Regular Article

A Hamiltonian model of the Fibonacci quasicrystal using non-local interactions: simulations and spectral analysis^★,★★

¹ School of Mathematics, Thapar Institute of Engineering & Technology, Patiala, Punjab, India
² Centre for Theoretical and Physical Sciences, Clark Atlanta University, Atlanta, Georgia, USA
³ Ronin Institute, 127 Haddon Pl., Montclair, NJ 07043, USA

^a e-mail: amriksen@thapar.edu

Received: 9 November 2019
Received in final form: 4 February 2020
Published online: 8 April 2020

Abstract

This article presents a Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order in one dimension by a suitable prescription for breaking translation symmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods of constructing empires of vertex configurations of a given lattice. These empires have non-local scope and form the building blocks of the proposed lattice model. This model is tested via Monte Carlo simulations beginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration, which is a one-dimensional quasicrystal of length N, as the final relaxed state of the system. The Hamiltonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interacting empire vectors followed by a summation over all permissible configurations. A spectral analysis of the Hamiltonian matrix is performed and a theoretical method is presented to find the exact solution of the attractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a precise theoretical explanation is provided which shows that the Fibonacci chain is the most probable ground state. The proposed Hamiltonian is a mathematical model of the one dimensional Fibonacci quasicrystal.