# Aperiodicity in Low Dimensions

## TL;DR

This paper reviews the structural and physical properties of low-dimensional aperiodic crystals, focusing on their origins and effects in nanoscale materials.

## Contribution

The paper introduces a comparative analysis of mechanisms behind aperiodicity in low-dimensional systems and their unique electronic and magnetic properties.

## Key findings

- Low-dimensional aperiodic solids form due to suppressed force constants, thermodynamic instabilities, and topological constraints.
- Decahedral and icosahedral multiply twinned particles exhibit distinct electronic and spin properties based on their symmetry.
- Finite aperiodic solids experience structural strain due to solid-angle mismatch at twin junctions.

## Abstract

This review provides a comparative analysis of the structure and physical properties of low-dimensional aperiodic crystalline solids, aiming to elucidate the origin and nature of aperiodicity in reduced-dimensional lattices. The breakdown of periodicity in low-dimensional systems arises from several mechanisms, including the suppression of specific force constants, thermodynamic instabilities, and topological constraints associated with imperfect space filling. At the nanoscale, certain cubic crystalline materials can form finite, zero-dimensional multiply twinned particles (MTPs) with decahedral or icosahedral symmetry. These clusters lack translational invariance and experience intrinsic structural strain due to solid-angle mismatch at twin junctions, which limits their characteristic size and renders them finite aperiodic solids. Particular attention is devoted to the electronic and spin properties of pentagonally symmetric MTPs: icosahedral particles exhibit symmetry-protected spin degeneracy consistent with centrosymmetric lattices, whereas non-centrosymmetric decahedral particles may display spin polarization and emergent low-dimensional magnetism. Collectively, these systems illustrate the diverse physical origins, manifestations, and consequences of aperiodicity in low-dimensional crystalline matter.