Abstract Before we dive into the calculation of the first of our critical exponents, it will be helpful for us to consider a little more closely some of the problems that arise from the perturbation expansion we have developed for our correlation functions. This expansion is an expansion in powers of the coupling constant A, whose coefficients are integrals given by the Feynman rules. In §9.4, we were concerned with the behaviour of some of these integrals as we let the wavevector cutoff A tend to infinity.
Abstract Before we dive into the calculation of the first of our critical exponents, it will be helpful for us to consider a little more closely some of the problems that arise from the perturbation expansion we have developed for our correlation functions. This expansion is an expansion in powers of the coupling constant A, whose coefficients are integrals given by the Feynman rules. In §9.4, we were concerned with the behaviour of some of these integrals as we let the wavevector cutoff A tend to infinity.
his book explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition. Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a m
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
The zero-field, two-point correlation function of an n-vector system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions is calculated to order ${\ensuremath{\epsilon}}^{2}$ for T>Tc. The scaling function is obtained as a closed, cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector q, the leading variation is ${{\stackrel{^}{E}}_{1}}^{n,d}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{{\stackrel{^}{E}}_{2}}^{n,d}(q)t$, where $\ensuremath{\alpha}$ is the specific-heat exponent; thence the maximum in the scattering above Tc is located, in good agreement with high-T series-expansion estimates.
The zero-field two-point correlation function of an n-vector system in d dimensions is calculated to order n1 for $T\ensuremath{\ge}{T}_{c}$ and 2<d<4. The critical-point scaling function is obtained as a closed cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector q, the variation of the Fourier transform of the correlation function is ${\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}$, where $\ensuremath{\alpha}$ is the specific-heat exponent and ${\stackrel{^}{E}}_{2}$, ${\stackrel{^}{E}}_{3}$ are of order n1. At d=3 the term t2 is modified by logarithmic corrections. As $q\ensuremath{\rightarrow}0$ at fixed nonzero t, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}$ are of order $\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\ensuremath{\epsilon}$-expansion results of previous work.
This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.
Abstract Until the early 1970s all calculations of critical exponents were made either by exactly solving a model for its thermodynamic properties and then examining its behaviour in the critical regime (as with the Ising model in one and two dimensions) or by direct numerical simulations (as with the three-dimensional Ising model) or by extrapolation from approximate solutions which become invalid in the vicinity of a phase transition (high-temperature expansions, for example). However, in a now-famous paper published in 1966, Kadanoff presented arguments which, he maintained, would allow one to simplify calculations in the critical regime to the point at which critical exponents could be extracted, without ever working out the partition function for the problem. However, though Kadanoff’s ideas showed great physical intuition about the processes giving rise to critical phenomena, they lacked the mathematical precision which might have given one faith in his results.