Griffiths inequality

In statistical mechanics, the Griffiths inequality, sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality, named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.

The inequality was proved by Griffiths for Ising ferromagnets with two-body interactions,^[1] then generalised by Kelly and Sherman to interactions involving an arbitrary number of spins,^[2] and then by Griffiths to systems with arbitrary spins.^[3] A more general formulation was given by Ginibre,^[4] and is now called the Ginibre inequality.

Let ${\displaystyle \textstyle \sigma =\{\sigma _{j}\}_{j\in \Lambda }}$ be a configuration of (continuous or discrete) spins on a lattice Λ. If A ⊂ Λ is a list of lattice sites, possibly with duplicates, let ${\displaystyle \textstyle \sigma _{A}=\prod _{j\in A}\sigma _{j}}$ be the product of the spins in A.

Assign an a-priori measure dμ(σ) on the spins; let H be an energy functional of the form

${\displaystyle H(\sigma )=-\sum _{A}J_{A}\sigma _{A}~,}$

where the sum is over lists of sites A, and let

${\displaystyle Z=\int d\mu (\sigma )e^{-H(\sigma )}}$

be the partition function. As usual,

${\displaystyle \langle \cdot \rangle ={\frac {1}{Z}}\sum _{\sigma }\cdot (\sigma )e^{-H(\sigma )}}$

stands for the ensemble average.

The system is called ferromagnetic if, for any list of sites A, J_A ≥ 0. The system is called invariant under spin flipping if, for any j in Λ, the measure μ is preserved under the sign flipping map σ → τ, where

${\displaystyle \tau _{k}={\begin{cases}\sigma _{k},&k\neq j,\\-\sigma _{k},&k=j.\end{cases}}}$

In a ferromagnetic spin system which is invariant under spin flipping,

${\displaystyle \langle \sigma _{A}\rangle \geq 0}$

for any list of spins A.

In a ferromagnetic spin system which is invariant under spin flipping,

${\displaystyle \langle \sigma _{A}\sigma _{B}\rangle \geq \langle \sigma _{A}\rangle \langle \sigma _{B}\rangle }$

for any lists of spins A and B.

The first inequality is a special case of the second one, corresponding to B = ∅.

Observe that the partition function is non-negative by definition.

Proof of first inequality: Expand

${\displaystyle e^{-H(\sigma )}=\prod _{B}\sum _{k\geq 0}{\frac {J_{B}^{k}\sigma _{B}^{k}}{k!}}=\sum _{\{k_{C}\}_{C}}\prod _{B}{\frac {J_{B}^{k_{B}}\sigma _{B}^{k_{B}}}{k_{B}!}}~,}$

then

${\displaystyle {\begin{aligned}Z\langle \sigma _{A}\rangle &=\int d\mu (\sigma )\sigma _{A}e^{-H(\sigma )}=\sum _{\{k_{C}\}_{C}}\prod _{B}{\frac {J_{B}^{k_{B}}}{k_{B}!}}\int d\mu (\sigma )\sigma _{A}\sigma _{B}^{k_{B}}\\&=\sum _{\{k_{C}\}_{C}}\prod _{B}{\frac {J_{B}^{k_{B}}}{k_{B}!}}\int d\mu (\sigma )\prod _{j\in \Lambda }\sigma _{j}^{n_{A}(j)+k_{B}n_{B}(j)}~,\end{aligned}}}$

where n_A(j) stands for the number of times that j appears in A. Now, by invariance under spin flipping,

${\displaystyle \int d\mu (\sigma )\prod _{j}\sigma _{j}^{n(j)}=0}$

if at least one n(j) is odd, and the same expression is obviously non-negative for even values of n. Therefore, Z<σ_A>≥0, hence also <σ_A>≥0.

Proof of second inequality. For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin, ${\displaystyle \sigma '}$, with the same distribution of ${\displaystyle \sigma }$. Then

${\displaystyle \langle \sigma _{A}\sigma _{B}\rangle -\langle \sigma _{A}\rangle \langle \sigma _{B}\rangle =\langle \langle \sigma _{A}(\sigma _{B}-\sigma '_{B})\rangle \rangle ~.}$

Introduce the new variables

${\displaystyle \sigma _{j}=\tau _{j}+\tau _{j}'~,\qquad \sigma '_{j}=\tau _{j}-\tau _{j}'~.}$

The doubled system ${\displaystyle \langle \langle \;\cdot \;\rangle \rangle }$ is ferromagnetic in ${\displaystyle \tau ,\tau '}$ because ${\displaystyle -H(\sigma )-H(\sigma ')}$ is a polynomial in ${\displaystyle \tau ,\tau '}$ with positive coefficients

${\displaystyle {\begin{aligned}\sum _{A}J_{A}(\sigma _{A}+\sigma '_{A})&=\sum _{A}J_{A}\sum _{X\subset A}\left[1+(-1)^{|X|}\right]\tau _{A\setminus X}\tau '_{X}\end{aligned}}}$

Besides the measure on ${\displaystyle \tau ,\tau '}$ is invariant under spin flipping because ${\displaystyle d\mu (\sigma )d\mu (\sigma ')}$ is. Finally the monomials ${\displaystyle \sigma _{A}}$, ${\displaystyle \sigma _{B}-\sigma '_{B}}$ are polynomials in ${\displaystyle \tau ,\tau '}$ with positive coefficients

${\displaystyle {\begin{aligned}\sigma _{A}&=\sum _{X\subset A}\tau _{A\setminus X}\tau '_{X}~,\\\sigma _{B}-\sigma '_{B}&=\sum _{X\subset B}\left[1-(-1)^{|X|}\right]\tau _{B\setminus X}\tau '_{X}~.\end{aligned}}}$

The first Griffiths inequality applied to ${\displaystyle \langle \langle \sigma _{A}(\sigma _{B}-\sigma '_{B})\rangle \rangle }$ gives the result.

