Finite-temperature superconducting correlations of the Hubbard model

We utilize numerical linked-cluster expansions (NLCEs) and the determinantal quantum Monte Carlo algorithm to study pairing correlations in the square-lattice Hubbard model. To benchmark the NLCE, we first locate the finite-temperature phase transition of the attractive model to a superconducting state away from half filling. We then explore the superconducting properties of the repulsive model for the $d$-wave and extended $s$-wave pairing symmetries. The pairing structure factor shows a strong tendency to $d$-wave pairing and peaks at an interaction strength comparable to the bandwidth. The extended $s$-wave structure factor and correlation length are larger at higher temperatures but clearly saturate as temperature is lowered, whereas the $d$-wave counterparts, which start off lower at high temperatures, continue to rise near half filling. This rise is even more dramatic in the $d$-wave susceptibility. The convergence of NLCEs breaks down as the susceptibilities and correlation lengths become large, so we are unable to determine the onset of long-range order. However, our results extend the conclusion, previously restricted to only magnetic and charge correlations, that NLCEs offer a unique window into pairing in the Hubbard model at strong coupling.

Figure 1

(a) Temperature dependence of the $s$-wave pairing structure factor at $n=0.85$ for the attractive model with $U=-8$. The line is from NLCE and symbols are from DQMC. The inset shows the inverse of the same function vs $T$ and a low-temperature fit to $A\exp (B/\sqrt{T-T_c})$ with $T_c=0.11$. (b), (c) The corresponding correlation length and short-range correlation functions vs $T$. In the main panel of (a) and in (b), bare NLCE results before resummations for the last two orders, $8\mathrm{th}$ and $9\mathrm{th}$, are shown as thin dotted and dashed lines, respectively.

Figure 2

The uniform $d$-wave and extended $s$-wave pairing structure factors for the repulsive model with $U=8$ at densities 1.00, 0.95, 0.90, and 0.85 vs temperature. Lines are from the NLCE and symbols are from the DQMC. Bare NLCE results before resummations for the $8\mathrm{th}$ and $9\mathrm{th}$ orders are shown as thin dotted and dashed lines, respectively. The insets show the correlation lengths from the NLCE vs temperature for each case.

Figure 3

Temperature dependence of the $d$-wave pairing structure factor at $n=0.90$ for $U=4$, 6, 8, and 12. Symbols are the DQMC results. Top inset shows the same structure factor vs $U$ at a fixed temperature $T=0.43$. The bottom inset shows the inverse of the structure factor vs $T$, along with a fit to the function $A\exp (B/\sqrt{T-T_c})$ for $T<0.6$, which results in $T_c=0.048$.

Figure 4

The $d$-wave pairing susceptibility at $n=0.90$ vs temperature for $U=4$, 6, 8, and 12. Bare NLCE results before resummations for the last two orders, $6\mathrm{th}$ and $7\mathrm{th}$, are shown as thin dotted and dashed lines, respectively. Symbols are the DQMC results. The inset shows the inverse of the susceptibility vs temperature for the same values of the interaction strength.