THE LANCZOS METHOD APPLIED TO THE HEISENBERG ANTIFERROMAGNETIC

A square planar lattice of 2N spins of 1/2 is investigated. An initial state which has zero spin angular momentum is chosen, rather than the commonly used Ising state. For a finite value of N this state gives a lower expectation value for the Hamiltonian than does the Ising state, and it is to be preferred on symmetry grounds. It is used as initial state in the Lanczos method, which produces, in principle, a set of orthonormal states all of which also have zero spin. The matrix elements of H in this representation can be found from the expectation values of its powers taken over the initial state and the first three matrix elements are evaluated. Their dependences on N are such that if the Lanczos matrix is truncated, as in perturbation theory, the ground state energy is reduced by an amount independent of N, which implies that to obtain a good approximation more matrix elements are needed. It is also indicated that the same method could be used to obtain a spin-wave spectrum of states having total spin of unity. The method is systematic and could, in principle, be applied to any two sub-lattice system. However, it is also found that there are many other low-lying states, in which the moments of the sub-lattices are inclined and so have a non-zero total spin. It appears that some of these manifolds show splittings due to dipolar interactions which give states with energies below that of the Heisenberg ground state. The Heisenberg model therefore has to be questioned as a model of physical relevance.