Correlation functions for the Z-invariant Ising model

Continuing our work hep-th/9609135 where a explicit formula for the two-point functions of the two dimensional Z-invariant Ising model were found. I obtain here different results for the higher correlation functions and several consistency checks are done.

The correlation functions and spontaneous magnetization are calculated for the three-dimensional Ising model and for the three-dimensional Z_2 electrodynamics.

The correlation functions are calculated for the two dimensional Ising model with free boundary conditions and the two dimensional Ising model with periodic boundary conditions.

Using the variational formula for operator product coefficients a method for perturbative calculation of the short-distance expansion of the Spin-Spin correlation function in the two dimensional Ising model is presented. Results of explicit calculation up to third order agree with known results from the scaling limit of the lattice calculation.

The correlation functions are calculated for the three - dimensional Z_2 electrodynamics for the particular values of the ineraction energies and for the free boundary conditions.

We introduce an exact algorithm for the computation of spin correlation functions for the two dimensional +/-J Ising spin glass in the ground state. Unlike with the transfer matrix method, there is no particular restriction on the shape of the lattice sample, and unlike Monte Carlo based methods it avoids extrapolation from finite temperatures. The computational requirements depend only on the number and distribution of frustrated plaquettes.

Consider the natural graph associated to a rhombus tiling of a polygonal regionin the plane. The spin correlations between boundary vertices of this graph inthe Z-invariant Ising model do not depend on the choice of the rhombus tilingbut only on the region. We provide a matrix formula depending on the regionwhich allows practical computations of boundary correlations in this setting,extending the results of Galashin in the critical case.

We derive and prove the connection formulas for the lambda generalized diagonal Ising model correlation functions.

A new approach to the correlation functions is presented for the XXZ model in the anti-ferroelectric regime. The method is based on the recent realization of the quantum affine symmetry using vertex operators. With the aid of a boson representation for the latter, an integral formula is found for correlation functions of arbitrary local operators. As a special case it reproduces the spontaneous staggered polarization obtained earlier by Baxter.

Exact integral representations of spin one-point functions (ground state expectation values) are reported for the spin-1 analog of the XXZ model in the region $-1<q<0$. The method enables one to calculate arbitrary $n$-point functions in principle. We also report a construction of level 2 irreducible highest weight representations of $U_q(\hat{sl}_2)$ in terms of boson and fermion operators, and explicit forms of related vertex operators.