Lattice φ⁴ Field Theory
1+1D scalar field Monte Carlo — symmetry breaking & continuum limit
Phase: symmetric | ⟨φ⟩ ≈ 0
The φ⁴ scalar field on a 1D lattice: S = Σ [½(φ_{i+1}−φ_i)²/a² + ½μ²φᵢ² + λφᵢ⁴]. For μ² < 0, the potential has two minima — the field spontaneously breaks the Z₂ symmetry (φ→−φ). The Monte Carlo simulation uses Metropolis updates. The upper panel shows the field configuration φ(x); the lower shows the double-well potential V(φ). The continuum limit is taken as lattice spacing a→0 with renormalized couplings.