A novel method for evaluating two-spin correlations of a one-dimensional Ising model is described. The method is based on an algebraic technique motivated by renormalization group theory and adapted for presentation in a graduate statistical physics course.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
We report some new results obtained for the thermodynamic properties and correlation functions of the one-dimensional Ising model with an inverse-square interaction between the spins. Using the Anderson and Yuval renormalization-group equations, we evaluate the correlation function and magnetic susceptibility for T<Tc. Monte Carlo data for the magnetization, specific heat, and spin-spin correlation function have also been obtained.