Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part VII, Chapter 3

Solitons & Topological Objects

Stable, localized field configurations protected by topology

What is a Soliton?

A soliton is a stable, localized, particle-like solution to a classical field equation. Unlike perturbative excitations (quantum particles), solitons are classical field configurations that maintain their shape as they propagate.

Topology often protects solitons from decay. They cannot continuously deform to the vacuum without infinite energy, creating stable, finite-energy configurations that behave like particles!

3.1 Kinks in (1+1) Dimensions

The simplest soliton is the kink in a 1+1 dimensional scalar field theory. Consider the Lagrangian:

$$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - V(\phi)$$

with a double-well potential:

$$V(\phi) = \frac{\lambda}{4}(\phi^2 - v^2)^2$$

Two degenerate vacua at Ο† = Β±v. The kink solution interpolates between them:

$$\phi_{\text{kink}}(x) = v \tanh\left(\frac{m(x-x_0)}{\sqrt{2}}\right)$$

where m = v√λ is the mass scale and xβ‚€ is the kink position.

Properties of the Kink

  • As x β†’ -∞: Ο† β†’ -v (left vacuum)
  • As x β†’ +∞: Ο† β†’ +v (right vacuum)
  • Width: Ξ”x ~ 1/m (localized!)
  • Energy (mass): M = (4√2/3)mvΒ²
  • Topologically stable: cannot decay to vacuum continuously

3.2 Topological Charge of Kinks

The kink carries a topological charge:

$$Q = \frac{1}{2v}\int_{-\infty}^{\infty} dx \, \partial_x \phi = \frac{1}{2v}[\phi(\infty) - \phi(-\infty)]$$

For the kink: Q = (v - (-v))/2v = 1. For the anti-kink (Ο†: v β†’ -v): Q = -1.

Why Topologically Stable?

The topological charge Q is an integer determined by boundary conditions. To go from Q = 1 to Q = 0 (vacuum), you must:

  • Either jump discontinuously (infinite gradient energy)
  • Or pass through Ο† = 0 (potential barrier at all points)

Both require infinite energy, so the kink is absolutely stable!

3.3 Vortices in (2+1) Dimensions

In 2+1 dimensions with a complex scalar field and U(1) gauge symmetry (Abelian Higgs model):

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + |D_\mu \phi|^2 - \lambda(|\phi|^2 - v^2)^2$$

where DΞΌ = βˆ‚ΞΌ - ieAΞΌ is the covariant derivative.

The vortex solution has the form (in cylindrical coordinates):

\begin{align*} \phi(r,\theta) &= v f(r) e^{in\theta} \\ A_\theta(r) &= \frac{n}{er} a(r) \end{align*}

where n ∈ β„€ is the winding number (vorticity).

Boundary Conditions

  • r β†’ 0: f(r) β†’ r|n| (field vanishes at origin to avoid singularity)
  • r β†’ ∞: f(r) β†’ 1, a(r) β†’ 1 (approaches vacuum)
  • Magnetic flux: Ξ¦ = ∫BΒ·dA = 2Ο€n/e
  • Energy: E ~ nΒ²vΒ² ln(R/rcore)

The vortex core (where Ο† = 0) has radius ~ 1/m. The magnetic field is confined to this region. This is analogous to flux tubes in type-II superconductors!

3.4 Magnetic Monopoles in (3+1) Dimensions

The most famous topological soliton is the 't Hooft-Polyakov magnetic monopolein a non-Abelian gauge theory with spontaneous symmetry breaking.

Consider SO(3) gauge theory broken to U(1):

$$\text{SO}(3) \to \text{U}(1)_{\text{EM}}$$

by a Higgs triplet Ο†a (a = 1,2,3) with βŸ¨Ο†a⟩ = vΞ΄a3.

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}^a F^{a,\mu\nu} + \frac{1}{2}(D_\mu \phi)^a(D^\mu \phi)^a - V(\phi)$$

The monopole solution (hedgehog configuration):

\begin{align*} \phi^a(r) &= v H(r) \frac{x^a}{r} \\ A_i^a &= \frac{1}{eg} \epsilon^{aij} \frac{x^j}{r^2} [1 - K(r)] \end{align*}

where H(r) and K(r) are profile functions satisfying:

  • r β†’ 0: H(r) β†’ 0, K(r) β†’ 1
  • r β†’ ∞: H(r) β†’ 1, K(r) β†’ 0

Properties of the Monopole

  • Magnetic charge: gm = 4Ο€/e (quantized!)
  • Dirac quantization: egm = 4Ο€ (n = 1)
  • Mass: M ~ v/Ξ± where Ξ± = eΒ²/4Ο€
  • Core size: ~ 1/mHiggs
  • Topological charge: Ο€β‚‚(SΒ²) = β„€ (second homotopy group)

3.5 Dirac Quantization Condition

The existence of a magnetic monopole with charge gm implies Dirac's quantization condition for electric charge:

$$eg_m = 2\pi n, \quad n \in \mathbb{Z}$$

This explains why electric charge is quantized! If even one monopole exists in the universe, all electric charges must be integer multiples of a fundamental unit.

