Quantum dynamics of phase transitions in broken symmetry $\lambda {\varphi }^4$ field theory

Abstract

We perform a detailed numerical investigation of the dynamics of broken symmetry $\lambda {\varphi }^4$ field theory in 1+1 dimensions using a Schwinger-Dyson equation truncation scheme based on ignoring vertex corrections. In an earlier paper, we called this the bare vertex approximation (BVA). We assume the initial state is described by a Gaussian density matrix peaked around some non-zero value of $<\varphi (0)>$, and characterized by a single particle Bose-Einstein distribution function at a given temperature. We compute the evolution of the system using three different approximations: Hartree, BVA and a related 2PI-1/N expansion, as a function of coupling strength and initial temperature. In the Hartree approximation, the static phase diagram shows that there is a first order phase transition for this system. As we change the initial starting temperature of the system, we find that the BVA relaxes to a new final temperature and exhibits a second order phase transition. We find that the average fields thermalize for arbitrary initial conditions in the BVA, unlike the behavior exhibited by the Hartree approximation, and we illustrate how $<\varphi (t)>$ and $<\chi (t)>$ depend on the initial temperature and on the coupling constant. We find that the 2PI-1/N expansion gives dramatically different results for $<\varphi (t)>$.

Comments: 28 pages, 21 figures; ver 2 -- additional comments on the nature of the phase transition in 1+1 dimensions