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Physical review. B, Solid state

New Inequalities among the Critical-Point Exponents for the Spin-Spin and Energy-Energy Correlation Functions

Luke Lokia Liu, Richard I. Joseph, Eugene Stanley

Published Sep 1, 1972• 5 citations
Publisher

Abstract

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 $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\mu}$.

DOI
10.1103/physrevb.6.1963

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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.
Three plausible postulates (rigorous for Ising systems) are shown to lead to three new inequalities: (i) $(2\ensuremath{-}\ensuremath{\eta}){\ensuremath{\mu}}_{\ensuremath{\varphi}}\ensuremath{\ge}\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}$, (ii)${\ensuremath{\mu}}_{\ensuremath{\varphi}}\ensuremath{\ge}\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}$, and (iii)$d{\ensuremath{\mu}}_{\ensuremath{\varphi}}\ensuremath{\ge}\frac{(\ensuremath{\delta}+1)}{\ensuremath{\delta}}$, concerning ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$, the critical-point exponent characterizing the divergence along the critical isotherm of the correlation length ${\ensuremath{\xi}}_{\ensuremath{\varphi}}(T,H)\ensuremath{\equiv}{[\frac{\ensuremath{\Sigma}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{2\ensuremath{\varphi}}{C}_{2}(T,H,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}{\ensuremath{\Sigma}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{}{C}_{2}(T,H,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}]}^{\frac{1}{2\ensuremath{\varphi}}}$. Result (iii) for ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ is an analog, for the critical isotherm, of the Josephson inequalities. If we make the plausible but unproved assumption that ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ is independent of $\ensuremath{\varphi}$, inequality (i) becomes an equality!
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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.