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arXiv:hep-ph/0209051 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum dynamics of phase transitions in broken symmetry λφ4 field theory

Fred Cooper, John F. Dawson, Bogdan Mihaila

Published Mar 28, 2003• 68 citations
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Abstract

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) $1/N$ 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.

DOI
10.1103/physrevd.67.056003

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We perform a detailed numerical investigation of the dynamics of a single component ``explicitly broken symmetry'' \ensuremathλ\ensuremathφ4\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}\ensuremathλ\ensuremathφ4 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 〈\ensuremathφ(0)〉,〈\ensuremath{\varphi}(0)〉,〈\ensuremathφ(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 〈\ensuremathφ(t)〉〈\ensuremath{\varphi}(t)〉〈\ensuremathφ(t)〉 and 〈\ensuremathχ(t)〉〈\ensuremath{\chi}(t)〉〈\ensuremathχ(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.
In this paper we discuss a Schwinger-Dyson (SD) approach for determining the time evolution of the unequal time correlation functions of a nonequilibrium classical field theory, where the classical system is described by an initial density matrix at time t=0.t=0.t=0. We focus on $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in 1+11+11+1 space-time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation (BVA), is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for 1/N1/N1/N expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\ensuremath{\varphi}$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.
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.
The quantum time evolution of ${\ensuremath{\varphi}}^{4}$-field theory for a spatially homogeneous system in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym equations including the tadpole and sunset self-energies. Whereas the tadpole self-energy yields a dynamical mass, the sunset self-energy is responsible for dissipation and an equilibration of the system. In particular we address the dynamics of the spectral (``off-shell'') distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. The investigation explicitly demonstrates that the only translation invariant solutions representing the stationary fixed points of the coupled equation of motions are those of full thermal equilibrium. They agree with those extracted from the time integration of the Kadanoff-Baym equations for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ Furthermore, a detailed comparison of the full quantum dynamics to more approximate and simple schemes such as that of a standard kinetic (on-shell) Boltzmann equation is performed. Our analysis shows that the consistent inclusion of the dynamical spectral function has a significant impact on relaxation phenomena. The different time scales that are involved in the dynamical quantum evolution towards a complete thermalized state are discussed in detail. We find that far off-shell $1\ensuremath{\leftrightarrow}3$ processes are responsible for chemical equilibration, which is missed in the Boltzmann limit. Finally, we briefly address the case of (bare) massless fields. For sufficiently large couplings $\ensuremath{\lambda}$ we observe the onset of Bose condensation, where our scheme within symmetric ${\ensuremath{\varphi}}^{4}$ theory breaks down.
We describe a Hartree ensemble method to approximately solve the Heisenberg equations for the ϕ 4 model in 1 + 1 dimensions. We compute the energies and number densities of the quantum particles described by the ϕ field and find that the particles initially thermalize with a Bose-Einstein distribution for the particle density. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved. 1. Hartree ensemble approximation Non-perturbative equilibrium properties of quantum field theories are studied with great success using numerical lattice techniques. Similarly one would like to investigate quantum field dynamics non-perturbatively. This has obvious applications to e.g. heavy ion collisions and early universe physics. Furthermore, it could be used to study equilibrium properties in cases where Monte Carlo methods do not work, such as in QCD at finite density or chiral gauge theories. Using Monte Carlo techniques to compute the Minkowski time path integral is, of course, not possible. Neither can one directly solve the Heisenberg equations for the quantum field operators. Hence one has to resort to approximations, such as classical dynamics [2] (for recent work see [3]), large n or Hartree (see e.g. [4]). Here we discuss a different type of Hartree or gaussian approximation than used previously, in which we use an ensemble of gaussian wavefunctions to compute Green functions. Ref. [1] also contains a discussion of this method and a detailed presentation will appear elsewhere. We use the lattice ϕ4 model in 1+1 dimensions as a test model, which has the following Heisenberg operator equations, ˙ˆϕ = ˆπ, ˙ ˆπ = ( ∆ − µ
We describe a Hartree ensemble method to approximately solve the Heisenberg equations for the ϕ 4 model in 1 + 1 dimensions. We compute the energies and number densities of the quantum particles described by the ϕ field and find that the particles initially thermalize with a Bose-Einstein distribution for the particle density. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved. 1. Hartree ensemble approximation Non-perturbative equilibrium properties of quantum field theories are studied with great success using numerical lattice techniques. Similarly one would like to investigate quantum field dynamics non-perturbatively. This has obvious applications to e.g. heavy ion collisions and early universe physics. Furthermore, it could be used to study equilibrium properties in cases where Monte Carlo methods do not work, such as in QCD at finite density or chiral gauge theories. Using Monte Carlo techniques to compute the Minkowski time path integral is, of course, not possible. Neither can one directly solve the Heisenberg equations for the quantum field operators. Hence one has to resort to approximations, such as classical dynamics [2] (for recent work see [3]), large n or Hartree (see e.g. [4]). Here we discuss a different type of Hartree or gaussian approximation than used previously, in which we use an ensemble of gaussian wavefunctions to compute Green functions. Ref. [1] also contains a discussion of this method and a detailed presentation will appear elsewhere. We use the lattice ϕ4 model in 1+1 dimensions as a test model, which has the following Heisenberg operator equations, ˙ˆϕ = ˆπ, ˙ ˆπ = ( ∆ − µ