Scaled Tight-Binding Crystal

TL;DR

This paper explores the spectral and transmission properties of a scaled tight-binding chain derived from local symmetry dynamics, revealing complex spectral features and localization behaviors through analytical and numerical methods.

Contribution

It introduces a scaled chain model based on local symmetry dynamics and analyzes its spectral, localization, and transmission properties using a resonator model and numerical tools.

Findings

Eigenvalue spectrum exhibits multiple branches with characteristic transitions.

Hierarchy of minigaps and energy spacings observed in energy level fluctuations.

Localization properties vary with coupling strength, affecting transmission profiles.

Abstract

The concept of local symmetry dynamics has recently been used to demonstrate the evolution of discrete symmetries in one-dimensional chains leading to emergent periodicity. Here we go one step further and show that the unboundedness of this dynamics can lead to chains that consist of subunits of ever increasing lengths which results in a scaled chain. Mapping this scaled chain onto a corresponding tight-binding Hamiltonian we investigate its spectral and transmission properties. Varying the off-diagonal coupling the eigenvalue spectrum shows different branches with characteristic transitions and peaks in the corresponding density of states. The fluctuations of the energy levels exhibit a hierarchy of minigaps each one accompanied by a characteristic sequence of energy spacings. We develop a local resonator model to describe the spectral properties and gain a deeper understanding of it in the weak to intermediate coupling regime. Eigenstate maps together with the inverse participation ratio are used to unravel the characteristic (de-)localization properties of the scaled chain with varying coupling strength. Finally we probe the energy-dependent transmission profile of the scaled chain.