Some Recent Results on Pair Correlation Functions and Susceptibilities in Exactly Solvable Models

Using detailed exact results on pair-correlation functions of Z-invariant Ising models, we can write and run algorithms of polynomial complexity to obtain wavevector-dependent susceptibilities for a variety of Ising systems. Reviewing recent work we compare various periodic and quasiperiodic models, where the couplings and/or the lattice may be aperiodic, and where the Ising couplings may be either ferromagnetic, or antiferromagnetic, or of mixed sign. We present some of our results on the square-lattice fully-frustrated Ising model. Finally, we make a few remarks on our recent works on the pentagrid Ising model and on overlapping unit cells in three dimensions and how these works can be utilized once more detailed results for pair correlations in, e.g., the eight-vertex model or the chiral Potts model or even three-dimensional Yang-Baxter integrable models become available.Comment: LaTeX2e using iopart.cls, 10 pages, 5 figures (5 eps files), Dunk Island conference in honor of 60th birthday of A.J. Guttman

Quantum Loop Subalgebra and Eigenvectors of the Superintegrable Chiral Potts Transfer Matrices

It has been shown in earlier works that for Q=0 and L a multiple of N, the ground state sector eigenspace of the superintegrable tau_2(t_q) model is highly degenerate and is generated by a quantum loop algebra L(sl_2). Furthermore, this loop algebra can be decomposed into r=(N-1)L/N simple sl_2 algebras. For Q not equal 0, we shall show here that the corresponding eigenspace of tau_2(t_q) is still highly degenerate, but splits into two spaces, each containing 2^{r-1} independent eigenvectors. The generators for the sl_2 subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q=0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q not equal 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.Comment: LaTeX 2E document, using iopart.cls with iopams packages. 28 pages, uses eufb10 and eurm10 fonts. Typeset twice! Version 2: Details added, improvements and minor corrections made, erratum to paper 2 included. Version 3: Small paragraph added in introductio

Correlation functions for the three state superintegrable chiral Potts spin chain of finite lengths

We compute the correlation functions of the three state superintegrable chiral Potts spin chain for chains of length 3,4,5. From these results we present conjectures for the form of the nearest neighbor correlation function.Comment: 10 pages; references update

Spontaneous Magnetization of the Integrable Chiral Potts Model

We show how $Z$-invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the $N$ asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.Comment: LaTeX 2E document, using iopart.cls with iopams packages, 22 pages, 1 eps figure. Presented at the Simons Center for Geometry and Physics Workshop on Correlation Functions for Integrable Models 2010: January 18-22, 2010. Version 2: The identity conjectured in version 1 is now proved and its proof is presented in arXiv:1108.4713; various small corrections and improvements have been made als