Sketches of Physics : The Celebration Collection

Lecture Notes in Physics: The Formative Years
Lecture Notes in Physics: The Renaissance Years
Preface
Contents
Contributors
A New Era of Quantum Materials Mastery and Quantum Simulators In and Out of Equilibrium
	Contents
	1 Preamble
	2 Introduction
		2.1 The Rise of (Twisted) van der Waals Heterostructures
		2.2 Towards Nonequilibrium Quantum Materials Design
	3 Manipulating Materials with Light
		3.1 Shooting at the Crystal Lattice
		3.2 Shooting at Electrons
		3.3 Amplifying the Light
	4 Novel Materials and Avenues of Time-Resolved Control
		4.1 Fabricating New Quantum Materials: A Novel Twist
		4.2 Exotic Collective Phenomena and Their Control
	5 The Road Ahead: The Future of 2D Materials Is Bright
		5.1 Polaritonic Chemistry
		5.2 High Energy Physics (and Beyond) in a Condensed Matter Lab
		5.3 (Quantum) Floquet Materials Engineering
		5.4 Twistronics for Ultrafast Quantum Materials Design
	6 Closing
	References
Evaluation and Utility of Wilsonian Naturalness
	Contents
	1 Naturalness and Theory Viability
	2 Indexed Theories of a Theory Class
	3 Threshold Finetuning and Naturalness Tests
	4 The Standard Model
	5 Adding a Heavy Singlet
	6 Supersymmetry
	7 Grand Unified Theories
	8 Extra Dimensions and the Hierarchy Problem
	9 Twin Higgs Theories
	10 Conclusions
	References
The Geometric Phase: Consequences in Classical and QuantumPhysics
	Contents
	1 Introduction
	2 The Geometric Phase in Foucault\'s Pendulum
		2.1 The Mathematical Model
		2.2 The Geometric Phase
		2.3 The Foucault\'s Pendulum
	3 The Aharonov-Bohm (AB) and the Berry Phases
	4 Polarization of Light in an Optical Fiber or in a Birefringent Sheet
	5 The Geometric Phase in Interferometry: Three-Level System
	6 The Quantum Pump
	7 The Quantum Spin Pump
	8 The Phase Battery
	9 Conclusion
	References
The Coming Decades of Quantum Simulation
	Contents
	1 Introduction and Outline
	2 Quantum Computing
		2.1 Classical Computers and Classical Information Processing
		2.2 Quantum Information Processing
		2.3 Universal Quantum Computers
		2.4 Noisy Intermediate Scale Quantum Devices
	3 Quantum Simulators
		3.1 Idea and Main Concepts
		3.2 Platforms and Architectures
		3.3 The Coming Decades of Quantum Simulation
	4 Quantum Simulation of Fundamental Problems of Physics
		4.1 Relevant Paradigmatic Systems
			4.1.1 Fermi-Hubbard Model
			4.1.2 Systems with Fine-Tuned Interactions: Supersolids and Quantum Liquid Droplets
		4.2 Fundamental Systems of Condensed Matter, High Energy Physics and Quantum Field Theory
			4.2.1 Schwinger Model
			4.2.2 Bosonic Schwinger Model
			4.2.3 Abelian-Higgs Mechanism
	5 Novel Quantum Simulators for Novel Physics (NOQIA)
		5.1 Quantum Simulation Based on Atto-Science and Ultra-Fast Processes
			5.1.1 Optical Schrödinger Cat States
		5.2 Twistronics and Moiré Patterns
			5.2.1 Plethora of States in Magic-Angle Graphene
			5.2.2 Photonic Moiré Patterns
		5.3 Quantum Simulation for Strongly Correlated Phases of Matter
			5.3.1 Checkerboard Insulator with Semiconductor Dipolar Excitons
			5.3.2 Quantum Simulation of Interaction Induced Topological Phases
		5.4 Quantum Simulation with Rydberg Atoms
			5.4.1  Rydberg Atoms for Optimization Algorithms
			5.4.2 Quantum Spin Liquids
			5.4.3 Quantum Scars
			5.4.4 Quantum Phases of Matter on a 256-Atom Programmable Quantum Simulator
	6 Design, Techniques and Diagnostics of Quantum Simulators
		6.1 Single Site and Single Particle Resolution
		6.2 Entanglement and Topology Characterization Using Random Unitaries
		6.3 Experiment-Friendly Approaches for Entanglement Characterization
		6.4 Experiment-Friendly Approaches for Characterization of Topology
		6.5 Synthetic Dimensions
		6.6 Methods for Verification and Certification of Quantum Simulators
			6.6.1 Tensor Networks
			6.6.2 Exact Diagonalization
			6.6.3 Polynomial Relaxations Based on Semi-Definite Programming
			6.6.4 Machine Learning
	7 Conclusions
	References
