Study of the superconducting order parameter in the two-dimensional negative-$U$ Hubbard model by grand-canonical twist-averaged boundary conditions

By using variational Monte Carlo and auxiliary-field quantum Monte Carlo methods, we perform an accurate finite-size scaling of the $s$-wave superconducting order parameter and the pairing correlations for the negative-$U$ Hubbard model at zero temperature in the square lattice. We show that the twist-averaged boundary conditions (TABCs) are extremely important to control finite-size effects and to achieve smooth and accurate extrapolations to the thermodynamic limit. We also show that TABCs are much more efficient in the grand-canonical ensemble rather than in the standard canonical ensemble with fixed number of electrons. The superconducting order parameter as a function of the doping is presented for several values of $\left|U\right|/t$ and is found to be significantly smaller than the mean-field BCS estimate already for moderate couplings. This reduction is understood by a variational ansatz able to describe the low-energy behavior of the superconducting phase by means of a suitably chosen Jastrow factor including long-range density-density correlations.

Figure 1

$s$-wave variational parameter ${\mathrm{\Delta }}_0$ as a function of $n$ at $U=-2$. Mean-field calculations are performed on $L=12$ and $L=384$ clusters, and VMC calculations are done in GCE without Jastrow correlator on $L=12$ with PBC and TABCs. PBC-PBC indicates PBC in both $x$ and $y$ directions, and w/o stands for without. The error bars in the VMC results are smaller than the symbol sizes.

Figure 2

$s$-wave variational parameter ${\mathrm{\Delta }}_0$ as a function of $n$ at $U=-2$ calculated by VMC. The system size and boundary conditions used are indicated in the figure. Here, GTABC represents the grand-canonical twist-averaged boundary conditions, TABC the canonical twist-averaged boundary conditions, PBC-APBC stands for PBC in one direction and APBC in the other one, whereas PBC-PBC indicates PBC in both directions. The error bars are smaller than the symbol sizes.

Figure 3

$s$-wave variational parameter ${\mathrm{\Delta }}_0$ as a function of $n$ at $U=-2$. The results for mean-field calculations on $L=384$ and VMC on $L=12$ with and without Jastrow correlator in GCE with TABCs are shown. The error bars in the VMC results are smaller than the symbol sizes.

Figure 4

Pairing correlations ${\varphi }^2$ as a function of $n$ at $U=-2$ calculated by VMC. The system size and boundary conditions used are indicated in the figure. The notations are the same as those in Fig. 2.

Figure 5

Finite-size-scaling analyses of the pairing correlation ${\varphi }^2\left(L\right)$ for (a) $U=-2$, (b) $U=-3$, and (c) $U=-4$ by VMC at half-filling with different boundary conditions. The solid lines are fits to the GTABC data. The extrapolated values to the thermodynamic limit ${lim}_{L\to \infty }{\varphi }^2\left(L\right)$ are indicated at $1/L=0$ for each panel and correspond to 0.01994(4) for $U=-2$, 0.0497(4) for $U=-3$, and 0.0758(5) for $U=-4$. The notations are the same as in Fig. 2.

Figure 6

Finite-size-scaling analyses of the pairing correlation ${\varphi }^2\left(L\right)$ for (a) $U=-2$, (b) $U=-3$, and (c) $U=-4$ by VMC at quarter-filling with different boundary conditions. The solid lines are fits to the GTABC data. The extrapolated values to the thermodynamic limit ${lim}_{L\to \infty }{\varphi }^2\left(L\right)$ are indicated at $1/L=0$ for each panel and correspond to 0.00095(25) for $U=-2$, 0.0143(1) for $U=-3$, and 0.0378(4) for $U=-4$. The notations are the same as those in Fig. 2.

Figure 7

Pairing correlation ${\varphi }^2$ as a function of $n$ at $U=-2$ by AFQMC and VMC. The system size and boundary conditions are indicated in the figure. The notations are the same as in Fig. 2.

Figure 8

Finite-size-scaling analyses of the pairing correlation ${\varphi }^2\left(L\right)$ for (a) $U=-2$, (b) $U=-3$, and (c) $U=-4$ by AFQMC at half-filling with different boundary conditions. The solid lines are the fit to the TABC data. The TABC extrapolated values to the thermodynamic limit ${lim}_{L\to \infty }{\varphi }^2\left(L\right)$ are indicated at $1/L=0$ for each panel and correspond to 0.0098(3) for $U=-2$, 0.0222(6) for $U=-3$, and 0.0368(2) for $U=-4$. The notations are the same as those in Fig. 2.

Figure 9

Finite-size-scaling analyses of the pairing correlation ${\varphi }^2\left(L\right)$ for (a) $U=-2$, (b) $U=-3$, and (c) $U=-4$ by AFQMC at quarter-filling with different boundary conditions. The solid lines are the fit to the GTABC data. The GTABC extrapolated values to the thermodynamic limit ${lim}_{L\to \infty }{\varphi }^2\left(L\right)$ are indicated at $1/L=0$ for each panel and correspond to 0.0009(3) for $U=-2$, 0.0088(1) for $U=-3$, and 0.0266(2) for $U=-4$. The notations are the same as those in Fig. 2.