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 nonperturbative 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 4 f theory but note that it applies in principle to any model.

Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation

“…We find that the sign problem is of the same order as for the stochastic construction of Γ[G] in Ref. 17. We performed extensive Monte Carlo simulations such that the error bars are exclusively determined by the extrapolation of the first deformation orders as is shown in Fig.…”

“…After nearly ten years and despite recent and tremendous progress [14][15][16], one may well fear that the combination of an asymptotic/divergent series, even with a mild sign problem, is as prohibitive as the standard approaches. Recently [17], we therefore suggested to use the more flexible Dyson-Schwinger equation (DSE) instead of self-consistent Feynman diagrams [18] to provide a fully self-consistent scheme on the one and two particle level. Furthermore, we extended the homotopy analysis method (HAM) [19,20] to φ 4 field theory in two dimensions (2D) (providing us with more tools to enhance the convergence properties in a systematic way), and showed how the expansion in terms of rooted trees is amenable to a systematic Monte Carlo sampling.…”

“…Clearly, this is a radically different way at looking at interacting field theories. However, in our previous work [17] we truncated the DSE at the level of the 6-point vertex. The infinite tower of equations for n-point correlation functions was not solved and differences with the full, exact answer could be seen when the correlation length increases.…”

“…This gives the physical 2-point correlation function G. The physical 4-point vertex function Γ follows by evaluating the universal functional Γ[G] for the physical G. In Ref. 17 we have already shown how to solve a realistic FT when only taking the first three terms on the right hand side of (5) into account i.e. working with a truncated version of Γ[G] which has a finite convergence radius.…”

“…We use an extension of the Monte Carlo algorithm developed in Ref. 17 to sample the expansion of the HAM in rooted tree diagrams. We will describe the detailed algorithm, especially the correct implementation of the functional derivatives, for an arbitrary many-body system with 2-body interactions elsewhere.…”