Journal article

Dynamical mean field approximation applied to quantum field theory

  • Akerlund, Oscar Institut für Theoretische Physik, ETH Zurich, Switzerland
  • Forcrand, Philippe de Institut für Theoretische Physik, ETH Zurich, Switzerland - Physics Department, CERN, TH Unit, Geneva, Switzerland
  • Georges, Antoine Centre de Physique Théorique, CNRS, École Polytechnique, Palaiseau, France - Collège de France, Paris, France - DPMC, Université de Genève, Switzerland
  • Werner, Philipp Department of Physics, University of Fribourg, Switzerland
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  • Physical Review D. - 2013, vol. 88, no. 12, p. 125006
English We apply the dynamical mean field theory (DMFT) approximation to the real, scalar φ4 quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents ν and β. We also map out the phase diagram in the most relevant case of four dimensions. In two and three dimensions, DMFT incorrectly predicts a first-order phase transition for all bare quartic couplings, which is problematic, because the second-order nature of the phase transition of lattice φ4 theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behavior away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is second order for small quartic couplings and turns weakly first order as the coupling increases beyond a tricritical value. In dimensions five and higher, DMFT gives qualitatively correct results, predicts reasonable values for the critical exponents and considerably more accurate critical couplings than standard mean field theory. The approximation works best for small values of the quartic coupling. We investigate the change from first- to second-order transition in the local limit of DMFT which is computationally much cheaper. We also discuss technical issues related to the convergence of the nonlinear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a φ4 interaction.
Faculté des sciences et de médecine
Département de Physique
  • English
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