A relationship between the two-point correlation function of the Ising model in the critical region and the $〈sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(0)sin\sqrt{\ensuremath{\pi}}\ensuremath{\varphi}(\ensuremath{\rho})〉$ Green's function of a sine-Gordon field is constructed. This relationship is tested by means of a mass perturbation expansion with an infrared cutoff and shown to agree, at the leading-logarithm approximation and up to the second order, with the exact result of the Ising model.
The two{dimensional (2D) Ising model is reviewed as a theory of free fermions on a lattice.The discussion includes the fermionization procedure based on the mirror{ordered factorization of the density matrix, Gaussian fermionic integral representation for partition function, the momentum{ space analysis and Onsager's result, the eective continuum{limit eld theories and the critical{ point singularities.The emergence of long{range fermionic correlations in a nonzero magnetic eld and the behaviour of the speci c heat along the critical isotherm are commented.Attention is given to the choice of rational computational devices.
The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field
h at
T=0. This model corresponds to the
XY model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of
h, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field
h=hc=1, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For
h<1,h>1 the results already obtained for the
XY model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at
T=0.
For both
T>Tc and
T<Tc, we report exact expressions for the
n-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields
n-point Schwinger functions of some relativistic quantum field theories.