Generalized dimer states: hierarchies

Abstract

We present variational calculations for quantum spin-half models based upon modifications to nearest-neighbour dimer states. Using a sequence of projections we can attempt a sort of finite-size scaling, in the number of variational parameters, towards the exact solution. For the simple nearest-neighbour Heisenberg model we achieve an error of only 0.005% in energy for our most accurate calculation, much better than an equivalent Lanczos calculation. With which we compare. Due to the fact that we control the total spin in our wavefunctions, we can evaluate the coherence lengths associated with cyclic-exchange permutations (the spin-half sector) and for spin-spin correlations (the spin-one sector), yielding, for example, a convincing numerical confirmation that the spin-one biquadratic-exchange Hamiltonian has a finite correlation length. We perform a study of the one-dimensional J1-J2 model, trying to predict the onset of the phase transition by searching directly for the divergence of the correlation length. Our calculations predict the phase transition near J2/J1 approximately 1/4, as expected. We also perform a study of the ladder geometry, giving further evidence that infinitesimal coupling between ladders yields an immediate gap in the spectrum. The only complication is that the minimization over our space of variational parameters is subtle: surprisingly complicated and "difficult to find" structure is observed in our low-energy solutions which might make our approach of limited use.