Altermagnetic-doping interplay as a route to enhanced d-wave pairing in the Hubbard model

Abstract

Altermagnets - collinear, zero-net-moment magnets with momentum-odd spin splitting protected by crystalline symmetries - offer a tunable route to suppress long-range antiferromagnetism while preserving strong short-range spin fluctuations. We show that this environment robustly stabilizes unconventional superconductivity and naturally produces mixed-symmetry pairing. Through a strong-coupling analysis of a spin-anisotropic Hubbard model, we derive an anisotropic t-J model where exchange interactions cooperatively enhance singlet d-wave pairing and promote triplet p-wave pairing. Our mean-field analysis reveals a pairing evolution driven by altermagnetic anisotropy: for small spin anisotropy, the d-wave channel is enhanced, closely resembling the dominant pairing symmetry in cuprate superconductors, which suggests that weak spin anisotropy may be an essential ingredient in realistic models of these materials. Constrained-path quantum Monte Carlo simulations confirm this picture, showing a regime where dominant d-wave correlations coexist with an emergent p-wave component near optimal doping. As spin anisotropy increases, strong C2 anisotropy and spin splitting activate the triplet channel, leading to a stable d+p mixed-pairing state. This synergistic state exhibits significantly enhanced overall pairing strength, suggesting the possibility of a higher superconducting transition temperature.

Figures (3)

• Figure 1: Spin-anisotropic Hubbard model, mapping to the $t$--$J$ model, and pairing analysis. (a) Schematic of the Hubbard model with spin-anisotropic nearest-neighbor hopping $t \pm t_A$ under on-site repulsion $U$. (b) In the strong-coupling limit, the system maps onto an anisotropic $t$--$J$ Hamiltonian characterized by $J_z$ and $J_\perp$. (c--f) Representative Fermi surfaces at different fillings $n$ and anisotropies $t_A$, illustrating the reconstruction from a perfectly nested square to distorted and spin-split contours. (g) QMC evaluation of the total pairing strength $\Delta_{\mathrm{tot}}$ in the $(n,t_A)$ parameter space, showing a broad enhancement with increasing anisotropy. (h,i) Comparison of extended $s$-wave (blue) and $d$-wave (red) states within the $t$--$J$ model. Although the nominal grand potential $\Omega_{\min}$ of the $s$-wave branch is slightly lower (h), its gap amplitude remains vanishingly small across the entire range (i), corresponding to a normal state. In contrast, the $d$-wave solution develops a robust finite gap whose onset filling decreases and magnitude grows with $t_A$, establishing $d$-wave as the dominant superconducting instability stabilized by spin-anisotropic hopping in the altermagnetic background.
• Figure 2: Enhancement of $d$-wave pairing restricted by competing density-wave channels. (a) Quantum Monte Carlo evaluation of the total vertex $d$-wave pairing function $N^{\mathrm{Vertex}}_{d_{x^2-y^2}}$ in the $(n,t_A)$ parameter space, showing a pronounced maximum around $n \approx 0.88$ and $t_A \approx 0.6$ (blue circle), with pairing strength increasing along the dashed trajectory. (b) Filling dependence of vertex functions for multiple pairing channels, demonstrating the dominance of $d_{x^2-y^2}$ and $p_{\uparrow\downarrow}$ over others. (c--f) Form factor $\eta^\zeta$ of the $d_{x^2-y^2},\ d_{xy},\ p\ (p_x$ or $p_y$) pairing symmetry. (g,h) the spin structure factor $N_{\mathrm{SDW}}^{local}$ and the charge structure factor $N^{local}_{\mathrm{CDW}}$ in the $(n,t_A)$ space. (i--k) Momentum-resolved vertex function for the $d_{x^2-y^2}$-wave channel at the representative parameter points indicated by the blue markers. Together, these results establish that $d$-wave pairing is strongly enhanced in the region where both SDW and CDW tendencies are weakened, consistent with the RVB picture in which short-range spin fluctuations mediate superconductivity once density-wave instabilities are suppressed.
• Figure 3: Enhancement of $p$-wave pairing and emergence of $d+p$ coexistence. (a--c) Free-energy landscape $\Omega(\Delta_d,\Delta_p)$ of the $t$--$J$ model for representative anisotropies $t_A=0.3,0.4,0.5$: the global minimum evolves from a pure $d$-wave state to a $d+p$ mixed state (star symbol in panel b) and finally towards a regime dominated by the $p$-wave component. (d--f) Free-energy landscape $\Omega(\Delta_d,\Delta_p)$ of the $t$--$J$ model for representative anisotropies $n=0.4,0.71,0.9$: the global minimum evolves from a pure $d$-wave state to a $d+p$ mixed state (star symbol in panel b) and finally towards a regime dominated by the $p$-wave component.(g) Quantum Monte Carlo results for the vertex pairing function $N^{\mathrm{Vertex}}_{p_{\uparrow\downarrow}}$ in the $(n,t_A)$ parameter space, showing a pronounced enhancement around $n \approx 0.85$--$0.90$ and $t_A \approx 0.45$--$0.55$. Taken together, QMC and $t$--$J$ mean-field results demonstrate that near optimal filling and intermediate anisotropy, $p$-wave pairing is strongly enhanced and cooperates with $d$-wave order, stabilizing a mixed-symmetry superconducting state.