  The one-dimensional XXZ model (s=1/2, N sites) with uniform long-range
interactions among the transverse components of the spins is considered. The
Hamiltonian of the model is explicitly given by
$H=J\sum_{j=1}^{N}(s_{j}^{x}s_{j+1}^{x}+s_{j}^{y}s_{j+1}^{y})
-(I/N)\sum_{j,k=1}^{N}s_{j}^{z}s_{k}^{z}-h\sum_{j=1}^{N}s_{j}^{z},$ where the
$s^{x,y,z}$ are half the Pauli spin matrices. The model is exactly solved by
applying the Jordan-Wigner fermionization, followed by a Gaussian
transformation. In the absence of the long-range 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-range interactions (I different from 0) the
nature of the transition is strongly affected. 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 transition is still present, although the system is
driven out of its usual universality class, and its critical exponents assume
typical mean-field values.
