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Physical review. B, Condensed matter

Resummation of the 1n expansion through a self-consistent approach

Imre Kondor, T. Temesvári, L. Herényi

Published Aug 1, 1980• 13 citations
Publisher

Abstract

Bray's self-consistent screening approximation for calculating critical exponents is extended to higher orders, resulting in a scheme equivalent to a resummation of the $\frac{1}{n}$ expansion. While the integration techniques borrowed from conformal invariant field theory reduce the task of evaluating Feynman integrals appearing in this scheme to about the level of a straight $\frac{1}{n}$ calculation, the second-order results obtained here for critical exponents as functions of the order-parameter dimensionality $n$ and space dimensionality $d$ are definitely superior to those of a direct $\frac{1}{n}$ calculation and produce acceptable estimates down to about $n\ensuremath{\simeq}1\ensuremath{-}2$ for $d=3$.

DOI
10.1103/physrevb.22.1451

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Bray's self-consistent screening approximation for calculating critical exponents is extended to higher orders, resulting in a scheme equivalent to a resummation of the 1n\frac{1}{n}n1​ expansion. While the integration techniques borrowed from conformal invariant field theory reduce the task of evaluating Feynman integrals appearing in this scheme to about the level of a straight 1n\frac{1}{n}n1​ calculation, the second-order results obtained here for critical exponents as functions of the order-parameter dimensionality nnn and space dimensionality ddd are definitely superior to those of a direct 1n\frac{1}{n}n1​ calculation and produce acceptable estimates down to about n\ensuremath≃1\ensuremath−2n\ensuremath{\simeq}1\ensuremath{-}2n\ensuremath≃1\ensuremath−2 for d=3d=3d=3.
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
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
Details of the calculation to O(1n2)O(\frac{1}{{n}^{2}})O(n21​) of the polarization part and of the vertex function with one ${\ensuremath{\phi}}^{2}$ insertion are given for an nnn-component, three-dimensional system with O(n)\mathrm{O}(n)O(n) symmetric, short-ranged interaction. The critical indices $\ensuremath{\lambda}$ and $\ensuremath{\mu}$ are deduced and are found to conform to other indices available to the same order. An analysis of how the requirements of exponentiation and universality are fulfilled shows that they are consequences of the fulfillment of a scaling law at the previous order. Cutoff-dependent contributions are seen to split off automatically. With possible higher-order calculations in mind considerable stress is put on the technical side: An apparently novel type of regularization technique allows one to pick up the relevant terms directly and is shown to be connected to a kind of analytic regularization. Graphs with diagonal insertions are calculated from generalized Feynman amplitudes associated with first-order diagrams. Those with vertex insertions, as well as irreducible graphs are handled by an integration technique taken over from conformal invariant field theory. The numerical performance of the 1n\frac{1}{n}n1​ expansion is discussed, and an attempt is made to obtain more meaningful numbers from the expansion through various Pad\'e- and Pad\'e---Borel-type resummations. The results are compared with information available from other sources.
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.
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.
Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.