Correlations quasi-long range

An alternative picture was first introduced by Aharony and Pytte in the context of random magnets. In this picture the order parameter correlation function exhibits algebraic decay with distance instead. This situation, intermediate between SRO and LRO, has come to be known as quasi-long-range order (QLRO). The most well-known example of QLRO, due to Berezinsky and to Kosterlitz and Thouless occurs in the low temperature phase of the two-dimensional XY model. A number of recent theoretical and computational studies have supported this point of view in random spin systems in a higher dimensionality . 

Thus, instead of the true Imig-range order we have a quasi-long-range order with the density correlation function (5.41b) qualitatively shown in Fig. 5.23b. 

Fig. 5.23 Diffraction intensity (a) and density correlation function (b) for the smectic A phase with a positional, quasi-long range, molecular order along the symmetry axis...

Fig. 8 Representation of correlation function, G(r), and X-ray intensity, /( ), for long-range order, algebraically decaying quasi-long-range order, and short-range order...

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

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