Alternating Linear Heisenberg Antiferromagnet: The Exciton Limit

The one-dimensional alternating antiferromagnet with H=2Jn^{S}₂₍{S}₂₍-₁+2J^'n^{S}₂₍{S}₂₍+₁ is studied for J^'. For ^-1k₁T the susceptibility is expanded in powers of the exciton density as (J^', J^{'}J) e^-2+B (J^', J^{'}J) e^-4+ and the coefficients A and B are calculated for J^{'}J0. The calculation of B (J^', 0) required the evaluation of the two-exciton scattering matrix. The interactions between excitons which affect the susceptibility are found to be repulsive. As a result, the coefficient B is correctly predicted by the usual assumption that excitons obey localized statistics. A general discussion relating statistics to the on-shell forward-scattering t matrix enables one to understand the difference between the statistical properties of spin waves and excitons. For opposite-spin excitons an attractive bound state is found to exist for all values of total momentum. Perturbation theory in J^{'}J is used to calculate the single-exciton dispersion relation at zero temperature as E (k) = (2J+5J^{'3}32{J^2}) - (J^{'+J^'2}2J-5J^{'3}32J) cosk- (J^{'2}4J+J^{'3}8{J^2}) cos^2k- (J^{'3}8{J^2}) cos^3k.