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arXiv:hep-ph/9811273 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Nonequilibrium Bose-Einstein condensates, dynamical scaling, and symmetric evolution in the largeN Φ4 theory

D. Boyanovsky, Hector J. de Vega, R. Holman, J. Salgado

Published May 13, 1999• 63 citations
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Abstract

We analyze the nonequilibrium dynamics of the $O(N) {\ensuremath{\Phi}}^{4}$ model in the large N limit with a broken symmetry tree level potential and for states of large energy density. The dynamics is dramatically different when the energy density is above the top of the tree level potential ${V}_{0}$ than when it is below it. When the energy density is below ${V}_{0},$ we find that nonperturbative particle production through spinodal instabilities provides a dynamical mechanism for the Maxwell construction. The asymptotic values of the order parameter only depend on the initial energy density and all values between the minima of the tree level potential are available; the asymptotic dynamical ``effective potential'' is flat between the minima. When the energy density is larger than ${V}_{0},$ the evolution samples ergodically the broken symmetry states, as a consequence of nonperturbative particle production via parametric amplification. Furthermore, we examine the quantum dynamics of phase ordering into the broken symmetry phase and find a novel scaling behavior of the correlation function. There is a crossover in the dynamical correlation length at a time scale ${t}_{s}\ensuremath{\approx}\mathrm{ln}(1/\ensuremath{\lambda}).$ For $t<{t}_{s}$ the dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}\sqrt{t}$ and the evolution is dominated by linear instabilities and spinodal decomposition, whereas for $t>{t}_{s}$ the evolution is nonlinear and dominated by the onset of nonequilibrium Bose-Einstein condensation of long-wavelength Goldstone bosons. In this regime a true scaling solution emerges with a nonperturbative anomalous scaling length dimension $z=1/2$ and a dynamical correlation length $\ensuremath{\xi}(t)\ensuremath{\propto}(t\ensuremath{-}{t}_{s}).$ The equal time correlation function in this scaling regime vanishes for $r>2(t\ensuremath{-}{t}_{s})$ by causality. For $t>{t}_{s}$ phase ordering proceeds by the formation of domains that grow at the speed of light, with nonperturbative condensates of Goldstone bosons and the equal time correlation function falls of as $1/r.$ A semiclassical but stochastic description emerges for time scales $t>{t}_{s}.$ Our results are compared to phase ordering in classical stochastic descriptions in condensed matter and cosmology.

DOI
10.1103/physrevd.59.125009

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The dynamics of a dense relativistic quantum fluid out of thermodynamic equilibrium is studied in the framework of the ${\ensuremath{\Phi}}^{4}$ scalar field theory in the large N limit. The time evolution of a particle distribution in momentum space (the tsunami) is computed. The effective mass felt by the particles in such a high density medium equals the tree level mass plus the expectation value of the squared field. The case of negative tree level squared mass is particularly interesting. In such a case dynamical symmetry restoration as well as dynamical symmetry breaking can happen. Furthermore, the symmetry may stay broken with a vanishing asymptotic squared mass showing the presence of out of equilibrium Goldstone bosons. We study these phenomena and identify the set of initial conditions that lead to each case. We compute the equation of state which turns out to depend on the initial state. Although the system does not thermalize, the equation of state for asymptotically broken symmetry is of radiation type. We compute the correlation functions at equal times. The two point correlator for late times is the sum of different terms. One stems from the initial particle distribution. Another term accounts for the out of equilibrium Goldstone bosons created by spinodal unstabilities when the symmetry is asymptotically broken. Both terms are of the order of the inverse of the coupling for distances where causal signals can connect the two points. The contribution of the out of equilibrium Goldstones exhibits scaling behavior in a generalized sense.
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.
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.
We study the detailed out-of-equilibrium time evolution of a homogeneous Bose-Einstein condensate (BEC). We consider a nonrelativistic quantum theory for a self-interacting complex scalar field, immersed in a thermal bath, as an effective microscopic model for the description of the BEC. The interaction between fluctuations proves to be crucial in the mechanism of instability generation. We show the existence of two regimes in k space, with a crossover for k(2)/2m approximately 2 lambda|phi(0)|(2), where lambda is the coupling constant and |phi(0)|(2) is the condensate density. We deduce and solve a set of coupled equations that completely determines the nonequilibrium dynamics of the condensate density.
Building on large-scale quantum Monte Carlo simulations, we investigate the zero-temperature phase diagram of hard-core bosons in a random potential on site-centered Cayley trees with branching number K=2K=2K=2. In order to follow how the Bose-Einstein condensate (BEC) is affected by the disorder, we focus on both the zero-momentum density, probing the quantum coherence, and the one-body density matrix (1BDM) whose largest eigenvalue monitors the off-diagonal long-range order. We further study its associated eigenstate which brings useful information about the real-space properties of this leading eigenmode. Upon increasing randomness, we find that the system undergoes a quantum phase transition at finite disorder strength between a long-range ordered BEC state, fully ergodic at large scale, and a new disordered Bose glass regime showing conventional localization for the coherence fraction while the 1BDM displays a nontrivial algebraic vanishing BEC density together with a nonergodic occupation in real space. These peculiar properties can be analytically captured by a simple phenomenological description on the Cayley tree which provides a physical picture of the Bose glass regime.
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.