A stochastic root finding approach: the homotopy analysis method applied to Dyson–Schwinger equations

Abstract

We present the construction and stochastic summation of rooted-tree diagrams, based on the expansion of a root finding algorithm applied to the Dyson–Schwinger equations. The mathematical formulation shows superior convergence properties compared to the bold diagrammatic Monte Carlo approach and the developed algorithm allows one to tackle generic high-dimensional integral equations, to avoid the curse of dealing explicitly with high-dimensional objects and to access non-perturbative regimes. The sign problem remains the limiting factor, but it is not found to be worse than in other approaches. We illustrate the method for ${\phi }^{4}$ theory but note that it applies in principle to any model.

Keywords

• diagrammatic Monte Carlo
• Dyson–Schwinger equations
• homotopy analysis method