Study on boundary–boundary correlation functions in the transverse Ising chain and XY chain

The topological interaction method proposed by Suzuki is applied to the transverse Ising chain and XY chain in order to study the behaviour of the boundary–boundary correlations. The correlation function 〈σ1xσNx〉 for the thermodynamic limit is derived by this method as a function of the topological interaction parameter J2 for finite temperatures. This analysis of the correlation function clarifies the emergence of the long-range order as temperature goes to zero. At zero temperature, a more general spin chain can be analysed analytically for N→∞ and numerically for finite N to obtain the following results. The correlation function 〈σ1xσNx〉 shows various kinds of behaviour, namely, (a) monotonic, (b) non-monotonic or (c) globally non-monotonic with small oscillation, with respect to the system-size N, in different regions of the parameters. The function 〈σ1yσNy〉 shows also the above behaviour (a), or shows (d) globally monotonic decay with small oscillation.