Dirac's Argument

Consider an electron with charge e circling a monopole with charge gm. The quantum phase picked up must be single-valued:

$$\Delta\phi = \frac{e}{\hbar}\oint \mathbf{A} \cdot d\mathbf{l} = \frac{eg_m}{\hbar} = 2\pi n$$

This requires egm = 2Ο€nℏ. Setting ℏ = 1 gives the quantization condition!

3.6 Domain Walls

In theories with discrete symmetry breaking, domain walls separate regions with different vacuum states. For example, with β„€β‚‚ symmetry breaking (Ο† β†’ -Ο†):

$$V(\phi) = \frac{\lambda}{4}(\phi^2 - v^2)^2$$

A domain wall interpolates between Ο† = +v and Ο† = -v regions. In 3+1 dimensions, this is a 2D surface (wall) in 3D space.

Cosmological Domain Walls

If the early universe underwent a phase transition with discrete symmetry breaking, domain walls would form:

  • Energy per unit area: Οƒ ~ Ξ»vΒ³
  • Can dominate energy density if not diluted
  • Domain wall problem: observed universe has no walls!
  • Solution: symmetry must be explicitly (not spontaneously) broken

3.7 Cosmic Strings

Cosmic strings are 1D topological defects that can form in the early universe from spontaneous breaking of U(1) symmetry:

$$\text{U}(1) \to \text{nothing}$$

They are essentially the 3+1D version of vortices, with:

  • Energy per unit length: ΞΌ ~ vΒ²
  • Core radius: ~ 1/m
  • Can be infinitely long or form closed loops
  • Classified by π₁(SΒΉ) = β„€ (first homotopy group)

Observational Signatures

Cosmic strings could be detected through:

  • Gravitational lensing (double images of galaxies)
  • CMB temperature discontinuities (Kaiser-Stebbins effect)
  • Gravitational wave signatures
  • Current constraints: GΞΌ < 10-7 (very stringent!)

3.8 Homotopy Groups and Topological Classification

Topological defects are classified by homotopy groups Ο€n(G/H) where G is the original symmetry and H is the unbroken symmetry:

DefectDimensionHomotopy GroupExample
Domain Wall2D in 3DΟ€β‚€(G/H)β„€β‚‚ breaking
Cosmic String1D in 3Dπ₁(G/H)U(1) breaking
Monopole0D (point)Ο€β‚‚(G/H)SO(3)β†’U(1)
Texture0D (point)π₃(G/H)SO(4)β†’SO(3)

3.9 BPS Bound and Supersymmetry

For solitons in theories with extended symmetry, the Bogomol'nyi-Prasad-Sommerfield (BPS) bound relates mass to topological charge:

$$M \geq |Z Q|$$

where Z is the central charge and Q is topological charge. BPS saturated states (M = |ZQ|) are absolutely stable and often preserve some supersymmetry.

BPS Solitons

  • Satisfy first-order differential equations (easier to solve!)
  • Exactly stable against all perturbations
  • In SUSY theories: preserve 1/2 of supersymmetries
  • Form multiplets with fermion zero modes
  • Examples: BPS monopoles, D-branes in string theory

Key Takeaways

  • Solitons: stable, localized classical solutions protected by topology
  • Kinks (1+1D): Q = (1/2v)[Ο†(∞) - Ο†(-∞)] ∈ β„€
  • Vortices (2+1D): winding number n, magnetic flux Ξ¦ = 2Ο€n/e
  • Monopoles (3+1D): gm = 4Ο€/e, Dirac quantization egm = 2Ο€n
  • Domain walls, cosmic strings: cosmological topological defects
  • Homotopy groups Ο€n(G/H) classify topological defects
  • BPS bound: M β‰₯ |ZQ|, equality for BPS states
  • Applications: cosmology, condensed matter, particle physics
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