Exponent Inequalities for Critical Point Spin Correlation Functions Available to Purchase

Simple but rigorous derivations are presented for the inequalities(Proofs of the first two inequalities have been given by Buckingham and Gunton.¹) Here d is the dimensionality and α′, β, γ, γ′ and δ are the exponents characterizing a ferromagnet near its critical point,² while η and ν describe the decay of the corresponding spin‐spin correlation functions² $Γ2(r)=〈S0zSrz〉$⁠, according to ,where the inverse range of correlation κ₁ may be defined by $κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]$⁠. The corresponding exponent η_E for the energy‐energy correlation function (essentially $Γ4(r, a)−[Γ(a)]2$⁠, where $Γ4(r, a)=〈S0zSazSrzSr+az〉$⁠) is proved to satisfy ,where the specific heat C_M at T=T_c diverges with the magnetization as M^−αc while the energy derivative $| ∂U/∂M |Tc$ varies as M^ζ. (Mean Field or classical values are α_c=0, ζ=1.) The proofs are based on the ``intuitively obvious'' properties: (a) positivity, namely, $Γ1=〈S0z〉≥0,Γ(r)−Γ12≥0,Γ4(r, a)−[Γ2(a)]2≥0$⁠, and (b) monotonic increase of Γ₁, Γ₂ and Γ₄ under increase of magnetic field and decrease of temperature. These properties are known³ to be rigorously valid for Ising models of arbitrary spin, lattice structure and ferromagnetic coupling (J_ij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.