Determination of Critical Exponents and Equation of State by Field Theory Methods

Abstract

We review here the methods, based on renormalized φ3 4 quantum field theory and renormalization group, which have led to an accurate calculation of critical exponents of the N-vector model, and more recently of the equation of state of the 3D Ising model. The starting point is the perturbative expansion for RG functions or the effective potential to the order presently available. Perturbation theory is known to be divergent and its divergence has been related to instanton contributions. This has allowed to characterize the large order behaviour of perturbation series, an information that can be used to efficiently “sum” them. Practical summation methods based on Borel transformation and conformal mapping has been developed, leading to the most accurate results available probing field theory in a non-perturbative regime. We illustrate the methods with a detailed discussion of the scaling equation of state of the 3D Ising model 1. Compared to exponents its determination involves a few additional (non-trivial) technical steps.A general reference on the topic is J. Zinn-Justin, 1989, Quantum Field Theory and Critical Phenomena, in particular chap. 28 of third ed., Clarendon Press (Oxford 1989, third ed. 1996). Many relevant articles are reprinted in Large Order Behaviour of Perturbation Theory, Current Physics vol. 7, J.C. Le Guillou and J. Zinn-Justin eds., (North-Holland, Amsterdam 1990). KeywordsRenormalization GroupCritical ExponentAmplitude RatioParametric RepresentationPerturbative ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.