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Physics and Chemistry of Liquids

Proposed relationship between the exponentsγandηat liquid–vapour critical pointviathe dimensionalityd

Zhang Zhi-dong, Norman Henry March

Published Mar 1, 2011• 5 citations
Publisher

Abstract

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.

DOI
10.1080/00319104.2010.499514

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arXiv:cond-mat/0211270•Nov 14, 2002•0 citations
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.
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.
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.
Critical behavior and exponents above the $\ensuremath{\lambda}$-transition points in charged and neutral Bose systems are examined with emphasis on the correlation of density fluctuations. Exponents are calculated with Wilson's expansion methods for arbitrary dtoO(1m)d \mathrm{to} O(\frac{1}{m})dtoO(m1​), where ddd is the dimension and mmm the number of components of the Bose field. Some results for arbitrary $m \mathrm{to} O({\ensuremath{\epsilon}}^{2})(\ensuremath{\epsilon}=4\ensuremath{-}d)$ are also obtained.
The definitions and determination of the thermodynamic magnetic critical point exponents α, α′, β, γ, γ′ and δ are reviewed, and ``gap exponents'' Δ and Δ′ for the divergences of higher derivatives of the free energy with respect to the field are introduced. The rigorous inequalities relating the exponents are discussed. The ``microdomain'' or ``cluster'' model of ferromagnetism leads to relations between these exponents since it expresses them in terms of two underlying exponents σ and τ. The exponent σ is directly related to the effective surface free energy of a large microdomain and thence to the dimensionality. This yields some insight into the meaning of the observed exponent values and of the differences between the critical behavior of the Ising and Heisenberg models. The recently developed homogeneity and scaling arguments are presented. These generate the same thermodynamic exponent relations and further ones for the correlation exponents η, ν and ν′. They also indicate that a general law of corresponding states might hold in the critical region. Theoretical and experimental evidence on the validity of these various ideas is briefly reviewed.
A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.