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arXiv:hep-ph/0303157 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Real-time dynamics of the O (N) model in 1+1 dimensions

Bogdan Mihaila

Published Aug 21, 2003• 12 citations
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Abstract

We study the nonequilibrium dynamics of the $\mathrm{O}(N)$ model in classical and quantum field theory in 1+1 dimensions, for $N>1.$ We compare numerical results obtained using the Hartree approximation and two next to leading order approximations: the bare vertex approximation and the two-particle irreducible $1/N$ expansion. The latter approximations differ through terms of order ${g}^{2},$ where g is the scaled coupling constant, $g=\ensuremath{\lambda}/N.$ In this paper we investigate the statement regarding the convergence with respect to g. The differences between these two approximation schemes are computed and found to be already small for relatively small values of N, when $\ensuremath{\lambda}$ is fixed.

DOI
10.1103/physrevd.68.036002

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We study the nonequilibrium dynamics of the O(N)\mathrm{O}(N)O(N) model in classical and quantum field theory in 1+1 dimensions, for N>1.N>1.N>1. We compare numerical results obtained using the Hartree approximation and two next to leading order approximations: the bare vertex approximation and the two-particle irreducible 1/N1/N1/N expansion. The latter approximations differ through terms of order g2,{g}^{2},g2, where g is the scaled coupling constant, g=\ensuremathλ/N.g=\ensuremath{\lambda}/N.g=\ensuremathλ/N. In this paper we investigate the statement regarding the convergence with respect to g. The differences between these two approximation schemes are computed and found to be already small for relatively small values of N, when \ensuremathλ\ensuremath{\lambda}\ensuremathλ is fixed.
The 1/N1/N1/N expansion of the two-particle irreducible effective action offers a powerful approach to study quantum field dynamics far from equilibrium. We investigate the effective convergence of the 1/N1/N1/N expansion in the O(N)O(N)O(N) model by comparing results obtained numerically in 1+11+11+1 dimensions at leading, next-to-leading and next-to-next-to-leading order in 1/N1/N1/N as well as in the weak coupling limit. A comparison in classical statistical field theory, where exact numerical results are available, is made as well. We focus on early-time dynamics and quasiparticle properties far from equilibrium and observe rapid effective convergence already for moderate values of 1/N1/N1/N or the coupling.
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We consider the time evolution of nonequilibrium quantum scalar fields in the O(N) model, using the next-to-leading order 1/N expansion of the two-particle irreducible effective action. A comparison with exact numerical simulations in 1+1 dimensions in the classical limit shows that the 1/N expansion gives quantitatively precise results already for moderate values of N. For sufficiently high initial occupation numbers the time evolution of quantum fields is shown to be accurately described by classical physics. Eventually the correspondence breaks down due to the difference between classical and quantum thermal equilibrium.
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.
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