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

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.

Figures (4)

• Figure 1: (Color online) Ground-state phase diagram of the spin-anisotropic attractive $t$--$U$--$V$ Hubbard model [Eq. \ref{['eq:Hamiltonian']}.] at $U=-3$ (with $t=1$) and filling $n\simeq0.85$, estimated from quantum Monte Carlo simulations. The horizontal axis is the nearest-neighbor attractive interaction $V$, and the vertical axis is the hopping anisotropy parameter $\alpha$. We identify an onsite $s$-wave superfluid ($s$-SF), a Cooper pair Bose metal (CPBM), and an oriented spin-triplet $p$-wave pairing phase ($p$-TP). Decreasing $\alpha$ suppresses $s$-wave coherence and drives the system into the CPBM regime at small $|V|$, while a sufficiently strong NN attraction, $|V|>|V_c(\alpha)|$, stabilizes the $p$-TP. Inset: spin-dependent hoppings $t_{x\uparrow}=t_{y\downarrow}=t$ and $t_{y\uparrow}=t_{x\downarrow}=\alpha t$. Symbols denote representative data parameters in Fig. \ref{['fig3']}.
• Figure 2: (Color online) Maximum vertex weight $N^{\rm eff}_\ell(\mathbf{k}_{\max})$ for the onsite $s$-wave (blue) and NN equal-spin $p$-wave (orange) channels versus NN attraction $V$. Panels (a)--(d) correspond to hopping anisotropy $\alpha=0.02$, $0.2$, $0.5$, and $0.7$, respectively. The error bars denote statistical uncertainties.
• Figure 3: (Color online) Pairing vertex distributions and correlators in the $t$--$U$--$V$ model. Panels (a)--(c) correspond to the CPBM (circle), $s$-SF (triangle), and p-TP (diamond) points in Fig. \ref{['fig1']}: (a) $N^{\rm eff}_{s\text{-pair}}(\mathbf{k})$ at $V=-0.2$, $\alpha=0.02$; (b) $N^{\rm eff}_{s\text{-pair}}(\mathbf{k})$ at $V=-0.2$, $\alpha=1.00$; (c) $N^{\rm eff}_{p\text{-pair}}(\mathbf{k})$ at $V=-0.6$, $\alpha=0.02$. (d) $C^{\rm eff}_{s\text{-pair}}(i)$ for (a) and (b); inset: scaling of $N^{\rm eff}_{s\text{-pair}}(k_{\max})$. (e) $C^{\rm eff}_{p\text{-pair}}(i)$ for (c); inset: scaling of $N^{\rm eff}_{p\text{-pair}}(k_{\max})$.
• Figure 4: (Color online) Momentum-space evolution of $N^{\mathrm{eff}}_{p\text{-pair}}(\mathbf{k})$. Top row: $\alpha = 0.02$. Panels (a)–(d) correspond to $V = 0, -0.2, -0.4, -0.6$, and panel (e) shows the associated spin-resolved Fermi surface for $\uparrow/\downarrow$ (blue/red dashed curves). Bottom row: $\alpha = 0.50$. Panels (f)–(i) show the same sequence of $V$, and panel (j) displays the corresponding Fermi surface. For $V=0$, the $p$-wave signal is absent; increasing $|V|$ redistributes spectral weight toward $\Gamma$ and produces oriented lobes signaling the emergence of odd-parity triplet correlations. For stronger anisotropy (smaller $\alpha$), the distribution elongates along a single polar axis ($p_x$ or $p_y$), indicating that anisotropy selects the orientation of the $p$-wave channel via Fermi surface deformation. At fixed $V$, the total magnitude of the $p$-wave signal remains comparable between the two anisotropies, with only minor numerical variations.