🔬Quantum Field Theory Unit 9 – Symmetry Breaking & Topological Defects

Key Concepts

• Symmetry plays a fundamental role in physics describes invariance under certain transformations
• Spontaneous symmetry breaking occurs when a system's ground state has lower symmetry than its Lagrangian
  • Leads to degenerate ground states connected by the broken symmetry
• Goldstone's theorem states that spontaneous breaking of a continuous symmetry leads to massless scalar bosons (Goldstone bosons)
• Topological defects are stable configurations that cannot be continuously deformed into a uniform state
  • Arise from spontaneous symmetry breaking in physical systems
• Types of topological defects include domain walls, cosmic strings, monopoles, and textures
  • Characterized by their dimensionality and topological properties
• Mathematical framework for describing topological defects involves homotopy theory and topological invariants
  • Homotopy groups classify defects based on mappings between manifolds
• Condensed matter systems (liquid crystals, superconductors, superfluids) exhibit rich variety of topological defects
  • Vortices in superfluids and superconductors, disclinations in liquid crystals
• Particle physics and cosmology also feature topological defects
  • Cosmic strings, domain walls, and magnetic monopoles in early universe theories

Symmetry in Physics

• Symmetry refers to invariance of a system under certain transformations (rotations, translations, gauge transformations)
• Continuous symmetries are described by Lie groups parameterized by continuous variables
  • Examples include rotation group SO(3), translation group, and unitary groups U(1) and SU(N)
• Discrete symmetries include reflection, time reversal, and charge conjugation
• Noether's theorem connects continuous symmetries to conservation laws
  • Translational symmetry leads to conservation of momentum
  • Rotational symmetry leads to conservation of angular momentum
• Gauge symmetries are local symmetries that depend on spacetime position
  • Basis for theories of fundamental interactions (electromagnetism, weak and strong nuclear forces)
• Symmetries can be global (same transformation at all points) or local (spacetime-dependent transformations)
• Symmetry breaking occurs when a system's ground state has lower symmetry than its Lagrangian or equations of motion

Spontaneous Symmetry Breaking

• Spontaneous symmetry breaking (SSB) happens when a system's ground state has lower symmetry than its Lagrangian
• Leads to degenerate ground states related by the broken symmetry transformation
  • Ground states form a manifold with topology determined by the broken symmetry group
• Occurs in many physical systems (ferromagnets, superconductors, superfluids, particle physics)
• Characterized by an order parameter that acquires a non-zero value in the broken-symmetry phase
  • Magnetization in ferromagnets, Cooper pair wavefunction in superconductors
• SSB can be triggered by small perturbations or fluctuations that break the symmetry
  • Analogous to a ball rolling off the top of a Mexican hat potential
• Goldstone's theorem is a key consequence of SSB for continuous symmetries
• SSB is a crucial ingredient in the Higgs mechanism for generating particle masses in the Standard Model

Goldstone's Theorem

• Goldstone's theorem states that spontaneous breaking of a continuous symmetry leads to massless scalar bosons (Goldstone bosons)
• Number of Goldstone bosons equals the number of broken symmetry generators
  • For U(1) symmetry breaking, there is one Goldstone boson
  • For SU(N) symmetry breaking, there are N^2-1 Goldstone bosons
• Goldstone bosons are excitations of the order parameter around the degenerate ground states
  • Represent long-wavelength fluctuations that restore the broken symmetry locally
• Examples include phonons in solids, magnons in ferromagnets, and pions in particle physics
• In relativistic theories, Goldstone bosons have a linear dispersion relation $E = c|k|$
• Goldstone bosons can acquire mass through explicit symmetry breaking or gauge interactions (Higgs mechanism)
• Experimental detection of Goldstone bosons provides evidence for SSB in physical systems

Topological Defects: An Introduction

• Topological defects are stable configurations that cannot be continuously deformed into a uniform state
• Arise from spontaneous symmetry breaking in physical systems with degenerate ground states
  • Ground state manifold has non-trivial topology characterized by homotopy groups
• Defects are classified by their dimensionality and topological properties
  • Characterized by topological invariants (winding numbers, topological charges)
• Examples include domain walls, cosmic strings, monopoles, and textures
  • Domain walls are 2D defects separating regions of different ground states
  • Cosmic strings are 1D defects with a non-zero winding number
  • Monopoles are 0D defects with a non-zero topological charge
• Formation of topological defects depends on the topology of the ground state manifold
  • Determined by the homotopy groups of the vacuum manifold
• Topological defects are stable against small perturbations due to topological protection
  • Cannot be removed by continuous deformations or local fluctuations
• Defects can interact with each other and with other fields (electromagnetic, gravitational)
  • Dynamics governed by effective field theories and symmetry considerations

