Contents -- Introduction and Overview J. V. Jose -- References -- 1. Early Work on Defect Driven Phase Transitions J. M. Kosterlitz and D. J. Thouless -- 1.1. Introduction -- 1.2. One-Dimensional Ising Model -- 1.3. Vortex Driven Transitions in Superfluid Films -- 1.4. Other Systems with Defect-Mediated Transitions -- 1.4.1. Two-dimensional magnetic systems -- 1.4.2. Isotropic Heisenberg model -- 1.4.3. Two-dimensional Coulomb plasma -- 1.4.4. Two-dimensional crystals -- 1.4.5. Thin film superconductors -- 1.5. Scaling Theory -- 1.6. Scaling Theory in Analogous Systems -- 1.6.1. Duality and the roughening of crystal facets -- 1.6.2. Substrate effects -- 1.6.3. Melting of a 2D crystal -- 1.6.4. Substrate effects on 2D melting -- 1.6.5. Scaling in superconducting films -- 1.7. Experiments and Simulation -- 1.7.1. Measurements on superfluid films -- 1.7.2. Experimental measurements on 2D melting -- 1.7.3. Simulations of 2D melting -- References -- 2. Duality, Gauge Symmetries, Renormalization Groups and the BKT Transition J. V. Jose -- 2.1. Introduction -- 2.2. Duality Transformations in the 2D XY Model -- 2.3. Migdal Kadanoff RG Approximation of the Two-Dimensional XY Model -- 2.4. Correlation Function Calculations and Renormalization Group Equations -- 2.5. Symmetric Breaking Fields, Duality and RG Equations -- 2.6. An Early Experimental Confirmation of the BKT Theory -- 2.3. Conclusions -- Acknowledgments -- References -- 3. Berezinskii-Kosterlitz-Thouless Transition through the Eyes of Duality G. Ortiz, E. Cobanera and Z. Nussinov -- 3.1. Introduction -- 3.2. The XY Model: A Paradigm of BKT Phenomenology -- 3.2.1. A transfer operator for the XY model -- 3.2.2. Hamiltonian form of the XY model -- 3.2.3. Duality of the XY model without the Villain approximation -- 3.3. The p-Clock Model: A Close Relative of XY.

3.3.1. A transfer matrix for the p-clock model -- 3.3.2. Hamiltonian form of the p-clock model -- 3.3.3. Dualities of the p-clock model -- 3.3.4. Self-dual classical p-clock model -- 3.3.5. Exact and emergent symmetries of the p-clock model -- 3.3.5.1. Non-Abelian, discrete symmetries -- 3.3.5.2. Emergent U(1) symmetry -- 3.4. Phase Diagram: From the p-Clock to the XY Model -- Appendix A: Exponential of Shift Operators -- Appendix B: Duality of the XY Model to q-Deformed Bosons -- Appendix C: The Villain and Its Dual Solid-On-Solid Models -- Appendix D: The p-Clock Model for p = 2, 3, and 4 -- Appendix E: Peierls Argument for the p-Clock Model -- References -- 4. The Berezinskii-Kosterlitz-Thouless Transition in Superconductors A. M. Goldman -- 4.1. Introduction -- 4.2. Phenomenological Theory - Mostly Films -- 4.3. Experimental Evidence - Films -- 4.4. Phenomenological Theory - Arrays -- 4.5. Experiments - Arrays -- 4.6. Comments on Renormalization Effects -- 4.7. Summary -- Acknowledgments -- References -- 5. Berezinskii-Kosterlitz-Thouless Transition within the Sine-Gordon Approach: The Role of the Vortex-Core Energy L. Benfatto, C. Castellani and T. Giamarchi -- 5.1. Introduction -- 5.2. Mapping on the Sine-Gordon Model and the Vortex-Core Energy -- 5.3. The Universal Jump of the Superfluid Density -- 5.4. Inhomogeneity -- 5.5. The Case of a Finite Magnetic Field -- 5.6. Conclusions -- Acknowledgments -- References -- 6. The Two-Dimensional Fully Frustrated XY Model S. Teitel -- 6.1. Introduction -- 6.2. The Uniformly Frustrated XY Model -- 6.3. The Fully Frustrated XY Model -- 6.3.1. Phase angle ordering -- 6.3.2. Charge lattice ordering -- 6.3.3. Summary -- 6.4. Mapping to the Coulomb Gas -- 6.4.1. Duality transformation -- 6.4.2. Mechanisms for destruction of order -- 6.5. Numerical Results -- 6.6. Kink Antikink Unbinding Transition.

