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Physical Review Letters

Feynman-Graph Expansion for Critical Exponents

Kenneth Geddes Wilson

Published Feb 28, 1972• 654 citations
Publisher

Abstract

The critical exponents $\ensuremath{\gamma}$, $\ensuremath{\eta}$ and the "crossover index" $\ensuremath{\phi}$ are computed for generalized classical Heisenberg models with $n$ internal degrees of freedom as an exact expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ ($d$ is the number of space dimensions). Results are obtained to order ${\ensuremath{\epsilon}}^{2}$ for $\ensuremath{\gamma}$ and to order ${\ensuremath{\epsilon}}^{3}$ for $\ensuremath{\eta}$. The results to this order for the three-dimensional Ising case ($n=\ensuremath{\epsilon}=1$) are $\ensuremath{\gamma}=1.244$ and $\ensuremath{\eta}=0.037$.

DOI
10.1103/physrevlett.28.548

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Feynman-Graph Expansion for the Equation of State near the Critical Point (Ising-like Case)

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Feynman-Graph Expansion for the Equation of State near the Critical Point

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Critical Exponents from Other Expansions

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

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Critical exponents and equation of state of the three-dimensional Heisenberg universality class

Massimo Campostrini B, Martin Hasenbusch B, Andrea Pelissetto C +2 more

We improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find $\ensuremath{\gamma}=1.3960(9),$ $\ensuremath{\nu}=0.7112(5),$ $\ensuremath{\eta}=0.0375(5),$ $\ensuremath{\alpha}=\ensuremath{-}0.1336(15),$ $\ensuremath{\beta}=0.3689(3),$ and $\ensuremath{\delta}=4.783(3).$ We consider an improved lattice ${\ensuremath{\varphi}}^{4}$ Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the ${\ensuremath{\varphi}}^{4}$ improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.

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Accurate critical exponents from field theory

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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.

Journal de physique•Jan 1, 1989•122 citations
The critical exponents \ensuremathγ\ensuremath{\gamma}\ensuremathγ, \ensuremathη\ensuremath{\eta}\ensuremathη and the "crossover index" \ensuremathϕ\ensuremath{\phi}\ensuremathϕ are computed for generalized classical Heisenberg models with nnn internal degrees of freedom as an exact expansion in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d (ddd is the number of space dimensions). Results are obtained to order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2 for \ensuremathγ\ensuremath{\gamma}\ensuremathγ and to order \ensuremathϵ3{\ensuremath{\epsilon}}^{3}\ensuremathϵ3 for \ensuremathη\ensuremath{\eta}\ensuremathη. The results to this order for the three-dimensional Ising case (n=\ensuremathϵ=1n=\ensuremath{\epsilon}=1n=\ensuremathϵ=1) are \ensuremathγ=1.244\ensuremath{\gamma}=1.244\ensuremathγ=1.244 and \ensuremathη=0.037\ensuremath{\eta}=0.037\ensuremathη=0.037.
The scaling equation of state of an Ising-like ferromagnet is derived by an expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where ddd is the dimension of space. The result is compared with numerical calculations on the three-dimensional Ising model. It is also established that the "linear model" is exact up to order ${\ensuremath{\epsilon}}^{2}$.
The scaling equation of state for a generalized classical Heisenberg ferromagnet near the critical point is derived by an expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where ddd is the dimension of space. It is shown that, though infrared divergences are induced by the Goldstone modes, the equation of state is divergence free. The results are compared with previous numerical calculations. It is also shown that, for non-Ising-like systems the "linear model" cannot be exact, even at first order in $\ensuremath{\epsilon}$ (although the numerical deviations from linearity are small).
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
We improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find $\ensuremath{\gamma}=1.3960(9),$ $\ensuremath{\nu}=0.7112(5),$ $\ensuremath{\eta}=0.0375(5),$ $\ensuremath{\alpha}=\ensuremath{-}0.1336(15),$ $\ensuremath{\beta}=0.3689(3),$ and $\ensuremath{\delta}=4.783(3).$ We consider an improved lattice ${\ensuremath{\varphi}}^{4}$ Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the ${\ensuremath{\varphi}}^{4}$ improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.
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.