The Ground State of the One-Dimensional Ising Chain

Abstract

We consider a linear spin 1/2 Ising chain with pair interactions extending up to the nth neighbor. The following general theorem is proved: 'The energy of any arbitrary spin sequence may be written as a linear sum of cluster energies, the upper limit of the number of Ising spins in a cluster being 2n'. The term cluster energy here is used to mean the energy of a group of spins in a certain configuration, evaluated as if the same configuration repeated itself throughout the chain. The structure of the ground state of the system is investigated using the above theorem. It is shown that the ground state is a repetition of a certain cluster, and that the upper limit of the cluster size is 2n, except for some specific combinations of the interaction strengths when the ground state may admit mixing of different clusters. The possible ground state configurations are worked out explicitly for n = 2, 3, and 4 in the absence of the magnetic field using the above theorem. Previous attempts were confined only up to n = 3.