Insights Into Complex Functions
	Contents
	1 Introduction
		1.1 Our Tools
		1.2 In a Nutshell
	2 Continuous Newton Method
		2.1 Newton Trajectories
		2.2 Lines of Constant Height and Constant Phase from Newton Flows
			2.2.1 Constant Phase
			2.2.2 Constant Height
		2.3 Sources and Sinks of Phase Lines
			2.3.1 Poles and Zeros
			2.3.2 No Poles and No Zeros
		2.4 Zero of the Derivative
			2.4.1 Parameter Connecting Two Points on a Phase Line
			2.4.2 Violation of Non-crossing Rule
			2.4.3 Separatrices and the Inverse of the Function
	3 Application of Cauchy-Riemann Differential Equations
		3.1 Elementary Example
		3.2 Changes in Amplitude Determine Slopes of Phase Lines
		3.3 Expression for the Slope of a Phase Line
		3.4 Parabolic Contour Lines
		3.5 Expression for the Slope of a Contour Line
	4 Conclusions and Outlook
	Appendix: Cauchy-Riemann Differential Equations
		Key Relations
		Representation in Real and Imaginary Parts
			Derivation
			Dependence of f on s Rather Than s and s*
		Representation in Amplitude and Phase
			Derivation
			Direct Derivation
		Reciprocity Relation for Slopes of Phase and Contour Lines
	References
Exploring the Hottest Atmosphere with the Parker Solar Probe
	Contents
	1 Introduction
	2 The Parker Solar Probe Spacecraft
	3 Overview of Solar Wind Turbulence and Coronal Heating Models
	4 Magnetohydrodynamic Turbulence Models Applied to the Coronal Heating Problem
	5 First Observations near the Alfvén Critical Surface
	6 Conclusions
	References
A Primer on the Riemann Hypothesis
	Contents
	1 Introduction
		1.1 Riemann Hypothesis
		1.2 Outline
	2 Riemann Zeta Function
		2.1 Dirichlet Series as Interfering Probability Amplitudes
			2.1.1 Riemann Zeta Function and Wave Packet Dynamics
			2.1.2 Inverse Problem
			2.1.3 Factorization
		2.2 Sum Representation
		2.3 Analytic Continuation
		2.4 A First Glimpse at the Zeros
			2.4.1 Trivial Zeros
			2.4.2 Non-trivial Zeros
	3 Riemann Function
		3.1 Product Representation
		3.2 Functional Equation
		3.3 Symmetry Relations
		3.4 Elementary Building Block
			3.4.1 General Properties
			3.4.2 Two Special Phase Lines and the Non-trivial Zeros
			3.4.3 Riemann Hypothesis in Terms of Lines of Constant Height and Phase
		3.5 The Riemann Hypothesis at the Edge of the Complex Plane
		3.6 Exponential Representation
			3.6.1 Euler Product
			3.6.2 Exact Expression
			3.6.3 Asymptotic Expression
			3.6.4 Amplitude and Phase at the Right Edge of the Complex Plane
		3.7 Representation in Terms of a Switching Function
			3.7.1 Motivation
			3.7.2 Formula
			3.7.3 Connection to Sum Representation of ξ
			3.7.4 Asymptotic Expansion
			3.7.5 Connection to the Riemann Hypothesis
	4 Dirichlet Characters
	5 Gauss Sums
		5.1 Reduction Formula
		5.2 Normalized Gauss Sums
		5.3 Symmetry Relation for Phase
	6 Dirichlet L-functions
		6.1 Analytic Continuation
		6.2 Functional Equation
		6.3 Symmetry Relations
		6.4 Special Examples
			6.4.1 Building Blocks of Titchmarsh Counterexample
			6.4.2 Dirichlet L-function with Vanishing Phase
		6.5 Exponential Representation
			6.5.1 Euler Product
			6.5.2 Final Expression
	7 Titchmarsh Counterexample
		7.1 Definition and Functional Equation
		7.2 Exponential Representation
		7.3 Summary
	8 Conclusions and Outlook
	Appendix 1: Functional Equation of Jacobi Theta Function
	Appendix 2: Equivalent Condition for Non-trivial Zeros
	Appendix 3: Exponential Product
		General Expression
		Asymptotic Limit
		Special Case: Riemann Function
	Appendix 4: Lines of Constant Height and Constant Phase
		Decomposition in Amplitude and Phase
		Special Limits
			At the Center of the Complex Plane
			At the Right Edge of the Complex Plane
		Cauchy-Riemann Differential Equations
	Appendix 5: Special Examples of Normalized Gauss Sums
		Non-vanishing Phase
		Vanishing Phase
	Appendix 6: Functional Equation of Generalized Jacobi Theta Function
		Extension of Summation
		Emergence of Gauss Sum
		Comparison to Jacobi Theta Function of ξ
	References