Correlation function of the two-dimensional Ising model on a finite lattice. II

We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.

💡 Research Summary

The paper addresses the long‑standing problem of obtaining exact correlation functions for the two‑dimensional Ising model when the geometry is mixed: one direction is infinite while the other is finite with periodic boundary conditions. In addition, the authors allow anisotropic couplings (different interaction strengths along the two lattice axes), which makes the analysis more realistic for layered magnetic materials and for numerical simulations that often employ cylindrical or toroidal lattices.

The authors begin by recalling the Onsager solution for the infinite lattice, fixing the critical temperature and the standard critical exponents (β = 1/8, ν = 1). They then introduce the mixed geometry: the x‑axis is taken to be infinite, the y‑axis has length L and periodic boundary conditions. The anisotropy is encoded in the dimensionless couplings Kₓ = Jₓ/(k_B T) and Kᵧ = Jᵧ/(k_B T). By performing a Fourier transform along the infinite direction (continuous momentum kₓ) and a discrete Fourier series along the finite direction (quantized momentum k_y = 2π n/L), the transfer‑matrix formalism reduces the problem to a set of one‑particle excitations labelled by (kₓ,k_y).

A central technical achievement of the paper is the derivation of an exact “partial form factor” for these excitations. The authors obtain a closed‑form expression
F(kₓ,k_y) =