The method of Borel resummation is used to obtain an accurate representation of renormalization-group functions of the n -vector model within the minimal renormalization scheme. This permits to relate the nonuniversal critical behaviour of different quantities well away from criticality. The results can be used to estimate the range of applicability of the n -vector model. Application to the λ-transition of 4 He explains the measured effective exponent of the superfluid density along the λ-line.
The method of Borel resummation is used to obtain an accurate representation of renormalization-group functions of the n -vector model within the minimal renormalization scheme. This permits to relate the nonuniversal critical behaviour of different quantities well away from criticality. The results can be used to estimate the range of applicability of the n -vector model. Application to the λ-transition of 4 He explains the measured effective exponent of the superfluid density along the λ-line.
The dynamic renormalization-group flow equations for model F of Halperin, Hohenberg, and Siggia [Phys. Rev. B 13, 1299 (1976)] are calculated by means of renormalized field theory within the minimal-renormalization scheme up to two-loop order. These equations are combined with the Borel-resummation results for the static renormalization-group functions computed by Schloms and Dohm. The corresponding static fixed point destabilizes the dynamic-scaling fixed point in two-loop order. The nonuniversal initial values of the static and dynamic flow equations are identified for the \ensuremath{\lambda} transition of 4He at various pressures. Predictions are made for the bulk thermal conductivity very close to ${\mathit{T}}_{\ensuremath{\lambda}}$ where the departures from dynamic scaling should be observable. Effective static and dynamic parameters are computed that can be applied to other critical phenomena above and below the \ensuremath{\lambda} line of 4He.
Abstract Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.
Path integrals have played a fundamental role in emphasizing the profound analogies between Quantum Field Theory (QFT), and Classical as well as Quantum Statistical Physics. Ideas coming from Statistical Physics have then led to a deeper understanding of Quantum Field Theory and open the way for a wealth of non-perturbative methods. Conversely QFT methods are become essential for the description of the phase transitions and critical phenomena beyond mean field theory. This is the point we want to illustrate here. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to an accurate determination 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 have been developed, leading to the most accurate results available probing field theory in a non perturbative regime. We illustrate the methods with a short discussion of the scaling equation of state of the 3D Ising model. Compared to exponents its determination involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order-dependent mapping.
Power-law singularities and critical exponents in n-vector models are considered from different theoretical points of view. It includes a theoretical approach called the GFD (grouping of Feynman diagrams) theory, as well as the perturbative renormalization group (RG) treatment. A non-perturbative proof concerning corrections to scaling in the two-point correlation function of the phi^4 model is provided, showing that predictions of the GFD theory rather than those of the perturbative RG theory can be correct. Critical exponents determined from highly accurate experimental data very close to the lambda-transition point in liquid helium, as well as the Goldstone mode singularities in n-vector spin models, evaluated from Monte Carlo simulation results, are discussed with an aim to test the theoretical predictions. Our analysis shows that in both cases the data can be well interpreted within the GFD theory.
Second order phase transitions are investigated with theoretical methods of the phi^4 field theory. Thermodynamic quantities like the correlation length, the susceptibility or the specific heat have singularities in the critical point. The goal is the precise temperature dependence near these singularities. A cooperative many particel effect generates critical behavior. Differences in the short range microscopic interaction of the investigated systems are insignificant. The dimension d and the number of components of the order parameter determine the critical behavior. The renormalization group theory identifies universality classes of critical behavior and predicts that critical exponents and certain amplitude ratios are equal for all systems of the same universality class. The case d=3, n=2 is of special interest. Experimental results of the superfluid transition of 4^He in space are compared to MC simulations of spin systems and analytical calculations in the framework of the phi^4 theory. This is a high precision test of the RG prediction of universality. RG theory has a wide range of applications. Hence, this quantitative test of RG is of interest in statistical physics, condensed matter physics and elementary particle physics. A short summary of two main results: A significant improvement of former Borel summations of critical exponents gamma and alpha is achieved with the same information of perturbation series. This improvement stems from an enhancement of the method of resummation. Amplitude functions at the fixpoint are resummed. Analytical results for certain universal amplitude ratios are achieved. So far only numerical or experimental quantitative results for these ratios are known.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.