Metropolis updates for Diagrammatic Monte-Carlo algorithms from Schwinger-Dyson equations

TL;DR

This paper introduces a new method for constructing Metropolis updates in Diagrammatic Monte-Carlo algorithms using Schwinger-Dyson equations, enabling simulations in complex quantum field theories without explicit diagram enumeration.

Contribution

The paper presents a general recipe for Metropolis updates in DiagMC algorithms based on Schwinger-Dyson equations, avoiding the need for duality transformations or diagram classification.

Findings

Successfully applied to compute the 1/N expansion in the quartic matrix model

Achieved good agreement with analytic results away from critical points

Identified a different sign problem related to diagram cancellations

Abstract

We describe a general recipe for constructing Metropolis updates for Diagrammatic Monte-Carlo (DiagMC) algorithms, based on the Schwinger-Dyson equations in quantum field theory. This approach bypasses explicit duality transformations, enumeration or classification of diagrams and can be used for lattice quantum field theories with unknown or complicated dual representations (such as non-Abelian lattice gauge theories). DiagMC algorithms constructed in this way can still be plagued by the sign problem, which is, however, completely different from the sign problem in conventional Monte-Carlo simulations and has its origin in cancellations between diagrams with positive and negative weights. To test the presented approach, we apply DiagMC to calculate the first 7 orders of 1/N expansion in the quartic matrix model and find good agreement with analytic results, with the exception of the close vicinity of the critical coupling where the critical slowing down sets in.