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arXiv:hep-ph/9906554

Out-of-equilibrium dynamics of ϕ^4 QFT in finite volume

C. Destri, Manfredini, E.

Published Jun 30, 1999• 0 citations
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Abstract

The $λϕ^4$ model in a finite volume is studied in the infinite $N$ limit and within a non-gaussian Hartree-Fock approximation both at equilibrium and out of equilibrium, with particular attention to certain fundamental features of the broken symmetry phase. The numerical solution of the dynamical evolution equations show that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions. Thus we conclude that there is no evidence for a dynamical Bose-Einstein condensation. On the other hand, out of equilibrium the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones characteristic of the same approximation schemes. This result suggests the cause, and the possible remedy, of some unlikely features of the application to out--of--equilibrium dynamics of the standard HF factorization scheme, which coincides with the gaussian restriction of our Hartree--Fock approximation.

DOI
10.48550/arxiv.hep-ph/9906554

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The λϕ4λϕ^4λϕ4 model in a finite volume is studied in the infinite NNN limit and within a non-gaussian Hartree-Fock approximation both at equilibrium and out of equilibrium, with particular attention to certain fundamental features of the broken symmetry phase. The numerical solution of the dynamical evolution equations show that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions. Thus we conclude that there is no evidence for a dynamical Bose-Einstein condensation. On the other hand, out of equilibrium the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones characteristic of the same approximation schemes. This result suggests the cause, and the possible remedy, of some unlikely features of the application to out--of--equilibrium dynamics of the standard HF factorization scheme, which coincides with the gaussian restriction of our Hartree--Fock approximation.
The $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ model in a finite volume is studied in the infinite N limit both at equilibrium and out of equilibrium, with particular attention to certain fundamental features of the broken symmetry phase. The numerical solution of the dynamical evolution equations shows that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions. Thus we conclude that there is no evidence for a dynamical Bose-Einstein condensation, in the usual sense. On the other hand, out of equilibrium the long-wavelength fluctuations do scale with the linear size of the system, signaling dynamical infrared properties quite different from the equilibrium ones characteristic of the same approximation scheme.
This work is devoted to the study of relaxation--dissipation processes in systems described by Quantum Field Theory. In the first part, I focus on the phi^4 scalar quantum field theory in finite volume in the large N limit. I find that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions, while the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones. I also introduce a non-gaussian Hartree-Fock approximation (tdHF) and I derive the mean-field coupled time-dependent Schroedinger equations for the modes of the scalar field. The dynamical evolution of a uniform condensate is studied in a further controlled gaussian approximation, for N=1. Results similar to the large N case show an internal inconsistency of this approximation. I also study the dynamics of the system in infinite volume in the broken symmetry phase. The data show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential. I finally study the dynamical evolution for the O(N+1) nonlinear sigma model in 1+1 dimensions in the large N limit. I first of all verify that the large coupling limit of the O(N+1) phi^4 model, which renders the model non linear, commutes with the large N limit, so that the O(infty) nonlinear sigma model is uniquely defined. I then study the evolution of several observables, like the spectrum of produced particles, finding no evidence for parametric resonance. I conclude with some remarks on the ``state of art'' in gauge theories and some comments about the open issues in the subject.
The $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ model in a finite volume is studied within a non-Gaussian, time-dependent Hartree-Fock (TDHF) approximation both at equilibrium and out of equilibrium, with particular attention to the structure of the ground state and of certain dynamical features in the broken symmetry phase. The mean-field coupled time-dependent Schr\"odinger equations for the modes of the scalar field are derived and the suitable procedure to renormalize them is outlined. A further controlled Gaussian approximation of our TDHF approach is used in order to study the dynamical evolution of the system from nonequilibrium initial conditions characterized by a uniform condensate. We find that, during the slow rolling down, the long-wavelength quantum fluctuations do not grow to a macroscopic size but do scale with the linear size of the system, in accordance with similar results valid for the large-N approximation of the O(N)O(N)O(N) model. This behavior undermines in a precise way the Gaussian approximation within our TDHF approach, which therefore appears as a viable means to correct an unlikely feature of the standard HF factorization scheme, such as the so-called ``stopping at the spinodal line'' of the quantum fluctuations. We also study the dynamics of the system in infinite volume with particular attention to the asymptotic evolution in the broken symmetry phase. We are able to show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential.
Several mean-field computations have revealed the existence of an out-of-equilibrium dynamical transition induced by quantum-quenching an isolated system starting from its symmetry-broken phase. In this work we focus on the quantum ${\ensuremath{\phi}}^{4}$ NNN-component field theory. By taking into account dynamical fluctuations at the Hartree-Fock level, corresponding to the leading order of the 1/N1/N1/N expansion, we derive the critical properties of the dynamical transition beyond mean-field theory (including at finite temperature). We find diverging time and length scales, dynamic scaling, and aging. Finally, we unveil a relationship with critical coarsening, an off-equilibrium regime that can be induced by quenching from the symmetric toward the symmetry-broken phase.
We perform a detailed numerical investigation of the dynamics of a single component ``explicitly broken symmetry'' $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ 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 here that the initial state is described by a Gaussian density matrix peaked around some nonzero value of $〈\ensuremath{\varphi}(0)〉,$ 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: the Hartree approximation, the BVA, and a related two-particle irreducible (2PI) 1/N1/N1/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 behavior consistent with a second order phase transition. We find that the average fields equilibrate for arbitrary initial conditions in the BVA, unlike the behavior exhibited by the Hartree approximation, and we illustrate how $〈\ensuremath{\varphi}(t)〉$ and $〈\ensuremath{\chi}(t)〉$ depend on the initial temperature and on the coupling constant. The Fourier transform of the two-point functions at late times can be fitted by a Bose-Einstein distribution function whose temperature is independent of momentum. We interpret this as evidence for thermalization.
We consider the out-of-equilibrium evolution of a classical condensate field $\ensuremath{\varphi}=〈\ensuremath{\Phi}〉$ and its quantum fluctuations for a ${\ensuremath{\Phi}}^{4}$ model in 1+11+11+1 dimensions with a double well potential. We use the two-particle point-irreducible (2PPI) formalism in the two-loop approximation. We compare our results to those obtained in the Hartree approximation, in the bare vertex approximation, and in the two-particle irreducible next-to-leading order large-NNN approach, with thermal initial conditions. In the 2PPI scheme, we find that the system tends to the symmetric configuration at late times, as expected in the absence of spontaneous symmetry breaking.