Ring Exchange and the Heisenberg and Hubbard Models

We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R(n) approximately 1/ROOTn, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J2/J1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J2 = 4J1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n(k), is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U = infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U = infinity Hubbard model, the single-particle correlation function, n(k), for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t1-t2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.