In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1
It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in
A numerical technique, termed the scaling-field method, is developed for solving by successive approximation Wilson's exact renormalization-group equation for critical phenomena in three-dimensional spin systems. The approach uses the scaling-field representation of the Wilson equation derived by Riedel, Golner, and Newman. A procedure is proposed for generating in a nonperturbative and unbiased fashion sequences of successively larger truncations to the infinite hierarchy of scaling-field equations. A "principle of balance" is introduced and used to provide a self-consistency criterion. The approach is then applied to the isotropic
N-vector model. Truncations to order 13 (10, when
N=1) scaling-field equations yield the leading critical exponents, $\ensuremath{\nu}$ and $\ensuremath{\eta}$, and several of the correction-to-scaling exponents, ${\ensuremath{\Delta}}_{m}$, to high precision. Results for
N=0,1,2,and3 are tabulated. For the Ising case (
N=1), the estimates $\ensuremath{\nu}=0.626\ifmmode\pm\else\textpm\fi{}0.009$, $\ensuremath{\eta}=0.040\ifmmode\pm\else\textpm\fi{}0.007$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\equiv}{\ensuremath{\Delta}}_{400}=0.54\ifmmode\pm\else\textpm\fi{}0.05$ are in good agreement with recent high-temperature-series results, though exhibiting larger confidence limits at the present level of approximation. For the first time, estimates are obtained for the second and third correction-to-scaling exponents. For example, for the Ising model the second "even" and first "odd" correction-to-scaling exponents are ${\ensuremath{\Delta}}_{422}=1.67\ifmmode\pm\else\textpm\fi{}0.11$ and ${\ensuremath{\Delta}}_{500}=1.5\ifmmode\pm\else\textpm\fi{}0.3$, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic
N-vector models are described elsewhere. Finally, the scaling-field method is compared with other techniques for the high-precision calculation of critical phenomena in three dimensions, i.e., high-temperature-series, Monte Carlo renormalization-group, and field-theoretic perturbation expansions.
The critical behavior of the random-field Ising model has long been a puzzle. Different methods predict that its critical exponents in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components contrary to the experiments and simulations. We calculate the exponents of the random-field O(N) model with the (4+epsilon)-expansion and obtain values different from the exponents of the pure ferromagnet in 2+epsilon dimensions. An infinite set of relevant operators missed in previous studies leads to a breakdown of the (6-epsilon)-expansion.
The critical behavior of the random-field Ising model has been a puzzle for a long time. Different theoretical methods predict that the critical exponents of the random-field ferromagnet in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components. This result contradicts the experiments and simulations. We calculate the critical exponents of the random-field O(N) model with the (4+ε)-expansion and obtain values different from the critical exponents of the pure ferromagnet in 2+εdimensions. In contrast to the previous approaches we take into account an infinite set of relevant operators emerging in the problem. We demonstrate how these previously missed relevant operators lead to the breakdown of the (6-ε)-expansion for the random-field Ising model.
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.