Tapio Simula was awarded a D.Sc.(Tech.) degree 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.

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.

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 Phonon 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 Helicity Enstrophy Kauffman Bracket Polynomial Jones Polynomial
5 Topological Excitations Topological Defects Soliton Bright Soliton Grey and Dark Soliton Solitonic Vortex Plain Vortex Polynomial Vortex Coherence Vortex Fractional Vortex Baby Skyrmion Monopole Fluxon, Chargeon, and Dyon Alice Vortex and Cheshire Charge
6 Structure of a Plain Vortex Uncertainty Principle for Vortices Kelvon 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 Stirring 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 Vacuum Speed of Light Photon Particles and Antiparticles Positronium Pair Creation and Annihilation Photon Emission and Absorption Charge Spin Dipole Moment Electrodynamics Non-Abelian Fractional Charge Particles Quantum Chromodynamics Gravitation and Black Holes Cosmic Inflation