Ph.D Student | Wolff Gil |
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Subject | The Statistical Mechanics of Deterministic Disorder |

Department | Department of Physics |

Supervisor | Professor Dov Levine |

Full Thesis text |

How should one characterize order in solids? While Bragg peaks in a solid’s diffraction pattern indicate order, it is possible to generate ordered structures which not only lack Bragg peaks in their diffraction patterns, but also have a completely diffuse diffraction pattern. I will construct several such structures in this thesis. While the diffraction pattern (which contains the same information as the two-point correlations) does not indicate that these structures are ordered, a recently suggested quantity called the patch entropy does succeed in indicating that these structures are ordered.

I will construct a two-dimensional spin system with a flat diffraction pattern, that is generated by substitution rules. As I will show, it is also describable in terms of a tiling Hamiltonian that displays a sequence of phase transitions indicating ordering at increasing length scales. While the two-point correlation function is a delta function, the four point correlations are generically non-zero. The patch entropy in this case is subextensive, due to the system being described by substitution rules.

In atomic systems, it is still possible to construct examples with diffuse diffraction,

but it is not known whether they are describable in terms of substitutions, or a finite range Hamiltonian. The statistics of the diffraction pattern in this case are similar

to the statistics of the Hendricks-Teller and pentagonal glass models. A numerical

investigation of the patch entropy shows that it is subextensive.

Amorphous systems have lately been used to create materials with an isotropic photonic band gap. These are materials, composed of a dielectric network in vacuum, where there are no electromagnetic modes in a certain frequency range. The features of the dielectric network which are responsible for the creation of the photonic band gap are not fully understood. It has lately been suggested that a certain long-range property called stealthy hyperuniformity is important. A stealthy hyperuniform material is one in which there exists a wave-vector such that the diffraction pattern is completely dark inside the ring defined by this wave-vector.

I shall present an algorithm that produces materials with a large photonic band gap in two dimensions. I find that it is possible to find a photonic band gap both in materials that are hyperuniform and those that are not. Additionally, I make an analytical calculation of the frequencies of the band edge modes, and find that they depend on short range properties. This is the first analytical approach to understanding photonic band gaps in amorphous materials.