phi4

Phi^4 theory, commonly written as phi^4 theory, is a relativistic quantum field theory of a real scalar field with a quartic self-interaction. The standard Lagrangian density is L = 1/2 ∂μφ ∂^μφ − 1/2 m^2 φ^2 − λ/4! φ^4, where m is the mass parameter and λ is the self-coupling. In Euclidean signature or in statistical mechanics applications the action is S_E = ∫ d^d x [1/2 (∂φ)^2 + 1/2 m^2 φ^2 + λ/4! φ^4]. The potential V(φ) = 1/2 m^2 φ^2 + λ/4! φ^4 is Z2-symmetric and, for m^2 < 0 and λ > 0, supports spontaneous symmetry breaking with a double-well potential.

In four spacetime dimensions, phi^4 theory is renormalizable with a dimensionless coupling λ. The renormalization group flow shows a positive beta function for λ > 0 at perturbative level, which implies a Landau pole and, in the strict continuum limit, a possibility that the interacting theory is trivial (the coupling vanishes). Despite this, the model remains a standard playground for perturbation theory, renormalization, and lattice simulations. In dimensions d < 4 an interacting Wilson-Fisher fixed point appears, yielding nontrivial critical behavior in the Ising universality class for N = 1 and in O(N) generalizations.

Applications and generalizations: phi^4 theory is used as a minimal interacting quantum field theory, a toy model for studying renormalization and critical phenomena, and a lattice field theory for investigating phase transitions and critical exponents. Variants include O(N) symmetric theories with N-component fields, where the quartic interaction is (λ/4)(φ·φ)^2, studied with nonperturbative methods such as Monte Carlo simulations and functional renormalization group techniques.