Pair correlations in the attractive Hubbard model

The mechanism of fermionic pairing is the key to understanding various phenomena such as high-temperature superconductivity and the pseudogap phase in cuprate materials. We study the pair correlations in the attractive Hubbard model using ultracold fermions in a two-dimensional optical lattice. By combining the fluctuation-dissipation theorem and the compressibility equation of state, we extract the interacting pair-correlation functions and deduce a characteristic length scale of pairs as a function of interaction and density filling. At sufficiently low filling and weak onsite interaction, we observe that the pair correlations extend over a few lattice sites even at temperatures above the superfluid transition temperature.

Figure 1

Schematic diagram for pairing in a two-dimensional square lattice. The equal-spin correlation function $g_{\uparrow \uparrow }^{\left(2\right)}\left(\mathbf{r}\right)$ exhibits fermionic antibunching as a correlation hole. The unequal-spin correlation function $g_{\uparrow \downarrow }^{\left(2\right)}\left(\mathit{r}\right)$ depends strongly on the onsite interaction strength $U$, where attractive (repulsive) interaction raises (lowers) pair correlation at short distance from the classical value of one.

Figure 2

(a) Density structure factor at zero momentum $S(\mathit{q}=0)$ for various interaction strengths and filling. (b) $\int [g^{\left(2\right)}\left(\mathit{r}\right)-1]d^{2}r$ versus interaction strengths $U/t$. The black solid line shows the noninteracting expectation at the lowest temperature in the data sets. Data points above the horizontal dashed line signal that the unequal-spin $g_{\uparrow \downarrow }^{\left(2\right)}\left(\mathit{r}\right)$ outweighs its equal-spin counterpart, and vice versa. (c) Interacting contribution $\int [g_{\uparrow \downarrow }^{\left(2\right)}\left(\mathit{r}\right)-1]d^{2}r$ is obtained by subtracting the equal-spin contribution $\int [g_{\uparrow \uparrow }^{\left(2\right)}\left(\mathit{r}\right)-1]d^{2}r$. Data are for $U/t=-1.83, -3.90, -6.09, -7.62$, and $-9.61$, and the corresponding temperatures $k_{B}T/t$ of the data set are 1.26(4), 1.31(5), 1.73(8), 2.05(11), and 2.16(8), respectively. Solid lines show the result from DQMC simulation.

Figure 3

(a)–(d) Inferred pair-correlation length. (a) Correlation length $\xi$ versus filling $n$. As the interaction strength increases, we observe that the correlation length shrinks for low filling $n\lesssim 0.5$. For increasing filling, we observe that $\xi$ settles at approximately the limit of the continuous approximation. The inset exemplifies the estimation of the length scale, which is obtained by dividing area (integral) by height (amplitude). Solid lines show the result from DQMC simulation. (b) Correlation length $\xi$ for specific $n$ versus $U/t$. $n=0.1$, 0.15, 0.2, 0.3, and 0.5 correspond to square, up-triangle, diamond, down-triangle, and pentagon markers, respectively. Temperatures of data points shown in (a) and (b) are the same as those in Fig. 2. (c) Correlation length $\xi$ versus temperature $k_{B}T/t$ at $n=0.2$. For low filling, we observe in general a decreasing trend as temperature rises. While already at $n=0.5$, no significant temperature dependence is observed. The solid lines are linear fits to the data points. (d) Correlation length $\xi$ versus temperature $k_{B}T/\left|U\right|$ at $n=0.2$. When rescaled with respect to the interaction strength, the temperatures reached in the weakly interacting case are higher than the ones in the strongly interacting case.

Figure 4

(a), (b) In situ density profile of doubly occupied states $D(x,y)$ and singly occupied states $S_{\uparrow }(x,y)$, respectively. Images shown are obtained by averaging over 40 experimental realizations with atom number fluctuations within $5\%$. (c)–(e) Density equation of state $n\left(\mu \right)$ for different interaction strengths.