Q-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

TL;DR

This paper investigates the q-dependent susceptibility of a Z-invariant Ising model on a Penrose tiling, revealing aperiodic, densely peaked behavior that intensifies near the critical temperature, using novel computational methods.

Contribution

It introduces a new approach to compute the susceptibility in a quasiperiodic Ising model, demonstrating the aperiodic nature and dense peak structure of chi(q).

Findings

chi(q) is aperiodic with dense peaks

Peaks become more visible as temperature approaches critical point

New linear programming method simplifies joint probability calculations

Abstract

We study the q-dependent susceptibility chi(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform chi(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that chi(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases--that is, as the temperature approaches the critical temperature.