Investigations in 1+1 dimensional lattice $ϕ^4$ theory

Abstract

In this work we perform a detailed numerical analysis of (1+1) dimensional lattice $ϕ^4$ theory. We explore the phase diagram of the theory with two different parameterizations. We find that symmetry breaking occurs only with a negative mass-squared term in the Hamiltonian. The renormalized mass $m_R$ and the field renormalization constant $Z$ are calculated from both coordinate space and momentum space propagators in the broken symmetry phase. The critical coupling for the phase transition and the critical exponents associated with $m_R$, $Z$ and the order parameter are extracted using a finite size scaling analysis of the data for several volumes. The scaling behavior of $Z$ has the interesting consequence that $<ϕ_R>$ does not scale in 1+1 dimensions. We also calculate the renormalized coupling constant $ λ_R$ in the broken symmetry phase. The ratio $ λ_R/m_R^2 $ does not scale and appears to reach a value independent of the bare parameters in the critical region in the infinite volume limit.