Types of Topological Defects

• Domain walls are 2D defects separating regions of different ground states
  • Arise when discrete symmetries are spontaneously broken (Z2, ZN)
  • Example: Ising model with spontaneous magnetization
• Cosmic strings are 1D defects with a non-zero winding number
  • Arise when U(1) or higher-dimensional symmetries are spontaneously broken
  • Example: Vortices in superfluids and superconductors
• Monopoles are 0D defects with a non-zero topological charge
  • Arise when spherical symmetries (SO(3), SU(2)) are spontaneously broken
  • Example: 't Hooft-Polyakov monopole in Grand Unified Theories
• Textures are defects characterized by a non-uniform configuration of the order parameter
  • Arise when higher-dimensional symmetries (SU(N), N>2) are spontaneously broken
  • Example: Skyrmions in nuclear physics and condensed matter
• Hybrid defects combining different types can also exist (e.g., domain wall bounded by strings)
• Topological defects can be created in phase transitions or through dynamical processes
  • Kibble-Zurek mechanism describes defect formation in continuous phase transitions
• Interactions between defects depend on their dimensionality and topological properties
  • Example: String-monopole bound states in Grand Unified Theories

Mathematical Framework

• Topological defects are described using homotopy theory and differential geometry
• Homotopy groups classify topological defects based on mappings between manifolds
  • $\pi_n(M)$ classifies defects of codimension $n+1$ in a manifold $M$
  • Example: $\pi_1(S^1) = \mathbb{Z}$ classifies vortices in 2D systems with U(1) symmetry
• Order parameter fields are represented as mappings from physical space to the ground state manifold
  • Defects correspond to non-trivial mappings with topological invariants
• Topological invariants (winding numbers, topological charges) characterize defects
  • Computed using integrals of differential forms over the defect's boundary
  • Example: Winding number of a vortex given by $\oint \nabla \theta \cdot d\mathbf{l}$
• Effective field theories describe the dynamics and interactions of topological defects
  • Obtained by integrating out high-energy degrees of freedom
  • Symmetries and topological properties constrain the form of the effective Lagrangian
• Numerical simulations are used to study the formation and evolution of topological defects
  • Kibble-Zurek mechanism can be simulated using stochastic differential equations
  • Lattice field theory methods are used to study defects in gauge theories

Applications in Condensed Matter Physics

• Topological defects are ubiquitous in condensed matter systems with spontaneously broken symmetries
• Vortices in superfluids and superconductors are examples of topological defects
  • Arise from spontaneous breaking of U(1) gauge symmetry
  • Carry quantized circulation and magnetic flux
  • Vortex dynamics govern the properties of type-II superconductors
• Disclinations and dislocations in liquid crystals are topological defects
  • Arise from spontaneous breaking of rotational and translational symmetries
  • Control the mechanical and optical properties of liquid crystals
• Magnetic skyrmions are topological defects in chiral magnets
  • Characterized by a non-zero topological charge
  • Promising candidates for spintronics applications
• Topological insulators and superconductors host gapless edge states protected by topology
  • Described by topological invariants (Chern numbers, Z2 invariants)
  • Potential applications in quantum computing and spintronics
• Defect-mediated phase transitions and critical phenomena are studied using renormalization group methods
  • Kosterlitz-Thouless transition in 2D superfluids mediated by vortex unbinding
• Experimental techniques (STM, AFM, TEM) are used to image and manipulate topological defects
  • Provides insight into the microscopic properties and dynamics of defects

Connections to Particle Physics

• Topological defects are also relevant in particle physics and cosmology
• Spontaneous symmetry breaking is a key ingredient in the Standard Model of particle physics
  • Higgs mechanism generates masses for W and Z bosons through SSB of electroweak symmetry
  • Quarks and leptons acquire masses through Yukawa couplings to the Higgs field
• Grand Unified Theories (GUTs) predict the existence of topological defects in the early universe
  • Cosmic strings, domain walls, and monopoles can form during GUT phase transitions
  • Observational constraints on cosmic defects provide tests of GUT models
• Axion string network may have formed during Peccei-Quinn symmetry breaking in the early universe
  • Decay of axion strings and domain walls may contribute to dark matter abundance
• Magnetic monopoles are predicted by many GUTs as topological defects
  • 't Hooft-Polyakov monopole is a classical solution in theories with spontaneously broken SU(2) symmetry
  • Experimental searches for magnetic monopoles are ongoing
• Cosmic strings may have acted as seeds for large-scale structure formation in the early universe
  • Gravitational effects of cosmic strings can be probed through CMB anisotropies and gravitational lensing
• Topological defects may have played a role in the matter-antimatter asymmetry of the universe
  • Baryon number violation near defects may have generated a net baryon asymmetry