Vertex operators in solvable lattice models

TL;DR

This paper develops the theory of q-vertex operators in solvable lattice models, deriving their properties, q-difference equations, and connections to representation theory, with detailed analysis of the Ising model case.

Contribution

It introduces a formulation of q-vertex operators for face and edge-interaction models and computes matrix elements for the Ising model using Jordan-Wigner fermions.

Findings

Derived q-difference equations for correlation functions.

Established connections between vertex operators and representation theory.

Calculated all matrix elements of vertex operators in the Ising model.

Abstract

We formulate the basic properties of q-vertex operators in the context of the Andrews-Baxter-Forrester (ABF) series, as an example of face-interaction models, derive the q-difference equations satisfied by their correlation functions, and establish their connection with representation theory. We also discuss the q-difference equations of the Kashiwara-Miwa (KM) series, as an example of edge-interaction models. Next, the Ising model--the simplest special case of both ABF and KM series--is studied in more detail using the Jordan-Wigner fermions. In particular, all matrix elements of vertex operators are calculated.