Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- Part II Solitons and Topology; Non-Abelian Theory
- 20 Kink Solutions in ϕ4 and Sine-Gordon, Domain Walls and Topology
- 21 The Skyrmion Scalar Field Solution and Topology
- 22 Field Theory Solitons for Condensed Matter: The XY and Rotor Model, Spins, Superconductivity, and the KT Transition
- 23 Radiation of a Classical Scalar Field: The Heisenberg Model
- 24 Derrick’s Theorem, Bogomolnyi Bound, theAbelian-Higgs System, andSymmetryBreaking
- 25 The Nielsen-Olesen Vortex, Topology and Applications
- 26 Non-Abelian Gauge Theory and the Yang–Mills Equation
- 27 The Dirac Monopole and Dirac Quantization
- 28 The ’t Hooft–Polyakov Monopole Solution and Topology
- 29 The BPST-’t Hooft Instanton Solution and Topology
- 30 General Topology and Reduction on an Ansatz
- 31 Other Soliton Types. Nontopological Solitons: Q-Balls; Unstable Solitons: Sphalerons
- 32 Moduli Space; Soliton Scattering in Moduli Space Approximation; Collective Coordinates
- Part III Other Spins or Statistics; General Relativity
- References
- Index
22 - Field Theory Solitons for Condensed Matter: The XY and Rotor Model, Spins, Superconductivity, and the KT Transition
from Part II - Solitons and Topology; Non-Abelian Theory
Published online by Cambridge University Press: 04 March 2019
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- Part II Solitons and Topology; Non-Abelian Theory
- 20 Kink Solutions in ϕ4 and Sine-Gordon, Domain Walls and Topology
- 21 The Skyrmion Scalar Field Solution and Topology
- 22 Field Theory Solitons for Condensed Matter: The XY and Rotor Model, Spins, Superconductivity, and the KT Transition
- 23 Radiation of a Classical Scalar Field: The Heisenberg Model
- 24 Derrick’s Theorem, Bogomolnyi Bound, theAbelian-Higgs System, andSymmetryBreaking
- 25 The Nielsen-Olesen Vortex, Topology and Applications
- 26 Non-Abelian Gauge Theory and the Yang–Mills Equation
- 27 The Dirac Monopole and Dirac Quantization
- 28 The ’t Hooft–Polyakov Monopole Solution and Topology
- 29 The BPST-’t Hooft Instanton Solution and Topology
- 30 General Topology and Reduction on an Ansatz
- 31 Other Soliton Types. Nontopological Solitons: Q-Balls; Unstable Solitons: Sphalerons
- 32 Moduli Space; Soliton Scattering in Moduli Space Approximation; Collective Coordinates
- Part III Other Spins or Statistics; General Relativity
- References
- Index
Summary
We consider field theory solitons relevant for condensed matter. We start with a field theory arising from a two-dimensional system of spins, the XY model, leading to the “rotor model,” or “O(2) model”. From the bosonic Hubbard model, we show a representation that leads to the same quantum rotor model. In the continuum limit, we obtain a massless scalar that has a global vortex as its solution. The dynamics of these vortices is relevant for the Kosterlitz–Thouless (KT) phase transition, a quantum phase transition appearing for instance in 2+1 dimensional superconductivity. The bosonic Hubbard model leads, in the continuum limit, also to a relativistic Landau–Ginzburg model, that has a kink-like solution.
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- Classical Field Theory , pp. 203 - 208Publisher: Cambridge University PressPrint publication year: 2019