Precise determination of critical exponents and equation of state by field theory methods

Renormalization group, and in particular its quantum field theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized phi (4)(3) quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N-vector model (Le Guillou and Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95; Phys. Rev. B 21 (1980) 3976: Guida and Zinn-Justin. J. Phys. A 31 (1998) 8103; cond-mat/9803210) and of the equation of state of;he 3D Ising model (Guida and Zinn-Justin, Nucl. Phys. B 489 [FS] (1997) 626, hep-th/9610223), These results are among the most precise available probing field theory in a non-perturbative regime. Precise calculations first require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large-order behaviour of perturbation series has then allowed to develop efficient summation techniques, based on Borel transformation and conformal mapping (Le Guillou and Zinn-Justin (Eds.), Large Order Behaviour of Perturbation Theory: Current Physics, vol. 7, North-Holland, Amsterdam, 1990). We first discuss exponents and describe our recent results (Guida and Zinn-Justin, 1998). Compared to exponents, the determination of the scaling equation of stale of the 3D Ising, model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived. Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer resources, many new calculations will become feasible. In addition to the infinite volume quantities, finite size universal quantities would also be of interest, to provide a more direct contact with numerical simulations. Let us also mention dynamical observables, a largely unexplored territory. (C) 2001 Elsevier Science B.V. All rights reserved.