Homotopic Action: A Pathway to Convergent Diagrammatic Theories

TL;DR

The paper introduces the homotopic action framework, unifying and improving diagrammatic series convergence for many-fermion systems, demonstrated through the Hubbard model.

Contribution

It presents a systematic, universal approach that unifies existing methods and enables new techniques for convergent diagrammatic expansions in strongly correlated systems.

Findings

Eliminates the need for resummation techniques.

Allows introduction of effective interactions.

Reduces polynomial complexity in diagrammatic Monte Carlo.

Abstract

The major obstacle preventing Feynman diagrammatic expansions from accurately solving many-fermion systems in strongly correlated regimes is the series slow convergence or divergence problem. Several techniques have been proposed to address this issue: series resummation by conformal mapping, changing the nature of the starting point of the expansion by shifted action tools, and applying the homotopy analysis method to the Dyson-Schwinger equation. They emerge as dissimilar mathematical procedures aimed at different aspects of the problem. The proposed homotopic action offers a universal and systematic framework for unifying the existing -- and generating new -- methods and ideas to formulate a physical system in terms of a convergent diagrammatic series. It eliminates the need for resummation, allows one to introduce effective interactions, enables a controlled ultraviolet regularization of continuous-space theories, and reduces the intrinsic polynomial complexity of the diagrammatic Monte Carlo method. We illustrate this approach by an application to the Hubbard model.