It is shown that the isotropic Heisenberg model can be analysed in terms of a random walk on the permutation group. This approach makes it intuitively clear why the Heisenberg model exhibits long range order or ferrogmagnetic behavior in three dimensions and not in two and one dimensions. This approach to the Heisenberg model lends itself to computer analysis.
In this letter we discuss how the isotropic Heisenberg model can be analysed in terms of a random walk on the permutation group. We believe that this analysis makes it intuitively clear why the isotropic Heisenberg model exhibits long range order or spontaneous magnetization at sufficiently low temperatures in three dimensions and not in two and one dimensions. It is well known from the work of Mermin and Wagner [ 1 ] that the isotropic Heisenberg model does not exhibit spontaneous magnetization in one and two dimensions. It was shown by Ginibre and Robinson [2] that certain anisotropic Heisenberg models have a first order phase transition in two dimensions. The idea behind this work is the connection between the anisotropic Heisenberg model and the Ising model which is known to exhibit spontaneous magnetization in two dimensions (see e.g. [3]).
For the sake of definiteness we confine our attention to the simple isotropic Heisenberg model in three dimensions. We consider a system of N spin 89 particles arranged in a cubic lattice. Each particle interacts with its six nearest neighbors. The Hamiltonian is given by H = -J ~' O i 9 , where E' is the sum over all pairs of nearest neighbor lattice sites, and ~'~i denotes the three Pauli spin matrices for the lattice site i. We use periodic boundary conditions ( so that the particles on the face of the cubic array of particles are considered to be nearest neighbors on particles on the opposite face.) It is well known or can be shown by straight forward computation that ~i.~j = 2oij --I where oij is the exchange operator which Interchanges the spin wave functions at the lattice sites i and j (e.g. oij (up | down) = (down | up)). Using this relation the Hamiltonian becomes *Work supported in part by the National Science Foundation Letters in Mathematical Physics 1 (1976) 125-130. All Rights Reserved.