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arXiv:hep-ph/0001178 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Improved time-dependent Hartree-Fock approach for scalar φ4 QFT

C. Destri, E. Manfredini

Published Jun 21, 2000• 12 citations
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Abstract

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 $O(N)$ 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.

DOI
10.1103/physrevd.62.025008

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According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N .In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity.In this article we extend our results to systems of fermions with m-particle interactions (m > 2).

Communications in Mathematical Sciences•Jan 1, 2007•14 citations
The \ensuremathλ\ensuremathφ4\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}\ensuremathλ\ensuremathφ4 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.
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 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.
A quenched second order phase transition is modeled by an effective ${\ensuremath{\Phi}}^{4}$ theory with a time-dependent Hamiltonian $\ifmmode \hat{H}\else \^{H}\fi{}(t),$ whose symmetry is broken spontaneously in time. The quantum field evolves out of equilibrium (nonequilibrium) during the phase transition as the density operator significantly deviates from $\stackrel{^}{\ensuremath{\rho}}{(t)=e}^{\ensuremath{-}\ensuremath{\beta}\ifmmode \hat{H}\else \^{H}\fi{}(t)}{/Z}_{H}.$ The recently developed Liouville--von Neumann method provides various quantum states for the phase transition in terms of a complex solution to the mean-field equation, which is equivalent to the Gaussian effective potential in the static case and to the time-dependent Hartree-Fock equation in the nonequilibrium case. Using multiple-scale perturbation theory we solve the mean-field equation analytically to the first order of the coupling constant and find the quantum states during the quenched second order phase transition. We propose a renormalization scheme during the process of phase transition to regularize the divergences, which originate from the mode coupling between hard and hard modes or between soft and hard modes. The effect of mode coupling is discussed.
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.
According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N .In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity.In this article we extend our results to systems of fermions with m-particle interactions (m > 2).