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Physics and Chemistry of Liquids

Crucial combinations of critical exponents for liquids–vapour and ferromagnetic second-order phase transitions

Norman Henry March

Published Apr 24, 2014• 7 citations
Publisher

Abstract

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.

DOI
10.1080/00319104.2014.906600

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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.
In earlier work, Ma [S.K. MA, Phys. Rev. Lett., 29, 1311 (1972)] has studied the critical exponents γ and η for charged and neutral Bose gases. Here we use the result of Ma, valid for general dimensionality d but only to O(m −1), where m is the number of components of the Bose field, to write a relation between γ(d) and η(d) to O(m −1). This then motivates, but now for the Ising model, a relationship between the critical exponents γ and η, via the dimensionality d. We finally demonstrate a connection between the two renormalisation group eigenvalues y t and y h , via the critical exponent δ with a dimensional dependence.
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.
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.
We have tested the theoretical values of critical exponents, predicted for the three--dimensional Heisenberg model, based on the published Monte Carlo (MC) simulation data for the susceptibility. Two different sets of the critical exponents have been considered - one provided by the usual (perturbative) renormalization group (RG) theory, and another predicted by grouping of Feynman diagrams in phi^4 model (our theory). The test consists of two steps. First we determine the critical coupling by fitting the MC data to the theoretical expression, including both confluent and analytical corrections to scaling, the values of critical exponents being taken from theory. Then we use the obtained value of critical coupling to test the agreement between theory and MC data at criticality. As a result, we have found that predictions of our theory (gamma=19/14, eta=1/10, omega=3/5) are consistent, whereas those of the perturbative RG theory (gamma=1.3895, eta=0.0355, omega=0.782) are inconsistent with the MC data. The seemable agreement between the RG prediction for eta and MC results at criticality, reported in literature, appears due to slightly overestimated value of the critical coupling. Estimation of critical exponents of 3D Ising model from complex zeroth of the partition function is discussed. A refined analysis yields the best estimate 1/nu=1.518. We conclude that the recent MC data can be completely explained within our theory (providing 1/nu=1.5 and omega=0.5) rather than within the conventional RG theory.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.