Computation of dynamical correlation functions for many fermion systems with auxiliary-field quantum Monte Carlo

TL;DR

This paper develops a quantum Monte Carlo method to compute dynamical correlation functions in many fermion systems, addressing finite size effects and proposing a new algorithm for efficient calculations, with applications to models like the Hubbard Hamiltonian.

Contribution

It introduces a new algorithm for calculating dynamical Green functions that varies particle number during Monte Carlo sampling, improving efficiency especially in dilute systems.

Findings

Accurate scaling of the charge gap with interaction strength U.

Effective reduction of finite size effects using twist angle minimization.

Demonstration of the algorithm's efficiency in dilute fermion systems.

Abstract

We address the calculation of dynamical correlation functions for many fermion systems at zero temperature, using the auxiliary-field quantum Monte Carlo method. The two-dimensional Hubbard hamiltonian is used as a model system. Although most of the calculations performed here are for cases where the sign problem is absent, the discussions are kept general for applications to physical problems when the sign problem does arise. We study the use of twisted boundary conditions to improve the extrapolation of the results to the thermodynamic limit. A strategy is proposed to drastically reduce finite size effects relying on a minimization among the twist angles. This approach is demonstrated by computing the charge gap at half-filling. We obtain accurate results showing the scaling of the gap with the interaction strength $U$, connecting to the scaling of the unrestricted Hartree-Fock method at small $U$ and Bethe Ansatz exact result in one dimension at large $U$. A new algorithm is then proposed to compute dynamical Green functions and correlation functions which explicitly varies the number of particles during the random walks in the manifold of Slater determinants. In dilute systems, such as ultracold Fermi gases, this algorithm enables calculations with much more favorable complexity, with computational cost proportional to basis size or the number of lattice sites.