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We derive the dominant contribution to the large-distance decay ofcorrelation functions for a spin chain model that exhibits both Haldane andNeel phases in its ground state phase diagram. The analytic results areobtained by means of an approximate mapping between a spin-1 anisotropicHamiltonian onto a fermionic model of noninteracting Bogolioubov quasiparticlesrelated in turn to the XY spin-1/2 chain in a transverse field. This approachallows us to express the spin-1 string operators in terms of fermionicoperators so that the dominant contribution to the string correlators at largedistances can be computed using the technique of Toeplitz determinants. Asexpected, we find long-range string order both in the longitudinal and in thetransverse channel in the Haldane phase, while in the Neel phase only thelongitudinal order survives. In this way, the long-range string order can beexplicitly related to the components of the magnetization of the XY model.Moreover, apart from the critical line, where the decay is algebraic, we findthat in the gapped phases the decay is governed by an exponential tailmultiplied by algebraic factors. As regards the usual two points correlationfunctions, we show that the longitudinal one behaves in a 'dual' fashion withrespect to the transverse string correlator, namely both the asymptotic valuesand the decay laws exchange when the transition line is crossed. For thetransverse spin-spin correlator, we find a finite characteristic length whichis an unexpected feature at the critical point. We also comment briefly theentanglement features of the original system versus those of the effectivemodel. The goodness of the approximation and the analytical predictions arechecked versus density-matrix renormalization group calculations.

Effective mapping of spin-1 chains onto integrable fermionic models. A study of string and Neel correlation functions