  We derive the dominant contribution to the large-distance decay of
correlation functions for a spin chain model that exhibits both Haldane and
Neel phases in its ground state phase diagram. The analytic results are
obtained by means of an approximate mapping between a spin-1 anisotropic
Hamiltonian onto a fermionic model of noninteracting Bogolioubov quasiparticles
related in turn to the XY spin-1/2 chain in a transverse field. This approach
allows us to express the spin-1 string operators in terms of fermionic
operators so that the dominant contribution to the string correlators at large
distances can be computed using the technique of Toeplitz determinants. As
expected, we find long-range string order both in the longitudinal and in the
transverse channel in the Haldane phase, while in the Neel phase only the
longitudinal order survives. In this way, the long-range string order can be
explicitly related to the components of the magnetization of the XY model.
Moreover, apart from the critical line, where the decay is algebraic, we find
that in the gapped phases the decay is governed by an exponential tail
multiplied by algebraic factors. As regards the usual two points correlation
functions, we show that the longitudinal one behaves in a 'dual' fashion with
respect to the transverse string correlator, namely both the asymptotic values
and the decay laws exchange when the transition line is crossed. For the
transverse spin-spin correlator, we find a finite characteristic length which
is an unexpected feature at the critical point. We also comment briefly the
entanglement features of the original system versus those of the effective
model. The goodness of the approximation and the analytical predictions are
checked versus density-matrix renormalization group calculations.
