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Physical review. B, Condensed matter

Calculation of critical exponents to O(1n2)

Imre Kondor, T. Temesvári

Published Jan 1, 1980• 6 citations
Publisher

Abstract

Details of the calculation to $O(\frac{1}{{n}^{2}})$ of the polarization part and of the vertex function with one ${\ensuremath{\phi}}^{2}$ insertion are given for an $n$-component, three-dimensional system with $\mathrm{O}(n)$ 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 $\frac{1}{n}$ 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.

DOI
10.1103/physrevb.21.260

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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 $\frac{1}{n}$ 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 $d=3$.

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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 behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

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The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of ${R}^{d}.$ Our aim is to sort out which ones of the previously published partly contradictory \ensuremath{\epsilon}-expansion results to second order in $\ensuremath{\epsilon}=4+m/2\ensuremath{-}d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+m/2\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents ${\ensuremath{\eta}}_{l2}$ and ${\ensuremath{\eta}}_{l4}$ and of the wave-vector exponent ${\ensuremath{\beta}}_{q}$ to order ${\ensuremath{\epsilon}}^{2}$ are reduced to single integrals, which for general $m=1,\dots{},d\ensuremath{-}1$ can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For $m=2$ and $m=6,$ we confirm the results of Sak and Grest [Phys. Rev. B 17, 3602 (1978)] and Mergulh\~ao and Carneiro's recent field-theory analysis [Phys. Rev. B 59, 13 954 (1999)].

arXiv:cond-mat/0006007•Physical review. B, Condensed matter•Nov 1, 2000•66 citations
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 \ensuremathϕ2{\ensuremath{\phi}}^{2}\ensuremathϕ2 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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ and \ensuremathμ\ensuremath{\mu}\ensuremathμ 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.
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 $n\ensuremath{\simeq}1\ensuremath{-}2$ for d=3d=3d=3.
Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad\'e-Borel technique is used for the resummation of the RG series and the Pad\'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.
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 critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of Rd.{R}^{d}.Rd. Our aim is to sort out which ones of the previously published partly contradictory \ensuremath{\epsilon}-expansion results to second order in $\ensuremath{\epsilon}=4+m/2\ensuremath{-}d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+m/2\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents ${\ensuremath{\eta}}_{l2}$ and ${\ensuremath{\eta}}_{l4}$ and of the wave-vector exponent ${\ensuremath{\beta}}_{q}$ to order ${\ensuremath{\epsilon}}^{2}$ are reduced to single integrals, which for general $m=1,\dots{},d\ensuremath{-}1$ can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For m=2m=2m=2 and m=6,m=6,m=6, we confirm the results of Sak and Grest [Phys. Rev. B 17, 3602 (1978)] and Mergulh\~ao and Carneiro's recent field-theory analysis [Phys. Rev. B 59, 13 954 (1999)].