More details are in ^[5] and.^[6]

The Ginibre inequality is an extension, found by Jean Ginibre,^[4] of the Griffiths inequality.

Let (Γ, μ) be a probability space. For functions f, h on Γ, denote

${\displaystyle \langle f\rangle _{h}=\int f(x)e^{-h(x)}\,d\mu (x){\Big /}\int e^{-h(x)}\,d\mu (x).}$

Let A be a set of real functions on Γ such that. for every f₁,f₂,...,f_n in A, and for any choice of signs ±,

${\displaystyle \iint d\mu (x)\,d\mu (y)\prod _{j=1}^{n}(f_{j}(x)\pm f_{j}(y))\geq 0.}$

Then, for any f,g,−h in the convex cone generated by A,

${\displaystyle \langle fg\rangle _{h}-\langle f\rangle _{h}\langle g\rangle _{h}\geq 0.}$

Let

${\displaystyle Z_{h}=\int e^{-h(x)}\,d\mu (x).}$

Then

${\displaystyle {\begin{aligned}&Z_{h}^{2}\left(\langle fg\rangle _{h}-\langle f\rangle _{h}\langle g\rangle _{h}\right)\\&\qquad =\iint d\mu (x)\,d\mu (y)f(x)(g(x)-g(y))e^{-h(x)-h(y)}\\&\qquad =\sum _{k=0}^{\infty }\iint d\mu (x)\,d\mu (y)f(x)(g(x)-g(y)){\frac {(-h(x)-h(y))^{k}}{k!}}.\end{aligned}}}$

Now the inequality follows from the assumption and from the identity

${\displaystyle f(x)={\frac {1}{2}}(f(x)+f(y))+{\frac {1}{2}}(f(x)-f(y)).}$

• To recover the (second) Griffiths inequality, take Γ = {−1, +1}^Λ, where Λ is a lattice, and let μ be a measure on Γ that is invariant under sign flipping. The cone A of polynomials with positive coefficients satisfies the assumptions of the Ginibre inequality.
• (Γ, μ) is a commutative compact group with the Haar measure, A is the cone of real positive definite functions on Γ.
• Γ is a totally ordered set, A is the cone of real positive non-decreasing functions on Γ. This yields Chebyshev's sum inequality. For extension to partially ordered sets, see FKG inequality.

• The thermodynamic limit of the correlations of the ferromagnetic Ising model (with non-negative external field h and free boundary conditions) exists.

This is because increasing the volume is the same as switching on new couplings J_B for a certain subset B. By the second Griffiths inequality

${\displaystyle {\frac {\partial }{\partial J_{B}}}\langle \sigma _{A}\rangle =\langle \sigma _{A}\sigma _{B}\rangle -\langle \sigma _{A}\rangle \langle \sigma _{B}\rangle \geq 0}$

Hence ${\displaystyle \langle \sigma _{A}\rangle }$ is monotonically increasing with the volume; then it converges since it is bounded by 1.

• The one-dimensional, ferromagnetic Ising model with interactions ${\displaystyle J_{x,y}\sim |x-y|^{-\alpha }}$ displays a phase transition if ${\displaystyle 1<\alpha <2}$.

This property can be shown in a hierarchical approximation, that differs from the full model by the absence of some interactions: arguing as above with the second Griffiths inequality, the results carries over the full model.^[7]

• The Ginibre inequality provides the existence of the thermodynamic limit for the free energy and spin correlations for the two-dimensional classical XY model.^[4] Besides, through Ginibre inequality, Kunz and Pfister proved the presence of a phase transition for the ferromagnetic XY model with interaction ${\displaystyle J_{x,y}\sim |x-y|^{-\alpha }}$ if ${\displaystyle 2<\alpha <4}$.
• Aizenman and Simon^[8] used the Ginibre inequality to prove that the two point spin correlation of the ferromagnetic classical XY model in dimension ${\displaystyle D}$, coupling ${\displaystyle J>0}$ and inverse temperature ${\displaystyle \beta }$ is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension ${\displaystyle D}$, coupling ${\displaystyle J>0}$, and inverse temperature ${\displaystyle \beta /2}$

${\displaystyle \langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle _{J,2\beta }\leq \langle \sigma _{i}\sigma _{j}\rangle _{J,\beta }}$

Hence the critical ${\displaystyle \beta }$ of the XY model cannot be smaller than the double of the critical temperature of the Ising model

${\displaystyle \beta _{c}^{XY}\geq 2\beta _{c}^{\rm {Is}}~;}$

in dimension D = 2 and coupling J = 1, this gives

${\displaystyle \beta _{c}^{XY}\geq \ln(1+{\sqrt {2}})\approx 0.88~.}$

• There exists a version of the Ginibre inequality for the Coulomb gas that implies the existence of thermodynamic limit of correlations.^[9]
• Other applications (phase transitions in spin systems, XY model, XYZ quantum chain) are reviewed in.^[10]