Multifaceted Skyrmion, The (Second Edition)

Mannque Rho, Ismail Zahed

• Publisher: World Scientific Publishing Company
• ISBN:
  9789814704427
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

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This book presents, in the form of reviews by world's leading physicists in wide-ranging fields in theoretical physics, the influence and prescience of Skyrme's daring idea of 1960, originally conceived for nuclear physics, that fermions can arise from bosons via topological solitons, pervasively playing a powerful role in wide-ranging areas of physics, from nuclear/astrophysics, to particle physics, to string theory and to condensed matter physics.

The skyrmion description, both from gauge theory and from gauge/gravity duality, offers solutions to some long-standing and extremely difficult problems at high baryonic density, inaccessible by QCD proper. It also offers explanations and makes startling predictions for fascinating new phenomena in condensed matter systems. In both cases, what is at the core is the topology although the phenomena are drastically different, even involving different spacetime dimensions.

This second edition has been expanded with addition of new reviews and extensively updated to take into account the latest developments in the field.

--> Contents:

• Hadrons and Nuclear Matter:
  • Skyrmions and Nuclei (R A Battye, N S Manton and P M Sutcliffe)
  • States of Carbon-12 in the Skyrme Model (P H C Lau and N S Manton)
  • Electromagnetic Form Factors of the Nucleon in Chiral Soliton Models (G Holzwarth)
  • Exotic Baryon Resonances in the Skyrme Model (D Diakonov and V Petrov)
  • Heavy-Quark Skyrmions (N N Scoccola)
  • Pentaquark Candidates P + c (4380) and P + c (4450) within the Soliton Picture of Baryons (N N Scoccola, D O Riska and M Rho)
  • Skyrmion Approach to Finite Density and Temperature (B-Y Park and V Vento)
  • Fractionized Skyrmions in Dense Compact-Star Matter (M Harada, Y-L Ma, H K Lee and M Rho)
  • The Skyrme Model in the BPS Limit (C Adam, C Naya, J Sánchez-Guillén, R Vazquez and A Wereszczyński)
  • Superqualitons: Baryons in Dense QCD (D K Hong)
• Condensed Matter:
  • Rotational Symmetry Breaking in Baby Skyrme Models (M Karliner and I Hen)
  • Emergent Gauge Fields and Their Nonperturbative Effects in Correlated Electrons (K-S Kim and A Tanaka)
  • Spin and Isospin: Exotic Order in Quantum Hall Ferromanets (S M Girvin)
  • Noncommutative Skyrmions in Quantum Hall Systems (Z F Ezawa and G Tsitsishvili)
  • Meron-Pair Excitations in Bilayer Quantum Hall System (K Moon)
  • Spin and Pseudospin Textures in Quantum Hall Systems (H A Fertig and L Brey)
  • Half-Skyrmion Theory for High-Temperature Superconductivity (T Morinari)
  • Deconfined Quantum Critical Points (T Senthil, A Vishwanath, L Balents, S Sachdev and M P A Fisher)
  • Skyrmions in a Density-Wave State: A Mechanism for Chiral Superconductivity (S Chakravarty and C-H Hsu)
• String Theory:
  • Skyrmion and String Theory (S Sugimoto)
  • Holographic Baryons (P Yi)
  • The Cheshire Cat Principle from Holography (H B Nielsen and I Zahed)
  • Baryon Physics in a Five-Dimensional Model of Hadrons (A Pomarol and A Wulzer)
  • Holographic Skyrmions (P M Sutcliffe)
  • Holographic Baryons and Instanton Crystal (V Kaplunovsky, D Melnikov and J Sonnenschein)

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Readership: Research scientists in the fields of condensed matter physics, nuclear and particle physics, and string theory.
--> Key Features:

• Written by leading experts in the fields
• Only book of review on current development of the matter, its pervasiveness in many fields of physics

Part 2

Condensed Matter

Chapter 11

Rotational Symmetry Breaking in Baby Skyrme Models

Marek Karliner and Itay Hen

Raymond and Beverly Sackler School of Physics and Astronomy
Tel-Aviv University, Tel-Aviv 69978, Israel
[email protected]

We discuss one of the most interesting phenomena exhibited by baby skyrmions – breaking of rotational symmetry. The topics we will deal with here include the appearance of rotational symmetry breaking in the static solutions of baby Skyrme models, both in flat as well as in curved spaces, the zero-temperature crystalline structure of baby skyrmions, and finally, the appearance of spontaneous breaking of rotational symmetry in rotating baby skyrmions.

