Strong-coupling expansion of chiral models

TL;DR

This paper reanalyzes the strong-coupling expansion of lattice chiral models, providing a complete algorithmic approach to identify contributions and evaluate group integrals, with applications to two-dimensional principal chiral models.

Contribution

It introduces a fully algorithmic method for strong-coupling expansions in lattice models, separating geometric and group-theoretical problems, and applies it to principal chiral models on various lattices.

Findings

Strong-coupling expansions derived up to 15th and 20th order.

Explicit large-N and N= results presented.

Method applicable to lattice gauge theories and models with similar symmetries.

Abstract

The strong-coupling character expansion of lattice models is reanalyzed in the perspective of its complete algorithmization. The geometric problem of identifying, counting, and grouping together all possible contributions is disentangled from the group-theoretical problem of weighting them properly. The first problem is completely solved for all spin models admitting a character-like expansion and for arbitrary lattice connectivity. The second problem is reduced to the evaluation of a class of invariant group integrals defined on simple graphs. Since these integrals only depend on the global symmetry of the model, results obtained for principal chiral models can be used without modifications in lattice gauge theories. By applying the techniques and results obtained we study the two-dimensional principal chiral models on the square and honeycomb lattice. These models are a prototype field theory sharing with QCD many properties. Strong-coupling expansions for Green's functions are derived up to 15th and 20th order respectively. Large-$N$ and $N=\infty$ results are presented explicitly. Related papers are devoted to a discussion of the results.