This article presents a novel Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order (quasicrystal growth) 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 quasi- lattice. These empires have non-local scope and form the building blocks of the new 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 one dimensional model of quasicrystal growth.
Comments: 16 Pages. Submitted to EPJB
[v1] 2019-11-17 14:40:21
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