Unified theory of critical exponents generated by the Ising Hamiltonian for discrete dimensionalities 2, 3 and 4 in terms of the critical exponent η. (2nd January 2016)

• Record Type: Journal Article
• Authors: March, N.H.
• Is Part Of: Physics and chemistry of liquids. Volume 54:Number 1(2016)
• Page Start: 127
• Page End: 129
• Publication Date: 2016-01-02
• Subjects: Ising model -- liquid‒vapour critical point -- critical exponents -- discrete dimensionalities; Liquids -- Periodicals; 530.4205
• DOI: 10.1080/00319104.2015.1058943
• Languages: English
• ISSNs: 0031-9104

Abstract

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.