Renormalization flow for unrooted forests on a triangular lattice

TL;DR

This paper calculates higher-order renormalization constants and beta-function coefficients for an O(N) sigma-model on a triangular lattice at N=-1, linking it to unrooted forests and extending diagram evaluation methods.

Contribution

It provides the first two-loop and three-loop renormalization constants for the sigma-model on a triangular lattice at N=-1, connecting field theory with combinatorial forest enumeration.

Findings

Computed two-loop renormalization constants.

Determined three-loop beta-function coefficient.

Extended coordinate space method to triangular lattice.

Abstract

We compute in small temperature expansion the two-loop renormalization constants and the three-loop coefficient of the beta-function, that is the first non-universal term, for the sigma-model with O(N) invariance on the triangular lattice at N=-1. The partition function of the corresponding Grassmann theory is, for negative temperature, the generating function of unrooted forests on such a lattice, where the temperature acts as a chemical potential for the number of trees in the forest. To evaluate Feynman diagrams we extend the coordinate space method to the triangular lattice.