Asymptotics and functional form of correlators in the XX-spin chain of finite length.

Asymptotics and functional form of correlators in the XX-spin chain of finite length. Abstract We verify the functional form of the asymptotics of the spin-spin equal-time correlation function for the XX-chain, predicted by the hypothesis of conformal invariance at large distances and by the bosonization procedure. We point out that bosonization pocedure also predicts the functional form of the correlators for the chains of finite length. We found the exact expression for the spin-spin equal-time correlator on finite lattice. We find the excellent agreement of the exact correlator with the prediction given by the leading asymptotics result up to the very small distances. We also establish the correspondence between the value of the constant before the asymptotics for the XX-chain with the expression for this constant proposed by Lukyanov and Zamolodchikov. We also evaluate the constant corresponding to the subleading term in the asymptotics. It is well known that the hypothesis of conformal invariance at large distances predicts both the critical exponents and the functional form of the correlation functions for massless one-dimensional systems on the circle of finite length L (for example see [1]). In this approach the conformal mapping w = L 2π ln(z) of the infinite plane z on the strip of finite width w = t + ix, x ∈ (0, L), is employed. It is known that in the case periodic boundary conditions (for the initial spin operators) the low-energy theory-is the Conformal Field Theory with cental charge c=1 (which can be computed for XXZ chain for an arbitrary boundary conditions), which is equivalent to the so called Gaussian model of CFT with c=1, with known spectrum of primary operators. The only parameter is the compactification radius (see below). I would like to stress also that application of CFT for calculation of e.g. critical indices is the hypothesis in a sense that it is not proved 1