The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

Abstract

The diagonal spin-spin correlations $⟨{\sigma }_{0,0}{\sigma }_{N,N}⟩$ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $N\in {\mathbb{Z}}_{\ge 0}$. These later recurrence relations are an example of the discrete Garnier equations which, in turn, are extensions to the "discrete Painlevé V" system.

23 pages, 1 figure

References in corpus (3)

• XXZ and Ising Spins on the Triangular Kagome Lattice
• Thermodynamics of Ising spins on the Triangular Kagome Lattice: Exact analytical method and Monte Carlo simulations
• Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model

Cited by in corpus (1)

• The anisotropic Ising correlations as elliptic integrals: duality and differential equations