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Quantised Vortices

A handbook of topological excitations
Tapio Simula


Vortices comprising swirling motion of matter are observable in classical systems at all scales, ranging from atomic size to that of galaxies. In quantum mechanical systems, such vortices are robust entities whose behaviours are governed by the strict rules of topology. The physics of quantum vortices is pivotal to the basic science of quantum turbulence and high-temperature superconductors, and underpins emerging quantum technologies including topological quantum computation. This handbook is aimed at providing a dictionary-style portal to the fascinating quantum world of vortices.

About Editors

Tapio Simula was awarded a DSc in 2003 by the Helsinki University of Technology. His research interests include the physics of quantum vortices and superfluidity in Bose–Einstein condensates. He is currently an Australian Research Council (ARC) Future Fellow at Swinburne University of Technology, Melbourne, Australia.

Table of Contents

I Vortices in Flatland

1 Vortices
Space-Time Symmetries
Quantum Liquids
Vorticity in Classical Fluids
Vorticity in Quantum Liquids

2 Quasiparticle Picture
Emergence of Quasiparticles
Boson Commutation Relations
Fermion Commutation Relations
Majorana Relations
Anyon Quasiparticles
Non-Abelian Anyon Quasiparticles
Bogoliubov{de Gennes Equations
Time-Reversal Symmetry
Particle-Hole Symmetry
Chiral Symmetry
Landau Critical Velocity
Roton-Maxon Spectrum
Edge Modes
Bulk Modes: Dipole, Breathing, Quadrupole and Scissors Modes
Kelvin Mode Vortex Waves
Tkachenko Mode Vortex Waves
Caroli{deGennes{Matricon Modes
Goldstone Zero Mode
Majorana Zero Mode
Magnon Spin Waves

3 Cold Atoms
Scalar Bose{Einstein Condensates
Bose Zero Temperature Energy Functional
Thomas{Fermi Relations
Healing Length
Thermodynamic Relations
Quantum Hydrodynamic Equations
Two-Component Bose{Einstein Condensates
Spin-1 Bose{Einstein Condensates
Spin-2 Bose{Einstein Condensates
High Spin Bose{Einstein Condensates
Representations of Spinor Bose{Einstein Condensates
Exotic Interactions
Bardeen{Cooper{Schrieffer Meaneld Theory
Ultracold Fermi Gases
Dirac{Bogoliubov{deGennes Systems
Gapless, Massless, Linear Spectra
Gapped, Massive, Quadratic Spectra

4 Topological Invariants and Quantities
Topology and Ordered Structures
A Game of Lines and Loops
Maps and Order Parameters
Homotopy Classication of Defects
Burgers Vector
Gauss-Bonnet Theorem
Winding Number
Berry Phase, Curvature, and Connection
Chern Number
Linking Number, Writhe and Twist
Kauffman Bracket Polynomial
Jones Polynomial

5 Topological Excitations
Topological Defects
Bright Soliton
Grey and Dark Soliton
Solitonic Vortex
Plain Vortex
Polynomial Vortex
Coherence Vortex
Fractional Vortex
Baby Skyrmion
Fluxon, Chargeon, and Dyon
Alice Vortex and Cheshire Charge

6 Structure of a Plain Vortex
Uncertainty Principle for Vortices
Circulation Quantum
Vortex Energy
Thermodynamic Stability
Spectral, Energetic Stability
Dynamical Lyapunov Stability
Inertial Vortex Mass
Gravitational Vortex Mass
Kelvon Based Vortex Mass
Hydrodynamic Induced Vortex Mass Component
Relativistic Vortex Mass Component
Baym{Chandler Vortex Mass
Kopnin Vortex Mass

7 Vortex Dynamics
Adiabatic Vortex Dynamics
Vortex Force and Velocity
Magnus Effect and Mutual Induction
Vortex Pair Creation and Annihilation
Onsager's Point-Vortex Model
Vortex-Particle Duality
Point Vortex Model with Cylindrical Boundary
Point Vortex Models with Square Boundaries
Point Vortex Models in General Domains
Vortex Classication Algorithm
Vortex Temperature
Winding Number Scaling Laws

8 Vortex Production in Bose{Einstein Condensates
Coherent Coupling of Internal States
Laguerre{Gauss Laser Modes
Topological Angular Momentum Conversion
Rotating Bucket
Rotating Thermal Cloud
Shaking Bucket
Snaking Instability
Many-Wave Interference
Vortex-Antivortex Honeycomb Lattices
Caustics and Diffraction Catastrophes
Vortex Quasicrystals
Vortex Phasons
Vortex Moire Superlattices
Synthetic Gauge Fields
Optical Flux Lattices
Filtered Speckle Fields
Kibble{Zurek Mechanism and Quenches
Berezinskii{Kosterlitz{Thouless Mechanism

9 Topological Quantum Computation
Non-Abelian Anyons
Topological Qubits
Quantum Dimension
Majorana Ising Anyon Model
Fibonacci Anyon Model
Model k Anyons
Non-Abelian Vortex Anyons
Annihilation, Pass-Through and Rungihilation
Non-Abelian Vortex Anyon Models
Vortex Anyon Creation, Pinning, Braiding, and Fusion
From Quantum Circuits to Anyon Braiding
Evaluation of Space-Time Knot Invariants

10 Two-Dimensional Quantum Turbulence
Regular and Chaotic Few-Vortex Dynamics
Inverse Energy and Direct Enstrophy Cascades
Vortex Near Field Spectrum
Vortex Far Field Spectrum
Vortex Dipole Spectrum
Kolmogorov-Obukhov Spectrum
Onsager Vortex Spectrum
Spin Turbulence Spectrum
Helmholtz Decomposition
Enstrophy Conservation and Non-Conservation
Evaporative Heating of Vortices
Point Vortex Model of Turbulence
Non-Abelian Two-Dimensional Quantum Turbulence
Super Fluid Reynolds Number
Eddy Turnover Time
Anomalous Hydrodynamics of Vortices
Negative Absolute Temperature
Negative Absolute Vortex Temperature
Non-Thermal Fixed Point
Dynamical Phase Transitions
Condensation of Onsager Vortices

11 Vortex States of Matter in Flatland
BCS Superconductivity
Meissner Effect
Type-II Superconductors
Abrikosov Vortex Lattice
Vortex Pinning and Creep Motion
Vortex Matter in Rotating Super Fluids
Vortex Nucleation and Hess{Fairbank Effect
Vortex Lattices in Neutral Super Fluids
Feynman Rule
Vortex Lattice Melting
Two-Dimensional Vortex Coulomb Gas
2D Coulomb Gas: Quantum Hall Effects
2D Coulomb Gas: Hauge{Hemmer Transition
2D Coulomb Gas: Berezinskii{Kosterlitz{Thouless Transition
2D Coulomb Gas: Supercondensation Transition
2D Coulomb Gas: Einstein{Bose Condensation Transition

12 Super Fluid Universe
Speed of Light
Particles and Antiparticles
Pair Creation and Annihilation
Photon Emission and Absorption
Dipole Moment
Non-Abelian Fractional Charge Particles
Quantum Chromodynamics
Gravitation and Black Holes
Cosmic Inflation


Paperback ISBN: 9780750330008

Ebook ISBN: 9781643271255

DOI: 10.1088/2053-2571/aafb9d

Publisher: Morgan & Claypool Publishers


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