Exact relations for correlation functions in the general Ising model

Gregory S. Um, P. J. Gambardella, Max Dresden

We propose a generalized compressible Ising model and solve it exactly, using the renormalization-group method. We obtain exact expressions for free energy and spin-spin correlation functions for periodic and free boundary condition in a finite chain or in the thermodynamic limit. Through this method, we develop functional recursion relations. The corresponding fixed points are investigated with the correlation length. Our method is applicable to higher-order correlation functions.

Masuo Suzuki

Michael Ellis Fisher

The "star-triangle" and "decoration" transformations are generalized so as to apply to arbitrary mechanical systems coupled to the spins of a standard Ising net. This leads to exact solutions for further plane Ising nets and also for lattices in which the spins on alternate sites have a magnitude greater than $S=\frac{1}{2}$. A general class of antiferromagnetic Ising models is constructed; exact closed expressions can be derived for all the thermodynamic and magnetic properties of these models in an arbitrary magnetic field.The magnetizations and susceptibilities of Ising nets in which different spins have different magnetic moments are investigated and a valuable relation between the susceptibilities of the honeycomb and triangular lattices is derived. It is shown how correlation functions involving a given spin can be expressed in terms of correlations involving the nearest-neighbor spins instead.

R.J. Boel, Piet Kasteleyn

Chin‐Kun Hu, N. Sh. Izmailian

We develop a transfer matrix method to compute exactly the spin-spin correlation functions $〈{s}_{0}{s}_{n}〉$ of Bethe lattice spin models in the external magnetic field $h$ and for any temperature $T.$ We first compute $〈{s}_{0}{s}_{n}〉$ for the most general spin-$S$ Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-$S$ Ising model includes the spin-$\frac{1}{2}$ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and $T$ approaches the critical temperature ${T}_{c}.$ Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.

C. Ruge, P. Zhu, F. Wagner