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Physics and Chemistry of Liquids

Generalisation of Zhang’s predictions for the critical exponents of the 3dIsing model ifα≠ 0

Norman Henry March

Published Jun 24, 2014• 3 citations
Publisher

Abstract

In an important study, Z. D. Zhang proposed a precise set of critical exponents for the 3d Ising model. The presently intractable mathematics of this model was bypassed via two conjectures. Because this has raised some controversy, we here propose a generalisation if the exponent should prove to be different from zero. As a result, the critical exponent may be worthy of further experimental study.

DOI
10.1080/00319104.2014.931009

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•Jan 1, 2008•12 citations
In an important study, Z. D. Zhang proposed a precise set of critical exponents for the 3d Ising model. The presently intractable mathematics of this model was bypassed via two conjectures. Because this has raised some controversy, we here propose a generalisation if the exponent should prove to be different from zero. As a result, the critical exponent may be worthy of further experimental study.
Experimental data for critical exponents in some magnetic materials are compared with recent theoretical results on the three-dimensional (3D) Ising model, as derived by one of us (ZDZ) based on two conjectures [Z.D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag. 87 (2007), pp. 5309–5419]. It is found that critical exponents in some bulk magnetic materials indeed form a 3D Ising universality. Our attention is then focused on the critical indices at fluid–fluid phase transition. We suggest to use Zhang's exponent β = 3/8 to fit the experimental data over the wider asymptotic region near the critical point of a fluid–fluid phase transition. The 3D Ising universality should exist for critical indices in a certain class of magnets and at fluid–fluid phase transition.
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.
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.
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
An analysis of the critical behavior of the three-dimensional Ising model using the coherent-anomaly method (CAM) is presented. Various sources of errors in CAM estimates of critical exponents are discussed, and an improved scheme for the CAM data analysis is tested. Using a set of mean-field type approximations based on the variational series expansion approach, accuracy comparable to the most precise conventional methods has been achieved. Our results for the critical exponents are given by α = 0.108(5), β = 0.327(4), γ = 1.237(4) and δ = 4.77(5).