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

Abstract

We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for φ 4 theory in 2D. It is based on an exact treatment of the functional derivative ∂Γ/∂G of the 4-point vertex function Γ with respect to the 2-point correlation function G within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around G = 0 in a HAM control parameter c0G, or even a convergent one up to the phase transition point if shifts in G can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving integro-differential equations.

Diagrammatic Monte Carlo algorithm for the resonant Fermi gas

“…It can also be advantageous to perform this exact summation by brute-force enumeration provided the momentum and time variables are chosen appropriately, as sucessfully demonstrated very recently for the electron gas 70 . A radically different approach would be to work with Schwinger-Dyson equations, for which new algorithms were introduced and applied to bosonic models [71][72][73][74] .…”

Homotopic Action: A Pathway to Convergent Diagrammatic Theories

“…The standard definition of homotopy is a continuous transformation of one function into another. In the shifted action formalism described above, we aim at optimizing the diagrammatic expansion by selecting an appropriate starting action S(ξ = 0) and its continuous transformation into the physical action S = S(ξ = 1), similarly to the homotopy analysis method [40,41]. If we distance ourselves from the specifics of how various shifts are implemented (with or without the Hubbard-Stratonovich transformations), we recognize that a far more intuitive and transparent way to cast the attempted transformation of the action would be to write…”

Four Loop Scalar ϕ4 Theory Using the Functional Renormalization Group

“…There are many systems of physical interest that are strongly coupled and must be described with non-perturbative methods. Schwinger-Dyson (SD) equations are often used, but one problem with this approach is that the hierarchy of coupled SD equations needs to be truncated, and several different truncations have been proposed [1]. The n-particleirreducible effective action is an alternative non-perturbative method.…”