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Journal of Applied Physics

Exponent Inequalities for Critical Point Spin Correlation Functions

Michael Ellis Fisher

Published Mar 1, 1969• 8 citations
Publisher

Abstract

Simple but rigorous derivations are presented for the inequalities2−η≤dγ′/(2β+γ′)≤dγ′/(2−α′),2−η≤d(δ−1)/(δ+1),γ≤(2−η)ν.(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 ∑ |r|<RΓ2(r)∼R2−η(T=Tc,H=0,R→∞),κ1∼(T−Tc)ν,(T→Tc,H=0),where the inverse range of correlation κ1 may be defined by κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]. The corresponding exponent ηE for the energy-energy correlation function (essentially Γ4(r, a)−[Γ(a)]2, where Γ4(r, a)=〈S0zSazSrzSr+az〉) is proved to satisfy 2−ηE≤dα′,2−ηE≤dαc/(1+ζ+αc),where the specific heat CM at T=Tc diverges with the magnetization as M−αc while the energy derivative | ∂U/∂M |Tc varies as Mζ. (Mean Field or classical values are αc=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 (Jij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.

DOI
10.1063/1.1657627

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Simple but rigorous derivations are presented for the inequalities2−η≤dγ′/(2β+γ′)≤dγ′/(2−α′),2−η≤d(δ−1)/(δ+1),γ≤(2−η)ν.(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 ∑ |r|<RΓ2(r)∼R2−η(T=Tc,H=0,R→∞),κ1∼(T−Tc)ν,(T→Tc,H=0),where the inverse range of correlation κ1 may be defined by κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]. The corresponding exponent ηE for the energy-energy correlation function (essentially Γ4(r, a)−[Γ(a)]2, where Γ4(r, a)=〈S0zSazSrzSr+az〉) is proved to satisfy 2−ηE≤dα′,2−ηE≤dαc/(1+ζ+αc),where the specific heat CM at T=Tc diverges with the magnetization as M−αc while the energy derivative | ∂U/∂M |Tc varies as Mζ. (Mean Field or classical values are αc=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 (Jij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.
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.
Two new inequalities, (i) ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\ge}\frac{2\ensuremath{\beta}(2\ensuremath{-}\ensuremath{\eta})}{(d\ensuremath{-}2+\ensuremath{\eta})}+2\ensuremath{\varphi}[\frac{2\ensuremath{\beta}}{(d\ensuremath{-}2+\ensuremath{\eta})}\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}]$ and (ii) $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}\frac{2(2\ensuremath{-}\ensuremath{\eta})}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}+2\ensuremath{\varphi}[\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}\ensuremath{-}{\ensuremath{\mu}}_{\ensuremath{\varphi}}]$, are derived among critical-point exponents that describe the behavior of the two-spin correlation function ${C}_{2}(T, H, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})\ensuremath{\equiv}〈{s}_{0}^{z}{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉\ensuremath{-}〈{s}_{0}^{z}〉〈{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉$, subject to plausible assumptions (rigorous for Ising magnets). Here ${\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}$ and ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ describe the divergence as $T\ensuremath{\rightarrow}{T}_{c}^{\ensuremath{-}}$ and as $H\ensuremath{\rightarrow}{0}^{+}$, respectively, of the "generalized correlation length" ${\ensuremath{\xi}}_{\ensuremath{\varphi}}(T, H)$, defined as the $2\ensuremath{\varphi}\mathrm{th}$ root of the normalized $2\ensuremath{\varphi}\mathrm{th}$ spatial moment of ${C}_{2}(T, H, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. Also derived are the corresponding inequalities among exponents that describe the behavior of the energy-energy correlation function. Inequality (i) is shown to lead to an inequality between primed and unprimed exponents. Moreover, if ${\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}$ is independent of $\ensuremath{\varphi}$, then (i) implies that ${\ensuremath{\nu}}^{\ensuremath{'}}\ensuremath{\ge}\frac{2\ensuremath{\beta}}{(d\ensuremath{-}2+\ensuremath{\eta})}$ and ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta}){\ensuremath{\nu}}^{\ensuremath{'}}$, while if ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ is independent of $\ensuremath{\varphi}$, then (ii) implies $\ensuremath{\mu}\ensuremath{\ge}\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}$ and $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\mu}$.
An analysis is made of the fluctuations in a finite part of an infinite, ddd-dimensional ferromagnetic Ising lattice at and near the critical point. This leads to a formal proof that $\frac{2d}{(\ensuremath{\delta}+1)}$ and $\frac{2d\ensuremath{\beta}}{({\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta})}$ are lower bounds on the exponent characterizing the asymptotic decrease with distance of the two-body correlation function at the critical point, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ are exponents representing the asymptotic forms of the coexistence curve, critical isotherm, and isothermal susceptibility.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
FINDING: 2D Ising model critical exponents are rational fractions derived from exact solution, with √2 appearing in the spontaneous magnetization exponent. MATH: - Critical exponents: α = 0 (log divergence), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4 - Spontaneous magnetization: M ∝ (T_c - T)^β, β = 1/8 = 0.125 - Correlation length exponent: ν = 1 - Susceptibility exponent: γ = 7/4 = 1.75 - Specific heat exponent: α = 0 (logarithmic) - Exact solution via Onsager (1944): free energy expressed in terms of elliptic integrals, critical temperature T_c = 2/ln(1+√2) ≈ 2.269 J/k_B - √2 appears in T_c formula and in β = 1/8 (since 1/8 = (1/2)^3, but deeper: β = 1/8 emerges from algebraic order parameter scaling involving √2 in the lattice combinatorics) CONNECTION: - β = 1/8 = 0.125 is a rational fraction, not directly a golden ratio, but the critical temperature T_c = 2/ln(1+√2) involves √2, linking to square lattice symmetry (crystallographic point group D4). - The exponent Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com