Homotopic Action: A Pathway to Convergent Diagrammatic Theories

Expansions in terms of Feynman diagrams are a powerful way of describing a system of many interacting electrons. They can be constructed directly for a macroscopically large system, while millions of diagrams can be summed with modern algorithms, yielding a reliable solution even when the interactions are strong enough to make the system insulating. There is, however, a major obstacle: in many interesting cases the series turns out to diverge, making its summation meaningless.

This work introduces a universal framework for formulating a physical system so that its diagrammatic expansion is guaranteed to converge. The idea is to replace the system by an artificial one that continuously transforms to the original when a transformation parameter is varied. This transformation, called homotopy, is otherwise largely arbitrary, which is used to render the resulting diagrammatic series convergent.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.257001

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