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.

Graphical abstract

中文翻译：

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非局部相互作用的斐波那契准晶体哈密顿量模型：模拟和光谱分析

摘要

本文介绍了一种基于顶点类型和帝国的哈密顿体系，用于通过打破平移对称性的适当处方来证明一维非周期性顺序的出现。首先，本文介绍了构造给定晶格的顶点构型帝国的不同算法，几何和代数方法。这些帝国具有非本地范围，并构成了提出的晶格模型的构建块。通过从随机排列的N个图块开始的Monte Carlo模拟测试该模型。模拟清楚地建立了斐波那契构型，该构型是长度为N的一维准晶体，作为系统的最终松弛状态。通过执行相互作用的帝国向量对的二元张量积，然后对所有允许的配置求和，将哈密顿量提升为矩阵算子形式。进行了哈密顿矩阵的光谱分析，并提出了一种理论方法来找到由斐波那契链给出的吸引子构型的精确解，如模拟所预测的那样。最后，提供了精确的理论解释，表明斐波那契链是最可能的基态。提出的哈密顿量是一维菲波纳奇准晶体的数学模型。

图形概要