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 ${\varphi }^4$ theory in 2D. It is based on an exact treatment of the functional derivative $\partial \Gamma /\partial G$ of the 4-point vertex function $\Gamma$ 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 $c_{0}G$, 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.

arXiv comment5 pages, 4 figures and Supplemental Material

References in corpus (30)

• Continuous-time Monte Carlo methods for quantum impurity models
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• The Infrared Behavior of QCD Green's Functions - Confinement, Dynamical Symmetry Breaking, and Hadrons as Relativistic Bound States
• Quantum Monte Carlo with Directed Loops
• Functional renormalization group approach to correlated fermion systems
• Continuous Time Quantum Monte Carlo method for fermions
• Worm algorithms for classical statistical models
• Exact solution of the Falicov-Kimball model with dynamical mean-field theory
• Polaron Problem by Diagrammatic Quantum Monte Carlo
• Fermi-Polaron: Diagrammatic Monte Carlo for Divergent Sign-Alternating Series
• Taming the dynamical sign problem in real-time evolution of quantum many-body problems
• Bold Diagrammatic Monte Carlo: When Sign Problem is Welcome
• Controlling Feynman diagrammatic expansions: physical nature of the pseudo gap in the two-dimensional Hubbard model
• Recent developments in Quantum Monte-Carlo simulations with applications for cold gases
• Determinant Diagrammatic Monte Carlo in the Thermodynamic Limit
• Quantum Monte-Carlo for correlated out-of-equilibrium nanoelectronics devices
• Shifted-Action Expansion and Applicability of Dressed Diagrammatic Schemes
• The Power of Perturbation Theory
• Diagrammatic Monte Carlo study of the Fermi polaron in two dimensions
• Correlation functions of three-dimensional Yang-Mills theory from Dyson-Schwinger equations
• Polynomial complexity despite the fermionic sign
• Diagrammatic Monte Carlo study of quasi-two-dimensional Fermi-polarons
• Vertex functions of Coulomb gauge Yang--Mills theory
• Regularization of Diagrammatic Series with Zero Convergence Radius
• A method for resummation of perturbative series based on the stochastic solution of Schwinger-Dyson equations
• Lecture notes on Diagrammatic Monte Carlo for the Fröhlich polaron
• A stochastic root finding approach: The Homotopy Analysis Method applied to Dyson-Schwinger Equations
• Grassmannization of classical models
• Diagrammatic Monte-Carlo for weak-coupling expansion of non-Abelian lattice field theories: large-N U(N)xU(N) principal chiral model

Cited by in corpus (2)

• Diagrammatic Monte Carlo algorithm for the resonant Fermi gas
• Homotopic Action: A Pathway to Convergent Diagrammatic Theories