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arXiv:2601.11267

Oriented Triplet $p$-Wave Pairing from Fermi surface Anisotropy and Nonlocal Attraction

Shuning Tan, Ji Liu, Minghuan Zeng, Tao Ying, Zhangkai Cao, Ho Kin Tang

Published Jan 16, 2026• 0 citations
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Abstract

Using constrained-path quantum Monte Carlo, we map the ground-state phase diagram versus the nearest-neighbor (NN) attraction $V$ and spin-dependent hopping anisotropy $α$ for the two-dimensional attractive $t$--$U$--$V$ Hubbard model at filling $n\simeq0.85$. We identify an onsite $s$-wave superfluid, a Cooper pair Bose metal with an uncondensed Bose surface, and an oriented equal-spin triplet $p$-wave pairing phase. The NN attraction activates the odd-parity channel, while hopping anisotropy suppresses the competing $s$-wave coherence and selects a $p_x/p_y$ polar axis, and thus lowers the critical $|V_c|$ for the onset of triplet-dominant $p$-wave pairing. A channel-resolved Landau analysis provides a criterion for the Landau $p$-wave scale $V_c^{\mathrm L}(α)$, consistent with the observed anisotropy dependence of $|V_c|$. Our results establish how NN interaction and Fermi surface anisotropy cooperate to generate the oriented triplet $p$-wave pairing, and suggest that cold-atom and altermagnetic platforms could potentially realize this mechanism.

DOI
10.48550/arxiv.2601.11267

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Oriented Triplet ppp-Wave Pairing from Fermi surface Anisotropy and Nonlocal Attraction
Using constrained-path quantum Monte Carlo, we map the ground-state phase diagram versus the nearest-neighbor (NN) attraction VVV and spin-dependent hopping anisotropy ααα for the two-dimensional attractive ttt--UUU--VVV Hubbard model at filling n≃0.85n\simeq0.85n≃0.85. We identify an onsite sss-wave superfluid, a Cooper pair Bose metal with an uncondensed Bose surface, and an oriented equal-spin triplet ppp-wave pairing phase. The NN attraction activates the odd-parity channel, while hopping anisotropy suppresses the competing sss-wave coherence and selects a px/pyp_x/p_ypx​/py​ polar axis, and thus lowers the critical ∣Vc∣|V_c|∣Vc​∣ for the onset of triplet-dominant ppp-wave pairing. A channel-resolved Landau analysis provides a criterion for the Landau ppp-wave scale VcL(α)V_c^{\mathrm L}(α)VcL​(α), consistent with the observed anisotropy dependence of ∣Vc∣|V_c|∣Vc​∣. Our results establish how NN interaction and Fermi surface anisotropy cooperate to generate the oriented triplet ppp-wave pairing, and suggest that cold-atom and altermagnetic platforms could potentially realize this mechanism.
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The Cooper pair Bose metal (CPBM) is a non-superfluid quantum phase in which uncondensed fermion pairs form a "Bose surface" in momentum space. We investigate the CPBM in the two-dimensional spin-anisotropic attractive Hubbard model by tuning the next-nearest-neighbor (NNN) hopping t', carrier filling n, and spin anisotropy alpha, using large-scale constrained-path quantum Monte Carlo simulations. A moderate NNN hopping (t'/t = 0.2) substantially enlarges the CPBM region: the phase extends into weaker anisotropy regimes and coexists with a commensurate charge-density wave (CDW) near half-filling (n > 0.95), where CDW order would otherwise dominate at t' = 0. Interestingly, t' suppresses the overall CDW peak amplitude and introduces a geometric correlation between the orientations of the Fermi and Bose surfaces: for weak Fermi-surface rotations, the Bose surface remains aligned with the lattice axes, while larger distortions drive both surfaces to rotate in tandem. Momentum-resolved pairing distributions reveal that the bosonic pairing channels are jointly controlled by t' and carrier filling n. For small t', d_xy-wave correlations dominate across the entire filling range. In contrast, for larger t', the dominant pairing symmetry varies with n, reflecting a nontrivial interplay between frustration and density. These findings establish carrier filling and NNN hopping as complementary levers for manipulating CPBM stability and provide concrete criteria for identifying non-superfluid bosonic matter in cold-atom and correlated-electron systems.
Superconductivity arises when electrons form Cooper pairs with phase coherence. In contrast, a lack of phase coherence in Cooper pairs can lead to an uncondensed metallic ground state known as the Bose metal state. In this paper, we investigate an attractively interacting fermionic system with nearest-neighbor hopping (t) and next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$) anisotropy between two species of spins in a two-dimensional lattice. Utilizing the constrained path quantum Monte Carlo method, we demonstrate the existence of a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave Cooper pair Bose metal (CPBM) phase with ${t}^{\ensuremath{'}}/t>0.7$. The CPBM phase exhibits a domelike structure in the phase diagram of filling $n\ensuremath{\sim}0.65$, with the maximal region around an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\sim}0.2$, suggesting that an appropriate value of ${t}^{\ensuremath{'}}$ facilitates the formation of the Bose metal. Furthermore, we find that a Bose metal formed by fermions with a closed Fermi surface confirms that the crucial condition for this exotic phenomenon is primarily the anisotropy of the Fermi surface rather than its topology. Our finding of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave CPBM demonstrates the same pairing symmetry as the pseudogap behavior in cuprates, and its experimental realization in ultracold atom systems is also feasible.
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The two-dimensional Hubbard model is widely believed to capture the key ingredients of high-temperature superconductivity in cuprate materials. Here we report a constrained path quantum Monte Carlo study of the square-lattice extended Hubbard model with on-site Coulomb repulsion UUU and nearest-neighbor (NN) electron attraction VVV. Upon doping $\ensuremath{\delta}$ = 0.125, we find that NN electron attraction VVV significantly promotes an exotic spin-triplet (ppp-wave) pairing, with the ppp-wave pairing correlations growing stronger as VVV increases. Meanwhile, the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave (ddd-wave) pairing in the intermediate coupling regime shows an insignificant response to the increase of VVV. The spin density wave order, which is observed near half filling in the particle-hole channel, is also almost unaffected by the NN electron attraction VVV, reinforcing the established relationship between ddd-wave superconductivity (SC) and spin order. As the doping becomes heavier (i.e., $\ensuremath{\delta}$ increases), the dominant region of ppp-wave pairing expands, further suppressing the presence of ddd-wave pairing. Our paper suggests the ppp-wave pairing region can be induced and further broadened by the NN electron attraction VVV, offering a feasible mechanism to realize ppp-wave SC in realistic cuprate materials.
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