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Physics and Chemistry of Liquids

Unified theory of critical exponents generated by the Ising Hamiltonian for discrete dimensionalities 2, 3 and 4 in terms of the critical exponentη

Norman Henry March

Published Sep 4, 2015• 3 citations
Publisher

Abstract

We present here a unified theory embracing solely the discrete values d = 2, 3 and 4. The critical exponent η is central of this theory. The basic assumption made is that, for these three discrete values, we can express the critical exponent γ in the form . This formula contains the exact values for d = 2 and 4, namely 7/4 and 1, respectively. For d = 3, this formula yields the value 5/4 obtained earlier by Zhang (Philo. Mag. 87 (2007) 5309) from an analytical theory in which the mathematical difficulties associated with the Ising Hamiltonian were bypassed by two clearly stated conjectures. Earlier, K.G. Wilson (Phys. Rev. Lett. 28 (1972) 548), however, had proposed γ = 1.247 from the low-order terms in the -expansion. Kardar’s book contains the value γ = 1.238 obtained by Borel summation of the epsilon expansion. Thus, the critical exponents given here for d = 2, 3 and 4 in terms of η and of course d itself are either exact or of such an accuracy that experimental error should embrace them.

DOI
10.1080/00319104.2015.1058943

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We present here a unified theory embracing solely the discrete values d = 2, 3 and 4. The critical exponent η is central of this theory. The basic assumption made is that, for these three discrete values, we can express the critical exponent γ in the form . This formula contains the exact values for d = 2 and 4, namely 7/4 and 1, respectively. For d = 3, this formula yields the value 5/4 obtained earlier by Zhang (Philo. Mag. 87 (2007) 5309) from an analytical theory in which the mathematical difficulties associated with the Ising Hamiltonian were bypassed by two clearly stated conjectures. Earlier, K.G. Wilson (Phys. Rev. Lett. 28 (1972) 548), however, had proposed γ = 1.247 from the low-order terms in the -expansion. Kardar’s book contains the value γ = 1.238 obtained by Borel summation of the epsilon expansion. Thus, the critical exponents given here for d = 2, 3 and 4 in terms of η and of course d itself are either exact or of such an accuracy that experimental error should embrace them.
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.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.
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
Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.
Four combinations of critical exponents pertaining to the Ising model are constructed which yield simply the dimensionality d. One of these, connecting γ and η, is shown to lead to an immediately useful estimates of the sum α + η in three dimensions, about which exponents controversy still exists. This sum in three dimensions is shown to be very near or equal to 1/8, a condition which Zhang’s predictions, for the exact solution of the 3D Ising model, satisfy with α = 0 and η = 1/8. But the pioneering work of Wilson based on renormalisation group theory gave the estimate η = 0.037, while the Perk advocates α = 0.1, without however, proceeding a proof. We therefore finally stress the current importance of further experimental measurements. The critical exponent δ may be the most favourable focus, to distinguish between Zhang’s prediction of 13/3 and a value around 4.8 following for d = 3 from Wilson’s value of η quoted above.