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arXiv:2507.09295 · Physical Review A

Spin-triplet pairing instability in a two-dimensional repulsive Hubbard model

X. Liu, Yu-Feng Song, Yuan-Yao He, Tao Ying, Xueru Zhang

Published Nov 21, 2025• 0 citations
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Abstract

The search for superconductivity with unconventional pairing symmetry has been a central focus in the study of strongly correlated electron systems. In this work, we report a numerically exact study of the spin-triplet pairing in a two-dimensional Hubbard model with repulsive interactions, employing the determinant quantum Monte Carlo method. The model includes next-nearest-neighbor and third-nearest-neighbor hopping terms and maintains spin balance. By tuning the fermion filling close to a type-II Van Hove singularity (VHS) in the model, we numerically investigate the ordering tendencies of several possible pairing channels with different symmetries. Our numerical results provide clear evidence of the spin-triplet $p$-wave pairing instability approaching low temperatures, as revealed by the vertex contribution to the pairing susceptibility. This signature becomes increasingly pronounced as the interaction strength increases in the weak to intermediate regime. We further find that, near the type-II VHS, the dominant spin-spin correlations in the system are ferromagnetic, suggesting its close relation to the spin-triplet pairing instability. Our findings offer a reliable approach to realize spin-triplet $p$-wave superfluidity in the repulsive Hubbard model from an unbiased numerical perspective.

DOI
10.1103/9552-rz3h

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The search for superconductivity with unconventional pairing symmetry has been a central focus in the study of strongly correlated electron systems. In this work, we report a numerically exact study of the spin-triplet pairing in a two-dimensional Hubbard model with repulsive interactions, employing the determinant quantum Monte Carlo method. The model includes next-nearest-neighbor and third-nearest-neighbor hopping terms and maintains spin balance. By tuning the fermion filling close to a type-II Van Hove singularity (VHS) in the model, we numerically investigate the ordering tendencies of several possible pairing channels with different symmetries. Our numerical results provide clear evidence of the spin-triplet ppp-wave pairing instability approaching low temperatures, as revealed by the vertex contribution to the pairing susceptibility. This signature becomes increasingly pronounced as the interaction strength increases in the weak to intermediate regime. We further find that, near the type-II VHS, the dominant spin-spin correlations in the system are ferromagnetic, suggesting its close relation to the spin-triplet pairing instability. Our findings offer a reliable approach to realize spin-triplet ppp-wave superfluidity in the repulsive Hubbard model from an unbiased numerical perspective.
We use a novel temperature-flow renormalization group technique to analyze magnetic and superconducting instabilities in the two-dimensional ${t\ensuremath{-}t}^{\ensuremath{'}}$ Hubbard model for particle densities close to the Van Hove filling as a function of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. In the one-loop flow at the Van Hove filling, the characteristic temperature for the flow to strong coupling is suppressed drastically around ${t}_{c}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.33t$, suggesting a quantum critical point between ddd-wave pairing at moderate ${t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}$ and ferromagnetism for ${t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}$. Upon increasing the particle density in the latter regime, the leading instability occurs in the triplet pairing channel.
Using finite-temperature determinantal quantum Monte Carlo calculations, we reexamine the pairing susceptibilities in the Hubbard model on the honeycomb lattice, focusing on doping levels onto and away from the Van Hove singularity (VHS) filling. For this purpose, electronic densities of 0.75 (at the hole-doping VHS) and 0.4 (well below the VHS) are considered in detail, where due to a severe sign problem at strong coupling strengths, we focus on the weak-interaction region of the Hubbard model Hamiltonian. By analyzing the temperature dependence of pairing susceptibilities in various symmetry channels, we find the singlet d+idd+idd+id wave to be the dominant pairing channel both at and away from the VHS filling. We furthermore investigate the electronic susceptibility to a specific chiral spin density wave (SDW) order, which we find to be similarly relevant at the VHS, while it extenuates upon doping away from the VHS filling.
In nonperturbative regimes, the superfluid instability in the two-dimensional (2D) Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.3t$, interaction strength $0\ensuremath{\le}U\ensuremath{\le}3t$, and lattice filling $0\ensuremath{\le}n\ensuremath{\le}2$. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and ${t}^{\ensuremath{'}}$. At ${t}^{\ensuremath{'}}=0.3$, the weak-coupling picture with the dominant triplet pairing sharply peaked at $n\ensuremath{\approx}0.73$ due to the Van Hove singularity is replaced by a plateau of the singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ paring, while for $U\ensuremath{\gtrsim}3t$ the effective couplings are consistent with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
Using the finite-temperature determinant quantum Monte Carlo (DQMC) algorithm, we study the pairing symmetries of the Hubbard Hamiltonian with next-nearest-neighbor (NNN) hopping t' on square lattices. By varying the value of t', we find that the d-wave pairing is suppressed by the onset of t', while the p + ip-wave pairing tends to emerge for low electron density and t' around -0.7. Together with the calculation of the anti-ferromagnetic and ferromagnetic spin correlation function, we explore the relationship between anti-ferromagnetic order and the d-wave pairing symmetry, and the relationship between ferromagnetic order and the p + ip-wave pairing symmetry. Our results may be useful for the exploration of the mechanism of the electron pairing symmetries, and for the realization of the exotic p + ip-wave superconductivity.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
Superfluid ground-state phase diagram of the 2d2d2d Hubbard Model in the emergent BCS regime
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