Numerical Verification of Perturbative Schwinger-Dyson Resummation on Lattice Models

TL;DR

The paper addresses verifying a perturbative Schwinger-Dyson resummation for the collective Coulomb field in a large-$N$ Homogeneous Electron Fluid by applying a $1/N$-truncated SD hierarchy to small lattice toy models. It derives a closed set of equations for the 1- and 2-point correlators, using Legendre-transform-based approximations to express higher-point functions in terms of lower-order correlators. The authors test the approach on $2\times2$ and $3\times3$ lattices via Markov Chain Monte Carlo with Metropolis-Hastings, finding agreement to about $10^{-3}$ at $N\sim 100$, improving with more samples. This demonstrates the viability of SD resummation on lattice models and suggests potential for investigating phase structure in the Homogeneous Electron Fluid and related lattice field theories.

Abstract

We investigate an approximation to the Schwinger-Dyson (SD) equations of the collective Coulomb field of the large $N$ Homogeneous Electron Fluid. The large $N$ approximation transforms the infinite SD hierarchy into a set of closed, equations for 1 and 2-pt correlators. In this paper, the dynamics of a toy model -- a small, square Euclidean lattice with periodic boundary conditions -- are considered. The Markov Chain Monte Carlo numerical method evaluated the 1 and 2-pt correlation functions on a $2 \times 2$ and $3 \times 3$ lattice. The derived equations are checked with the correlator values, and an agreement at $N \sim 100$ to order $10^{-3}$ was found. The agreement can be further strengthened by increasing runs in the Markov Chain Monte Carlo method.

Figures (7)

• Figure 1: A $2 \times 2$ grid with periodic boundary conditions. The sites only connected via dashed lines represent mirrored values, so there are only $4$ unique sites to consider. When generalized, this means that a $\sqrt{M} \times \sqrt{M}$ grid will only possess $M$ unique sites to consider.
• Figure 2: Feynman diagrams for $S^{(2)}$, $S^{(3)}$, and $S^{(4)}$. The propagator here is $A^{-1}_{ij}$.
• Figure 3: A tree diagram showing the relationship between $W_{3}$ and $\Gamma_{3}$. For large enough $N$, $G_3 = S^{(3)}$.
• Figure 4: A Tree diagram illustrating the relation between $W_4$ and $\Gamma_4$, where two symmetric terms of the last diagram are omitted. Note that the $W_3$ term that appears in eq. \ref{['eqn: W4']} has already been solved for in terms of $\Gamma_3$. For large enough $N$, $\Gamma_4 = S^{(4)}$.
• Figure 5: Plot of $\langle x_3 \rangle$ for $M = 4$ and $N = 100$. All $x(i)$ of each run started with the same values, which were $0$ (blue/solid), $-2$ (red/dashed), and $4$ (black/dot-dashed) respectively. Note that after around $\sim 300$ updates, thermalization has been achieved.