Linearized auxiliary fields Monte Carlo technique: Efficient sampling of the fermion sign

We introduce a method that combines the power of both the lattice Green' function Monte Carlo (LGFMC) with the auxiliary field quantum Monte Carlo (AFQMC) techniques, and allows us to compute exact ground-state properties of the Hubbard model for $U\lesssim 4t$ on finite clusters. Thanks to LGFMC, one obtains unbiased zero temperature results, not affected by the so-called Trotter approximation of the imaginary time propagator $e^{-H\tau }$. At the same time, the AFQMC formalism yields a remarkably fast convergence in $\tau$ before the fermion sign problem becomes prohibitive. As an application we report ground-state energies of the Hubbard model at $U/t=4$ with up to 100 sites.

Figure 1

Comparison of exact ground-state energy and the one obtained by LAQMC projection for the Hubbard model with two holes ($N=16$) in the 18-site square lattice with ${\Gamma }_t=2.5U$. Notice the rapid convergence with projection time even for large $U/t$.

Figure 2

Energy as a function of the power method iterations using different mean-field wave functions ${\psi }_{\text{MF}}$ for defining the guiding function in LAQMC. They differ in the value of the chemical potential ${\mu }_0$ and the BCS $d$-wave parameter ${\Delta }_{\text{BCS}}^{x^2-y^2}$ (see Ref. 17). (a) ${\mu }_0=-0.198t$, (b) ${\Delta }_{\text{BCS}}^{x^2-y^2}=0.002t$.

Figure 3

Energy per hole for LAQMC release nodes (solid symbols) and CPQMC (open symbols). Lines are guides to the eye.