Study of the superconducting order parameter in the negative-$U$ 2D-Hubbard model by grand-canonical twist-averaged boundary conditions

TL;DR

This study uses advanced quantum Monte Carlo methods with twist-averaged boundary conditions to accurately analyze the superconducting order parameter in the negative-U 2D Hubbard model, revealing smaller values than mean-field predictions and emphasizing the importance of boundary conditions.

Contribution

It demonstrates the effectiveness of twist-averaged boundary conditions in finite-size scaling and shows the reduced superconducting order parameter compared to mean-field estimates in the negative-U Hubbard model.

Findings

Twist-averaged boundary conditions improve finite-size scaling accuracy.

Superconducting order parameter is smaller than mean-field estimates.

Grand-canonical ensemble is more efficient for simulations.

Abstract

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 is 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 $|U|/t$ 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 behaviour of the superconducting phase, by means of a suitably chosen Jastrow factor including long-range density-density correlations.