The one-dimensional XXZ model with long-range interactions

The one-dimensional XXZ model (s = (1)/(2), N sites) with uniform long-rang e interactions among the transverse components of the spins is considered. The Hamiltonian of the model is explicitly given by H = J Sigma (N)(j=1) (s (j)(x)s(j+1)(x) + s(j)(y)s(j+1)(y)) - (I/N)Sigma (N)(j,k=1) s(j)(z)s(k)(z) - h Sigma (N)(j=1) s(j)(z), where the s(x,y,z) are half the Pauli spin matr ices. The model is exactly solved by applying the Jordan-Wigner fermionizat ion, followed by a Gaussian transformation. In the absence of the long-rang e interactions (I = 0), the model, which reduces to the isotropic XY model. is known to exhibit a second-order quantum-phase transition driven by the field at zero temperature. It is shown that in the presence of the long-ran ge interactions (I not equal 0) the nature of the transition is strongly af fected. For I > 0, which favours the ordering of the transverse components of the spins, the transition is changed from second to first order, due to the competition between transverse and xy couplings. On the other hand, for I < 0, which induces complete frustration of the spins, a second-order tra nsition is still present, although the system is driven out of its usual un iversality class, and its critical exponents assume typical mean-field valu es. (C) 2001 Elsevier Science B.V. All rights reserved.