Wavevector-Dependent Susceptibility in Quasiperiodic Ising Models

Abstract

Using the various functional relations for correlation functions in planar Ising models, new results are obtained for the correlation functions and the q-dependent susceptibility for Ising models on a quadratic lattice with quasiperiodic coupling constants. The effects are clearest if the interactions are both attractive and repulsive according to a quasiperiodic pattern. In particular, an exact scaling limit result for the two-point correlation function of the Z-invariant inhomogeneous Ising model is presented and the q-dependent susceptibility is calculated for some cases where the coupling constants vary according to Fibonacci rules. It is found that the ferromagnetic case differs drastically from the case with both ferro- and antiferromagnetic bonds. In the mixed case, the peaks of the q-dependent susceptibility are everywhere dense for temperature T both above or below the critical temperature Tc, but due to overlap only a finite number of peaks is visible. This number of visible peaks decreases as T moves away from Tc. In the ferromagnetic case, there is typically only one single peak at q=0, in spite of the aperiodicity present in the lattice. These results provide evidence that in real systems, even if the atoms arrange themselves aperiodically, there will be no dramatic difference in the diffraction pattern, unless the pair correlation function has clear aperiodic oscillations. The number of oscillations per correlation length determines the number of visible peaks.