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

TL;DR

This work probes spin-triplet superconductivity in a 2D repulsive Hubbard model near a type-II van Hove singularity using numerically exactDeterminant Quantum Monte Carlo. By evaluating static pairing susceptibilities and their vertex contributions across $d_{x^2-y^2}$, $d_{xy}$, $p_x+ip_y$, and $p_{xy}$ channels, the authors find that spin-triplet $p$-wave pairings dominate near the $vHs$, while spin-singlet channels are suppressed by repulsive interactions. They observe ferromagnetic spin fluctuations peaking at $oldsymbol{q}=oldsymbol{ abla} ext{Γ}$ that correlate with triplet tendencies, suggesting a mechanism for triplet pairing. An estimate of the spin-triplet transition temperature $T_c/t o 0.006(4)$ at $U/t=2$ is provided, though the sign problem limits a definitive thermodynamic-limit extrapolation. Overall, the results corroborate RG and DMFT predictions and offer an unbiased numerical pathway toward realizing spin-triplet superfluidity in correlated 2D systems, with potential experimental realization in ultracold-atom setups.

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 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 for 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 the spin-triplet $p$-wave superfluidity in the repulsive Hubbard model, from an unbiased numerical perspective.

Figures (12)

• Figure 1: The plots of (a) energy dispersion $\varepsilon_{\mathbf{k}}/t$, and (b) local density of states (LDOS) $\rho(\varepsilon)$, for the noninteracting part of the model (\ref{['eq:2DHamlt']}), with $t_2/t=+0.5$ and $t_3/t=-0.1$. The Fermi surface for $\mu=(t_1+2t_2)^2/(4t_3)-2t_1=2t$ is plotted in inset of panel (b). The saddle points located at $\mathbf{K}_1=(\pm\pi/2,0)$ and $\mathbf{K}_2=(0,\pm\pi/2)$ induce the type-II vHs at $\varepsilon=-2t$ (vertical green dashed line) in (b), and the corresponding electron filling is $n_{\rm v}=0.3757$.
• Figure 2: Illustration of the $\mathbf{r}$-space form factor $f(\boldsymbol{\delta})$ for the spin-singlet $d_{x^2-y^2}$ and $d_{xy}$ pairing, as well as the spin-triplet $p_x$+$ip_y$ and $p_{xy}$ pairing. The center point (green circle) denotes the $\mathbf{i}=(i_x,i_y)$ site, and the surrounding points (black circles) represent the $\mathbf{i}+\boldsymbol{\delta}$ site in $\hat{\Lambda}_{\mathbf{i},\boldsymbol{\delta}}$ [see Eq. (\ref{['eq:PairOp2']})].
• Figure 3: Static pairing susceptibilities $S(\mathbf{q}=\boldsymbol{\Gamma},i\omega=0)$ as a function of electron filling $n$ for the noninteracting system with $t_2/t=+0.5,t_3/t=-0.1$. Panel (a) plots the results for $d_{x^2-y^2}$, $d_{xy}$, $p_x$+$ip_y$, and $p_{xy}$ pairing channels at $T/t=0.0067$, while panels (b) and (c) are the results for $p_x$+$ip_y$ and $p_{xy}$ pairings, respectively, with four different temperatures $T/t=0.05,0.02,0.01,0.0067$. The vertical gray dashed lines mark the electron filling $n_{\rm v}=0.3757$ at type-II vHs.
• Figure 4: Numerical results of the sign average $\langle\mathcal{S}\rangle$ in DQMC simulations for the model (\ref{['eq:2DHamlt']}) with $U/t=2$ and $t_2/t=+0.5,t_3/t=-0.1$, from the systems with $L=8,12,16$. Panel (a) plots $\langle\mathcal{S}\rangle$ as a function of electron filling $n$ at $T/t=0.10$, while panel (b) shows $\langle\mathcal{S}\rangle$ versus temperature $T/t$ with fixed $n=n_{\rm v}=0.3757$.
• Figure 5: DQMC results of (a) pairing susceptibilities $S(\mathbf{q}=\boldsymbol{\Gamma},i\omega=0)$, and (b) their vertex contributions, in $d_{x^2-y^2}$-wave, $d_{xy}$-wave, $p_x$+$ip_y$-wave, and $p_{xy}$-wave pairing channels as a function of electron filling $n$, for the model (\ref{['eq:2DHamlt']}) with $U/t=2$ and $t_2/t=+0.5,t_3/t=-0.1$. These results are from calculations for $L=16$ system at $T/t=0.20$. The vertical gray dashed lines in both panels mark the electron filling $n_{\rm v}=0.3757$ at type-II vHs.