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.

• The paper demonstrates that altermagnetic spin anisotropy combined with carrier doping enhances d-wave pairing in the Hubbard model.
• It employs a mapping to an anisotropic t–J model and uses QMC to reveal optimal pairing near n≈0.88 and t_A≈0.6.
• The study reveals a mixed d+p phase with a predicted Leggett mode, offering design principles for higher T_c superconductors.

Altermagnetic-Doping Cooperation for Enhanced $d$-Wave Pairing in the Hubbard Model

Introduction

The study by Liu et al. investigates the cooperation of altermagnetic order and carrier doping as a mechanism for stabilizing and enhancing unconventional superconducting pairing within the paradigmatic Hubbard model. With altermagnetism established as a distinct collinear magnetic phase characterized by zero net moment but robust, momentum-odd spin splitting protected by crystalline symmetry, the interplay between altermagnetic correlations and charge doping is analyzed using a combination of analytical and numerical approaches. The results delineate the emergent phase behavior and provide insight into the relevance of spin anisotropy for $d$-wave and $p$-wave superconductivity—an issue directly connected to cuprates and altermagnetic candidate materials.

Figure 1: The Hubbard model with spin-anisotropic hopping $t\pm t_A$ under onsite repulsion $U$, its mapping to an anisotropic $t$–$J$ model with couplings $J_z$ and $J_\perp$, Fermi surface reconstructions, and pairing analysis as a function of filling and anisotropy.

Model Construction and Analytical Mapping

The core of the approach is a square-lattice Hubbard model with spin-dependent, anisotropic nearest-neighbor hoppings described by $t_A$, encoding the effect of symmetry-constrained altermagnetic spin splitting. For strong coupling ($d$0), the model maps onto an anisotropic $d$1–$d$2 Hamiltonian with superexchange couplings $d$3 and $d$4. The altermagnetic anisotropy suppresses long-range antiferromagnetism while preserving short-range fluctuations, creating a background favorable for superconductivity.

The mapping reveals two key interaction channels:

1. Singlet channel ($d$5): Enhanced by $d$6, promoting $d$7-wave and extended $d$8-wave pairing.
2. Triplet channel ($d$9): Activated by spin anisotropy and symmetry reduction, enabling $p$0-wave pairing.

These effects are illustrated by the Fermi surface evolution: increasing $p$1 drives a reconstruction from a perfectly nested, spin-degenerate Fermi surface to strongly distorted and spin-split topologies, as confirmed by the calculation of the total pairing strength $p$2.

Mean-Field and Quantum Monte Carlo Results

Mean-field analysis of the $p$3–$p$4 Hamiltonian demonstrates that moderate $p$5 stabilizes robust $p$6-wave superconductivity. For small $p$7, the $p$8-wave solution dominates, with a gap opening and increasing in magnitude with anisotropy. At larger $p$9, singlet and triplet channels coexist, and eventually, the $t\pm t_A$0-wave component dominates. The $t\pm t_A$1-wave instability is optimal at moderate anisotropy, after which the system evolves into a mixed $t\pm t_A$2 phase.

Quantum Monte Carlo (QMC), specifically constrained-path QMC, is used to non-perturbatively treat the full Hubbard model with doping. The QMC evaluation of the vertex $t\pm t_A$3-wave pairing correlation function reveals a pronounced enhancement at intermediate doping and $t\pm t_A$4, precisely where the density-wave (SDW and CDW) order parameters are concurrently suppressed—a prerequisite for robust unconventional superconductivity.

Figure 2: QMC vertex $t\pm t_A$5-wave pairing function $t\pm t_A$6 in the $t\pm t_A$7 plane; finite doping and moderate anisotropy yield a pronounced peak, flanked by suppressed SDW and CDW structure factors.

Additionally, the suppression of SDW and CDW correlations is quantified via the momentum-resolved structure factors, supporting the mechanism in which short-range spin fluctuations mediate superconductivity when density-wave instabilities are reduced. Notably, the peak $t\pm t_A$8-wave response is observed near $t\pm t_A$9 and $U$0.

Emergence of Mixed $U$1+$U$2 Pairing and Phase Diagram

Mean-field free energy landscapes as a function of both singlet and triplet order parameters, $U$3, elucidate the evolution of the superconducting state with increasing anisotropy and doping. At weak anisotropy, a pure $U$4-wave state is stable; at larger $U$5 and/or high filling, a demonstrable crossover (and eventual dominance) to $U$6-wave and $U$7 coexistence appears. In the coexistence regime, the order parameter is multi-component, and the phase stiffness between singlet and triplet components gives rise to a neutral, massive Leggett mode, whose detection is a target for Raman and THz spectroscopy.

Figure 3: Evolution of the mean-field free energy landscape in $U$8 space shows a transition from pure $U$9-wave, to a $t$0+$t$1 coexistence, to pure $t$2-wave with increasing $t$3 and/or filling; QMC confirms the enhancement and overlap of $t$4- and $t$5-wave vertex functions.

QMC results reinforce the presence of strong $t$6-wave vertex enhancement at moderate $t$7 and near-optimal doping, directly mirroring the regime where $t$8-wave correlations are also strong. Thus, the calculations robustly indicate that the $t$9 mixed phase is a robust feature of the altermagnetic-doped Hubbard model, and is expected to support higher $J$0 due to the cooperative enhancement of the total pairing amplitude.

Implications and Outlook

The study presents direct evidence that weak to moderate altermagnetic spin anisotropy, when combined with doping, is a potent route to optimal $J$1-wave superconductivity, and that the admixture of triplet ($J$2-wave) components naturally emerges due to the reduced (from $J$3 to $J$4) point group symmetry and spin-split Fermi surfaces. These results provide a unifying mechanism for the dominance of $J$5-wave pairing in cuprates and establish key design principles for engineered systems, e.g., oxide heterostructures or ultracold optical lattices, where spin-anisotropic hopping is tunable.

In the context of recent experimental progress in the observation and control of altermagnetic order [Phys. Rev. X 12, 040501 (2022); Nature 636, 348 (2024)], the results of this work point to direct routes for the realization of higher $J$6 unconventional superconductors. Furthermore, the predicted Leggett mode in the $J$7 regime provides a concrete dynamical observable to distinguish these multi-component condensates.

These findings motivate future work applying multiorbital extensions, incorporating longer-range interactions, and ab initio parameterizations to assess the quantitative relevance for cuprates, nickelates, and engineered heterostructures. The possible synergy with other emergent phenomena (pair density waves, topological superconductivity, and nonreciprocal transport) in altermagnetic systems warrants exhaustive exploration.

Conclusion

Liu et al. provide compelling theoretical and numerical evidence for the cooperative effect of altermagnetic order and charge doping as a stabilizing and enhancing factor for unconventional superconductivity—especially $J$8-wave—in the Hubbard model. The emergence of robust mixed $J$9 pairing, tunable via spin anisotropy and doping, is a direct consequence of the interplay between momentum-odd spin splitting and the suppression of competing density-wave orders, predictive of enhanced superconducting transition temperatures and emergent collective modes in altermagnetic materials and strongly correlated electron systems (2603.29377).