Efficient One-Loop-Renormalized Vertex Expansions with Connected Determinant Diagrammatic Monte Carlo

TL;DR

This paper introduces an efficient Monte Carlo method for evaluating high-order perturbative expansions in quantum many-body systems, improving convergence and variance reduction by summing over diagram topologies using determinants.

Contribution

The authors develop a novel algorithm that computes large-order corrections with exponential efficiency, integrating out long-range interactions to achieve polynomial error scaling.

Findings

Significant variance reduction in Monte Carlo simulations.

Enhanced convergence of perturbative series.

Successful application to the fermionic Hubbard model.

Abstract

We present a technique that enables the evaluation of perturbative expansions based on one-loop-renormalized vertices up to large expansion orders. Specifically, we show how to compute large-order corrections to the random phase approximation in either the particle-hole or particle-particle channels. The algorithm's efficiency is achieved by the summation over contributions of all symmetrized Feynman diagram topologies using determinants, and by integrating out analytically the two-body long-range interactions in order to yield an effective zero-range interaction. Notably, the exponential scaling of the algorithm as a function of perturbation order leads to a polynomial scaling of the approximation error with computational time for a convergent series. To assess the performance of our approach, we apply it to the non-perturbative regime of the square-lattice fermionic Hubbard model away from half-filling and report, as compared to the bare interaction expansion algorithm, significant improvements of the Monte Carlo variance as well as the convergence properties of the resulting perturbative series.