Finite-temperature signatures of underlying superconductivity in the electron-doped Hubbard model

Abstract

We perform numerically exact determinant quantum Monte Carlo simulations of the Hubbard model and analyze pairing tendencies by evaluating correlation functions at the imaginary-time midpoint ($τ=β/2$), which suppresses high-frequency weight and emphasizes low-energy physics. Using this diagnostic, we identify clear finite-temperature signatures of underlying $d$-wave superconductivity for electron doping, while finding no clear indication upon cooling for hole doping. Our analysis enables direct comparison with ground-state DMRG, revealing consistent real-space pairing patterns. These results provide a practical route to bridge the gap between finite-temperature and ground-state numerically exact simulations of the Hubbard model despite the fermion sign problem.

Figures (14)

• Figure 1: (A,B) Unequal-time $d$-wave pair-field susceptibility, $\chi_d \equiv \beta\langle \mathcal{T}_\tau\Delta_d(\tau=\beta/2)\Delta^\dagger_d\rangle/N_s$ for hole doping ($x > 0$; A) and electron doping ($x < 0$; B). The inset of (A) highlights half filling ($x=0$); the inset of (B) zooms in on the electron-doped regime $|x| \geq 15\%$. (C, D) Superconducting pairing vertex from the unequal-time susceptibility and its uncorrelated bubble, $\Gamma_d \bar{\chi_d} \equiv \bar{\chi_d}/ \chi_d -1$ (with $\bar{\chi_d}$ built from single-particle Green’s functions), for hole doping (C) and electron doping (D). Insets in (C, D) show the doping dependence of $\Gamma_d \bar{\chi_d}$ at fixed temperatures. Parameters: $U/t=8$, $t'/t=-0.25$; cluster $8\times 8$.
• Figure 2: Real-space pattern of the unequal-time pair-pair correlator $\beta S_{\alpha,\alpha'}(\mathbf{r},\tau=\beta/2) \equiv \beta \sum_i \langle \mathcal{T}_\tau \Delta_{\alpha}(\tau=\beta/2, \mathbf{r}_i + \mathbf{r}) \Delta_{\alpha'}^\dagger(0, {\mathbf{r}_i}) \rangle/{N_s}$ (see Supplementary Materials Formalism Section), at (A, D) hole doping $x=0.03$, (B) half filling $x=0$, and (C, E) electron doping $x=-0.03$. We take the vertical bond connecting $(0,0)$ and $(0,1)$ as the reference. Parameters: $U/t=6$, $t'/t=-0.25$; Temperatures and sizes: $T/t=0.2$ on $16\times 4$ (A-C); $T/t=0.222$ on $8\times 8$ (D, E).
• Figure 3: Superconducting pairing vertex $\Gamma_d \bar{\chi_d}$ for various interaction strengths $U$ at electron doping $x=-1\%$ (A) and $x=-7\%$ (B). Parameters: $t'/t=-0.25$; cluster $8\times 8$.
• Figure 4: Negative temperature slope of the pair susceptibility, $-\partial \chi_d/\partial T$, estimated from an Akima spline fit to $\chi_d(T)$. Parameters: $U/t=6$, $t'/t=-0.25$; cluster $8\times 8$.
• Figure S1: Unequal-time $d$-wave pair-field susceptibility and vertex (definitions as in main-text Fig. \ref{['fig:Pd_and_vertexU8']}), now for $U/t=6$ (other parameters unchanged). Trends are qualitatively consistent with the $U/t=8$ case in main-text Fig. \ref{['fig:Pd_and_vertexU8']}.