Analyticity and integrability in the chiral Potts model

Abstract

We study the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed.

Explicit evaluations and reciprocity theorems for finite trigonometric sums

“…In their study of the chiral Potts model and of certain integrable chiral quantum chains, McCoy and Orrick [61] showed that the high-and low-temperature expansions of the free energy serve as generating functions for certain classes of trigonometric sums. Several of these sums were subsequently studied in detail by A. Gervois and M. L. Mehta [39], [40].…”

“…In Section 4, we give another formulation of their theorem with a short proof by the same method of contour integration used in the two previous sections. Interesting cotangent and cosecant sums arise in the work of B. M. McCoy and W. P. Orrick [61] on the chiral Potts model in statisical mechanics. In Section 5, we explicitly evaluate a class of such sums.…”

Integrable Models in Statistical Mechanics: The Hidden Field With Unsolved Problems

“…From this he concludes that unless unexpected cancelations happen that there will be natural boundaries in the susceptibility on |v| = 1. This would indeed be a new effect which could make integrable models different from generic models, Such natural boundaries have been suggested by several authors in the past including Guttmann [52], and Orrick and myself [53] on the basis of perturbation studies of nonintegrable models which show ever increasingly complicated singularity structures as the order of perturbation increases; a complexity which magically disappears when an integrability condition is imposed. This connection between integrability and analyticity was first emphasized by Baxter [54] …”

A Few Finite Trigonometric Sums

“…The function f (z) also has a pole of order n at z = −b + 1. Using ( 4) and (8), as in the case of last theorem, we can obtain after a few steps of straight forward calculation,…”

“…Interestingly, this complicated-looking sum has the value d, which is remarkably simple, and is independent of m. Independence of m is intriguing, and may have deeper mathematical meaning. Finite trigonometric sums have also appeared in the study of chiral Potts model [8], theory of Dirac operators [9], Dedekind sums [10], theory of determinants and permanents [11,12], and many other places. There are many techniques for computing these sums; e.g., use of generating functions, Fourier analysis, method of residues, etc.…”