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.

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.

We study the finite-temperature superfluid transition in a modified two-dimensional (2D) XY model with power-law distributed "scratch"-like bond disorder. As its exponent decreases, the disorder grows stronger and the mechanism driving the superfluid transition changes from conventional vortex-pair unbinding to a strong randomness criticality (termed scratched-XY criticality) characterized by a non-universal jump of the superfluid stiffness. The existence of the scratched-XY criticality at finite temperature and its description by an asymptotically exact semi-renormalization group theory, previously developed for the superfluid-insulator transition in one-dimensional disordered quantum systems, is numerically proven by designing a model with minimal finite size effects. Possible experimental implementations are discussed.