Linear Spin Chain with Nearest and Next‐Nearest Neighbor Interactions

We study the properties of a linear chain of spins governed by the Hamiltonian $${\mathrm{H=J}}_1{\displaystyle \sum _{\text{j=1}}^N}{\mathbf{S}}_j·{\mathbf{S}}_{\text{j+1}}{\mathrm{+J}}_2{\displaystyle \sum _{\text{j=1}}^N}{\mathbf{S}}_j·{\mathbf{S}}_{\text{j+2}}$$ and derive equations for an upper bound of the free energy by means of a temperature dependent Hartree‐Fock approximation. These equations can be solved at zero temperature, yielding an approximate wavefunction for the ground state, of which the energy is an upper bound for the exact ground state energy. The upper bounds obtained are improvements of the results of Majumdar and Ghosh. There are several different cases to be considered, depending on the relative values of J₁ and J₂. In none of these is there a gap between the ground state and the first excited state in the thermodynamic limit. Finally, correlation functions are shortly discussed in connection with spiral structures in classical spin systems.