11.1. Breaking of Rotational Symmetry in Baby Skyrme Models

The Skyrme model^1,2 is an SU(2)-valued nonlinear theory for pions in (3+1) dimensions with topological soliton solutions called skyrmions. Apart from a kinetic term, the Lagrangian of the model contains a ‘Skyrme’ term which is of the fourth order in derivatives, and is used to introduce scale to the model.³ The existence of stable solutions in the Skyrme model is a consequence of the nontrivial topology of the mapping M of the physical space into the field space at a given time, M : S³ → SU(2)

S³, where the physical space R³ is compactified to S³ by requiring the spatial infinity to be equivalent in each direction. The topology which stems from this one-point compactification allows the classification of maps into equivalence classes, each of which has a unique conserved quantity called the topological charge.

The Skyrme model has an analogue in (2+1) dimensions known as the baby Skyrme model, also admitting stable field configurations of a solitonic nature.⁴ Due to its lower dimension, the baby Skyrme model serves as a simplification of the original model, but nonetheless it has a physical significance in its own right, having several applications in condensed-matter physics,⁵ specifically in ferromagnetic quantum Hall systems.^6–9 There, baby skyrmions describe the excitations relative to ferromagnetic quantum Hall states, in terms of a gradient expansion in the spin density, a field with properties analogous to the pion field in the 3D case.¹⁰

The target manifold in the baby model is described by a three-dimensional vector ϕ = (ϕ₁, ϕ₂, ϕ₃) with the constraint ϕ ⋅ ϕ = 1. In analogy with the (3 + 1)D case, the domain of this model R² is compactified to S², yielding the topology required for the classification of its field configurations into classes with conserved topological charges. The Lagrangian density of the baby Skyrme model is given by:

and consists of a kinetic term, a Skyrme term and a potential term.

While in (3+1) dimensions the latter term is optional,¹¹ its presence in the (2+1)D model is necessary for the stability of the solutions. However, aside from the requirement that the potential vanishes at infinity for a given vacuum field value (normally taken to be ϕ⁽⁰⁾ = (0,0,1)), its exact form is arbitrary and gives rise to a rich family of possible baby-Skyrme models, several of which have been studied in detail in the literature. The simplest potential is the ‘holomorphic’ model with U(ϕ₃) = μ² (1 −ϕ₃)⁴.^12–14 It is known to have a stable solution only in the charge-one sector (the name refers to the fact that the stable solution has an analytic form in terms of holomorphic functions). The model with the potential U(ϕ₃) = μ² (1 − ϕ₃) (commonly referred to as the ‘old’ model) has also been extensively studied. This potential gives rise to very structured non-rotationally-symmetric multi-skyrmions.^4,15 Another model with U(ϕ₃) = μ² (1 − ϕ₃²) produces ring-like multi-skyrmions.¹⁶ Other double-vacuum potentials which give rise to other types of solutions have also been studied.¹⁷

Clearly, the form of the potential term has a decisive effect on the properties of the minimal energy configurations of the model. It is then worthwhile to see how the multisolitons of the baby Skyrme model look like for the one-parametric family of potentials U = μ² (1 − ϕ₃)^s which generalizes the ‘old‘ model (s = 1) and the holomorphic model (s = 4).¹⁸ As it turns out, the value of the parameter s has d...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Contents
• Introduction
• Hadrons and Nuclear Matter
• Condensed Matter
• String Theory
• Author Index
• Subject Index