Inter-orbital spin-triplet superconductivity from altermagnetic fluctuations

TL;DR

The paper shows that inversion-symmetry-broken altermagnetic fluctuations generically favor spin-triplet superconductivity by enforcing momentum–orbital locking in a minimal two-orbital model near van Hove singularities. Using an RPA framework with Hubbard–Hund interactions, the authors identify a dominant AM$_z$ fluctuation that induces an inter-orbital spin-triplet state ($\tau_z$-triplet) through an internal $\pi$-phase Josephson coupling provided by a subdominant inter-orbital channel, distinguishing it from the $\tau_0$-triplet state mediated by ferromagnetic fluctuations. Their numerical results map out a phase diagram where AM fluctuations yield $\tau_z$-triplet pairing while FM fluctuations yield $\tau_0$-triplet pairing, with a clear experimental signature in triplet–triplet Josephson junctions: a pronounced suppression of the total supercurrent in $\tau_z$-$\tau_0$ junctions due to the internal phase. This work expands the landscape of spin-fluctuation-mediated superconductivity, offering a path to novel triplet superconductors with nontrivial orbital structure and testable predictions for Josephson experiments.

Abstract

Altermagnetic (AM) fluctuations are a new class of collinear spin fluctuations whose role in mediating superconductivity faces a fundamental tension: their $Γ$-point peak favors intra-orbital spin-triplet pairing, while their spin compensation favors inter-orbital singlets. Here, we demonstrate that inversion-symmetry-broken AM fluctuations generically resolve this competition in favor of spin-triplet pairing. As a proof of concept, we study a minimal two-orbital model with two van Hove singularities. The broken inversion symmetry induces momentum-orbital locking: the same orbital dominates at opposite momenta, enhancing the triplet channel. Crucially, a subdominant fluctuation channel arising from inter-van-Hove nesting provides an internal Josephson coupling that locks the phase difference between triplet pairs on different orbitals. We find this coupling changes sign ($+$ to $-$) upon a crossover from AM-dominant to ferromagnetic-dominant fluctuations. The resulting $π$-phase difference manifests as a $τ_z$-type order parameter, $c_{k,1\uparrow}c_{-k,1\uparrow} - c_{k,2\uparrow}c_{-k,2\uparrow}$. Although intra-orbital in the original basis, its orbital-nontrivial character, as manifested by its equivalence to inter-orbital pairing under rotation, defines a general \textit{inter-orbital spin-triplet superconductivity}. This state is distinct from the $τ_0$-triplet pairing mediated by ferromagnetic fluctuations, as evidenced by the canceled intra-orbital supercurrent in a Josephson junction between them.

Figures (5)

• Figure 1: Band structure and dominant magnetic fluctuations. (a) Schematic of the two-orbital square lattice model. (b) Band structure along high-symmetry paths, with the Fermi level (gray dashed line) tuned near van Hove singularities. (c) Momentum distribution of the largest eigenvalue of the static bare susceptibility $[\chi^{(0)}(\bm{k},i\omega= 0)]^{l_1 l_1}_{l_2 l_2}$, showing a pronounced peak at the $\Gamma$ point. (d) Three possible $\bm{Q}=\bm{0}$ onsite magnetic orders: $\tau_z$- and $\tau_x$-type altermagnetism (AM), and ferromagnetism (FM). Blue ($\color{blue}\uparrow$) and red ($\color{red}\downarrow$) arrows represent the spin polarization of local moments. (e) Momentum dependence of the bare susceptibility for the AM$_z$, AM$_x$, and FM channels, revealing the AM$_z$ fluctuation as the dominant fluctuation.
• Figure 2: Pairing mechanism. (a) Orbital-polarized Fermi surfaces near van Hove singularities: $d_{xz}$ ($d_{yz}$) character around the X (Y) point. (b) Feynman diagrams for the two key pairing channels: intra-orbital pairing mediated by $[\chi^{\text{RPA}}_{\text{spin}}(\bm{q})]^{11}_{11}$ and the pair-hopping mediated by $[\chi^{\text{RPA}}_{\text{spin}}(\bm{q})]^{12}_{12}$. (c) Evolution of the pairing vertices $[\chi^{\text{RPA}}_{\text{spin}}(\Gamma)]^{11}_{11}$ (black), $[\chi^{\text{RPA}}_{\text{spin}}(\Gamma)]^{11}_{22}$ (blue), and $[\chi^{\text{RPA}}_{\text{spin}}(\text{M})]^{12}_{12}$ with $J_H/U$ and $U=0.99U_c$.
• Figure 3: Fluctuation-pairing correspondences. The orbital-resolved gap functions, $\Delta^x_{d_{xz},d_{xz}}(\bm{k})$ and $\Delta^x_{d_{yz},d_{yz}}(\bm{k})$, for (a,b) AM$_z$-mediated ($J_H/U=0.02$) and (c,d) FM-mediated ($J_H/U=0.3$) pairings. The phase relation between orbitals differentiates the $\tau_z$-triplet from the $\tau_0$-triplet state, with the corresponding real-space patterns shown in the insets.
• Figure 4: Phase diagram. (a) Evolution of the pairing amplitude ratio $\tilde{\Delta}{xx}/\tilde{\Delta}{yy}$ with $J_H/U$ at $U=0.99U_c$, signaling the transition between $\tau_z$- and $\tau_0$-triplet states. (b) Full $U$–$J_H/U$ phase diagram, mapping the superconducting domes ($U<U_c$) of $\tau_z$-triplet (orange) and $\tau_0$-triplet (green) pairing against the altermagnetic (AM, blue) and ferromagnetic (FM, gray) ordered phases ($U \geq U_c$). The red dashed line marks the point where $\chi^{\text{RPA}}_{\text{AM}}(\Gamma)$ equals $\chi^{\text{RPA}}_{\text{FM}}(\Gamma)$.
• Figure 5: Possible experimental signatures. (a) Schematic of a Josephson junction between two spin-triplet superconductors (tSCs) separated by a normal metal (NM). The size $(L_{\text{SC}},L_{\text{NM}})=(200,30)$. (b-d) Current-phase relations $I(\phi_J)$ for different configurations: (b) $\tau_z$-$\tau_0$, (c) $\tau_0$-$\tau_0$, and (d) $\tau_z$-$\tau_z$. The pronounced suppression of the supercurrent uniquely in the $\tau_z$-$\tau_0$ junction (b) provides a clear fingerprint of the $\tau_z$-triplet state, distinguishing it from the conventional $\tau_0$-triplet.