Scalar φ⁴ Field Theory in 1+1D
Spontaneous symmetry breaking · Kink solitons · Effective potential · RG flow
Lagrangian: ℒ = ½(∂φ)² − V(φ), with V(φ) = ½m²φ² + ¼λφ⁴.
SSB: For m² < 0, the potential has two minima at φ = ±v, v = √(−m²/λ). The field spontaneously picks one vacuum — Z₂ symmetry is broken.
Kink soliton: φ_kink(x) = v·tanh((x−x₀)/ξ), where ξ = 1/√(−m²/2) is the kink width. Kinks interpolate between the two vacua and are topologically stable.
RG flow: In 2D, λ is marginally relevant; the Wilson-Fisher fixed point governs the Ising universality class. Flow toward the fixed point controls the continuum limit.