6.7. FFXY on a Honeycomb Lattice -- 6.8. Conclusions -- Acknowledgments -- References -- 7. Charges and Vortices in Josephson Junction Arrays R. Fazio and G. Schon -- 7.1. Introduction -- 7.2. Model of a Josephson Junction Array in the Quantum Regime -- 7.3. Charge-Vortex Duality -- 7.3.1. Self-duality and phase diagram -- 7.3.2. Magnetic and charge frustration -- 7.4. Vortex Dynamics -- 7.4.1. Ballistic vortex motion -- 7.4.2. Quantum vortices -- 7.5. Charge Dynamics -- 7.6. Conclusions -- Acknowledgment -- References -- 8. Superinsulator-Superconductor Duality in Two Dimensions and Berezinskii-Kosterlitz-Thouless Transition V. M. Vinokur and T. I. Baturina -- 8.1. Introduction -- 8.2. Superconductivity and Superinsulation from the Uncertainty Principle -- 8.3. Two-Dimensional Superconducting Systems -- 8.4. Superinsulation and Berezinskii Kosterlitz Thouless Transition -- 8.5. Two-Dimensional Universe in Superconducting Films -- 8.6. Microscopic Mechanism of Conductivity in the Insulating State -- 8.7. Conductivity of the Cooper-Pair Insulator and Superinsulator: Qualitative Consideration -- 8.8. Applications -- 8.9. Conclusions -- Summary -- Acknowledgments -- References -- 9. BKT Physics with Two-Dimensional Atomic Gases Z. Hadzibabic and J. Dalibard -- 9.1. Introduction -- 9.2. Experimental Realizations -- 9.2.1. Freezing-out one spatial degree of freedom -- 9.2.1.1. The dipole potential -- 9.2.1.2. A single atomic plane -- 9.2.1.3. Parallel atomic layers -- 9.2.2. Experimental procedure and parameters -- 9.2.3. Interactions -- 9.2.3.1. The interaction strength g -- 9.2.3.2. The healing length -- 9.2.3.3. Fano-Feshbach resonances -- 9.2.4. Direct implementation of the XY model -- 9.3. Theoretical Understanding of Interacting Atomic Gases -- 9.3.1. The equation of state of a homogeneous 2D Bose gas -- 9.3.2. The superfluid transition.

9.3.2.1. Vortex pairing -- 9.3.2.2. The critical point for the superfluid transition -- 9.3.2.3. Density fluctuations -- 9.3.3. The case of trapped gases -- 9.3.3.1. The local density approximation -- 9.3.3.2. The competition between BKT physics and Bose-Einstein condensation -- 9.3.3.3. The condensed fraction -- 9.4. Experimental Investigation of 2D Physics in Atomic Gases -- 9.4.1. Measurements of density and momentum distributions -- 9.4.1.1. In situ images -- 9.4.1.2. 2D TOF -- 9.4.1.3. 1D and 3D TOFs -- 9.4.2. Visualizing phase uctuations -- 9.4.2.1. Phonons -- 9.4.2.2. Vortices -- 9.4.3. Correlation functions -- 9.5. Outlook -- References -- 10. Vortex Physics in the Quantum Hall Bilayer H. A. Fertig and G. Murthy -- 10.1. Introduction -- 10.2. Exciton Condensation and Two-Dimensional Superfluidity -- 10.3. Puddles from Potential Fluctuations: A Periodic Model -- 10.4. Dissipative Transport in a Puddle-Network Model -- 10.5. Drag Experiments and Interlayer Bias -- 10.6. Conclusion -- Acknowledgments -- References -- Index.