Dynamical Mean Field Approximation Applied to Quantum Field Theory

Abstract

We apply the Dynamical Mean Field (DMFT) approximation to the real, scalar phi^4 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 nu and beta. We also map out the phase diagram in 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 phi^4-theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behaviour 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 less intensive. We also discuss technical issues related to the convergence of the non-linear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a phi^4-interaction.

15 pages, 10 figures, Version to appear in PRD. Changes include: Comparison to cluster extensions of Mean Field Theory, one figure added and one replaced, clarified description of method and extended discussion on possible extensions

References in corpus (4)

• Continuous-time Monte Carlo methods for quantum impurity models
• Correlated bosons on a lattice: Dynamical mean-field theory for Bose-Einstein condensed and normal phases
• A cluster algorithm for resistively shunted Josephson junctions
• Quantum-Classical Transition in Dissipative Double-Well Systems --A Numerical Study by a New Monte Carlo Scheme--

Cited by in corpus (9)

• Nonequilibrium dynamical mean-field theory for bosonic lattice models
• Dynamical Mean-Field Theory Simulations with the Adaptive Sampling Configuration Interaction Method
• Non-local correlations in the orbital selective Mott phase of a one dimensional multi-orbital Hubbard model
• Mean-field phase diagram of the Bose-Fermi Hubbard model
• Dynamical mean-field study of a photon-mediated ferroelectric phase transition
• Spin liquid to spin glass crossover in the random quantum Heisenberg magnet
• Phase transition of four-dimensional lattice ${\varphi }^4$ theory with tensor renormalization group
• Extended Mean Field study of complex ${\varphi }^4$-theory at finite density and temperature
• Higgs-Yukawa model with higher dimension operators via extended mean field theory