Ring exchange and the Heisenberg and Hubbard models
M W Long; C W M Castleton; C A Hayward; M W Long; Sch. of Phys., Birmingham Univ., UK; C W M Castleton; Sch. of Phys., Birmingham Univ., UK; C A Hayward; Sch. of Phys., Birmingham Univ., UK
Журнал:
Journal of Physics: Condensed Matter
Дата:
1994-10-31
Аннотация:
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<sub>n</sub> approximately 1/ square root n, 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 J<sub>2</sub>/J<sub>1</sub> at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J<sub>2</sub>=4J<sub>1</sub> as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n<sub>k</sub>, 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<sub>k</sub>, 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 t<sub>1</sub>-t<sub>2</sub> 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.
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