $U=7.5$. The y-axis denotes the strength of RSOC $\lambda_{R}$, varying from 0.1 to 0.55 for $\lambda_{so}=0$ and from 0.1 to 1 for $\lambda_{so}=3$. Cyan points indicate a sign-changing $s$-wave superconducting gap, with $\tau^0\sigma^0$ as the leading channel contributing most to the gap. Dark blue points represent a similar type of gap, except triplet channels as the leading contributors. Yellow points denote an ordinary $d$-wave gap, while red points signify a $d_{\pm}$-wave gap, where the gap exhibits opposite signs on different Fermi surfaces. Green points represent SDW states, characterized by RPA suceptibility that tends toward divergence.
• Figure 4: (a)-(f) The plottings illustrate the contribution weights from six channels to the $A_1 (S_{\pm})$-wave as functions as some specific parameters. The other fixed parameters are set as flollows: (a) $\lambda_{so}=3$, $n=0.26$ and $\lambda_{R}=0.35$, corresponding to Fig. \ref{['fs']} (p), (b) $\lambda_{so}=3$, $n=0.26$ and $U=7.5$, (c) $\lambda_{R}=0.35$, $n=0.26$ and $U=7.5$, (d) $\lambda_{so}=3$, $n=0.5$ and $\lambda_{R}=0.9$, (e) $\lambda_{so}=3$, $n=0.5$ and $U=3.5$, (f) $\lambda_{R}=0.725$, $n=0.5$ and $U=3.5$. Note that in (a), (b), (d) and (e), when the condition requiring the absolute value of the weight is not imposed, the pairing in $\tau^3\sigma^0$ channel can acquire a minus sign under a direct projection. In (f), the right part of the green-line minimum is also taken in absolute value for the same reason.
• Figure 5: (a) The Wilson loopyu2011 calculation for the $S_{\pm}$ superconducting state. (b) The quasi-particle spectrum of $S_{\pm}$ superconducting state. Note that the in-gap edge states are double-degeneracy with opposite chriality. Periodic and open boundary conditions are applied along $x$ and $y$ direction, respectively. In both (a) and (b), the parameters are the same as those in Fig. \ref{['fs']} (p). The parameter $\Delta_0$ represents the coefficient of the form factors $\phi_{i,k}$ and $\bm{d_{j,k}}$ in the superconducting gap function and is set to 0.1.