# Narrative draft

* over-arching element: disorder, delocalization

## title suggestion:
* delocalization in disordered media

## Introduction

### overview
- In condensed matter applications, we send signals, these are delocalized.
- Disorder localizes things.
- We find phases of matter that delocalize due to disorder.

- Gaussian disorder is a special kind of disorder because it leads to the preservation of average spatial symmetries, like mirror and rotation. Average symmetries and therefore Gaussian disorder protect electronic phases of matter that are unique to disordered systems: statistical topological insulators.
- Topological phases are special phases of condensed matter because they are robust in the presence of disorder, and their observables (boundary modes) persist largely unchanged for any disorder that doesn't close the bulk gap.
- Topological phases have counterparts in non-Hermitian systems. Non-Hermitian systems also have phases that have no Hermitian counterpart. 1D non-Hermitian systems are considered to generically host a topological phase known as the non-Hermitian skin effect (NHSE). It is characterized by the existence of a bulk invariant that predict the location of boundary modes. In higher dimensions, the NHSE persists but a general topological formalism describing it does not currently exist.

### disorder
* amorphous systems
### topological phases
* topological invariants
### statistical topological insulators
* average symmetries
### non-hermitian systems
* NHSE + winding number

### this thesis

* In the first two chapters of the thesis, we study disordered Hermitian systems and find topological phases protected by structural disorder.
* The surface modes of crystalline topological insulators gap out on surfaces that break the relevant spatial symmetries. In amorphous systems, average spatial symmetries are preserved on every surface. In chapter 1, we study 2D amorphous systems and find statistical topological insulator phases that host delocalized gapless edge modes on any surface orientation.
* In 3D systems, inversion, a spatial symmetry, is commonly accompanied by time-reversal symmetry. However, it is not mandatory for these symmetries to coexist. In Chapter 2, we propose a system that possesses inversion symmetry without time-reversal symmetry. This system exhibits a topological phase that also persists in amorphous systems. While the single-Dirac cone model remains delocalized, the double-Dirac cone model localizes.
* In the last two chapters, we study non-Hermitian systems and their responses to disorder, which differ significantly from the response in Hermitian systems.
* In 1D non-Hermitian systems, the modes of the system exhibit localization at the boundaries, which is known as the non-Hermitian skin effect (NHSE). The NHSE is associated with the properties of the bulk and is regarded as a topological phase. In Chapter 3, we demonstrate that due to the violation of the near-sightedness principle by non-Hermitian systems, a single impurity in the bulk has the ability to delocalize and attract all the modes of the system, thereby breaching the concept of bulk-edge correspondence.
* In the presence of numerous non-Hermitian impurities forming a disorder landscape, the system undergoes delocalization in reciprocal space due to the absence of energy conservation during scattering processes. In Chapter 4, we further investigate non-Hermitian systems beyond a topological framework and explore non-Hermitian dynamics in disordered systems. Scattering in the disorder landscape causes the initial wave packet to evolve universally towards the maximally amplified eigenstates of the system over time. As a result of this phenomenon, we observe dynamic phase transitions that differ from those observed in static studies of non-Hermitian systems.

## Chapter II : 2D amorphous statistical topological insulators

## Chapter III : 3D amorphous statistical topological insulators

## Chapter IV : lack of the near-sightedness principle in non-Hermitian systems

## Chapter V : dynamics of disordered non-Hermitian systems