Spectral Approach to Transport Problems in Two-Dimensional Disordered Lattices: Physical Interpretation and Applications

ISBN-13: 9783030022112 / Angielski / Twarda / 2018 / 107 str.

Evdokiya Georgieva Kostadinova

Opis

This book introduces the spectral approach to transport problems in infinite disordered systems characterized by Anderson-type Hamiltonians. The spectral approach determines (with probability one) the existence of extended states for nonzero disorder in infinite lattices of any dimension and geometry. Here, the author focuses on the critical 2D case, where previous numerical and experimental results have shown disagreement with theory. Not being based on scaling theory, the proposed method avoids issues related to boundary conditions and provides an alternative approach to transport problems where interaction with various types of disorder is considered. Beginning with a general overview of Anderson-type transport problems and their relevance to physical systems, it goes on to discuss in more detail the most relevant theoretical, numerical, and experimental developments in this field of research. The mathematical formulation of the innovative spectral approach is introduced together with a physical interpretation and discussion of its applicability to physical systems, followed by a numerical study of delocalization in the 2D disordered honeycomb, triangular, and square lattices. Transport in the 2D honeycomb lattice with substitutional disorder is investigated employing a spectral analysis of the quantum percolation problem. Next, the applicability of the method is extended to the classical regime, with an examination of diffusion of lattice waves in 2D disordered complex plasma crystals, along with discussion of proposed future developments in the study of complex transport problems using spectral theory.

Spis treści

CHAPTER ONE ................................................................................................................. 1

Introduction ..................................................................................................................... 1

1.1. Formulation of the Transport Problem ................................................................ 4

1.2. Nature of Disorder ............................................................................................... 7

1.3. Relevance to Physical Systems ........................................................................... 10

CHAPTER TWO .............................................................................................................. 15

Theoretical Background ................................................................................................ 15

2.1. Localization Criteria .......................................................................................... 15

2.2. Anderson Model ................................................................................................. 20

2.3. Edwards and Thouless Model ............................................................................ 25

2.4. Scaling Theory ................................................................................................... 29

CHAPTER THREE .......................................................................................................... 32

Spectral Approach ......................................................................................................... 32

3.1. Essence of the Spectral Method ......................................................................... 32

3.2. Simplified Numerical Model (“Toy Model”) ..................................................... 40

3.3. Physical Interpretation ...................................................................................... 45

3.4. Scope and Limitations of the Spectral Analysis ................................................. 49

CHAPTER FOUR ............................................................................................................. 51

Delocalization in 2D Lattices of Various Geometries .................................................. 51

4.1. Transport in the Honeycomb, Triangular, and Square Lattices ........................ 51

4.2. Orthogonality Check .......................................................................................... 56

4.3. Equation Fitting ................................................................................................. 57

4.4. Cluster Analysis ................................................................................................. 60

4.5. Comparison Between the Honeycomb and the Triangular Lattices .................. 63

CHAPTER FIVE .............................................................................................................. 65

Transport in the Two-Dimensional Honeycomb Lattice with Substitutional Disorder 65

5.1. Discrete Percolation .......................................................................................... 66

5.2. Formulation of the Transport Problem .............................................................. 69

5.3. Distribution of Variables ................................................................................... 78

5.4. 2D Honeycomb Lattice with Substitutional Disorder ........................................ 83

CHAPTER SIX ................................................................................................................. 89

Transport in 2D Complex Plasma Crystals ................................................................... 89

6.1. Complex Plasma Preliminaries ......................................................................... 90

6.2. Two-Dimensional Dust Crystal Analogue ......................................................... 94

6.3. Transport in the Classical Regime ..................................................................... 95

6.4. Numerical Simulations of Dust Particle Dynamics ........................................... 98

6.5. Spectral Analysis .............................................................................................. 105

CHAPTER SEVEN ........................................................................................................ 107

Conclusions ................................................................................................................. 107

APPENDIX A ................................................................................................................. 113

Basic Materials Science Terms ................................................................................... 113

A.1. Comparison Between Fermi Energy and Fermi Level .................................... 114

APPENDIX B ................................................................................................................. 118

Mathematical Preliminaries ........................................................................................ 118

B.1. Kindergarten Math .......................................................................................... 118

B.2. Measure Theory ............................................................................................... 119

B.3. Point-set Topology ........................................................................................... 121

B.4. Group Theory .................................................................................................. 124

B.5. Probability Theory ........................................................................................... 127

BIBLIOGRAPHY ........................................................................................................... 129

O autorze

Evdokiya Georgieva Kostadinova is a research assistant professor in the Center for Astrophysics, Space Physics & Engineering Research at Baylor University. She received her PhD from Baylor University in 2017. 

Blurb

This thesis introduces the spectral approach to transport problems in infinite disordered systems characterized by Anderson-type Hamiltonians. The spectral approach determines (with probability one) the existence of extended states for nonzero disorder in infinite lattices of any dimension and geometry. Here, the author focuses on the critical 2D case, where previous numerical and experimental results have shown disagreement with theory. Not being based on scaling theory, the proposed method avoids issues related to boundary conditions and provides an alternative approach to transport problems where interaction with various types of disorder is considered.

Beginning with a general overview of Anderson-type transport problems and their relevance to physical systems, it goes on to discuss in more detail the most relevant theoretical, numerical, and experimental developments in this field of research. The mathematical formulation of the innovative spectral approach is introduced together with a physical interpretation and discussion of its applicability to physical systems, followed by a numerical study of delocalization in the 2D disordered honeycomb, triangular, and square lattices. Transport in the 2D honeycomb lattice with substitutional disorder is investigated employing a spectral analysis of the quantum percolation problem. Next, the applicability of the method is extended to the classical regime, with an examination of diffusion of lattice waves in 2D disordered complex plasma crystals, along with discussion of proposed future developments in the study of complex transport problems using spectral theory.