Investigations in 1+1 dimensional lattice $\phi^4$ theory

TL;DR

This paper provides a comprehensive numerical study of (1+1)D lattice $^4$ theory, analyzing phase transitions, critical exponents, and renormalized parameters, revealing unique scaling behaviors in the broken symmetry phase.

Contribution

It offers new insights into the phase diagram, critical phenomena, and renormalized quantities of lattice $^4$ theory in 1+1 dimensions through detailed finite size scaling analysis.

Findings

Symmetry breaking occurs only with negative mass-squared.

Renormalized mass and field renormalization constant are computed.

The ratio $ _ /m_ ^2$ approaches a volume-independent value.

Abstract

In this work we perform a detailed numerical analysis of (1+1) dimensional lattice $\phi^4$ 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 $<\phi_R>$ does not scale in 1+1 dimensions. We also calculate the renormalized coupling constant $ \lambda_R$ in the broken symmetry phase. The ratio $ \lambda_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.