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, then generalised by Kelly and Sherman to interactions involving an arbitrary number of spins, and then by Griffiths to systems with arbitrary spins. A more general formulation was given by Ginibre, and is now called the Ginibre inequality.

Definitions

Let $\textstyle \sigma=\{\sigma_j\}_{j \in \Lambda}$ 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 $\textstyle \sigma_A = \prod_{j \in A} \sigma_j$ 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

$H(\sigma)=-\sum_{A} J_A \sigma_A ~,$

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

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

be the partition function. As usual,

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

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

$\tau_k = \begin{cases} \sigma_k, &k\neq j, \\ - \sigma_k, &k = j. \end{cases}$

Statement of inequalities

First Griffiths inequality

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

$\langle \sigma_A\rangle \geq 0$

for any list of spins A.

Second Griffiths inequality

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

$\langle \sigma_A\sigma_B\rangle \geq \langle \sigma_A\rangle \langle \sigma_B\rangle$

for any lists of spins A and B.

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

Proof

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

Proof of first inequality: Expand

$e^{-H(\sigma)} = \prod_{B} \sum_{k \geq 0} \frac{J_B^k \sigma_B^k}{k!} = \sum_{\{k_C\}_C} \prod_B \frac{J_B^{k_B} \sigma_B^{k_B}}{k_B!}~,$

then

$\begin{align}Z \langle \sigma_A \rangle &= \int d\mu(\sigma) \sigma_A e^{-H(\sigma)} = \sum_{\{k_C\}_C} \prod_B \frac{J_B^{k_B}}{k_B!} \int d\mu(\sigma) \sigma_A \sigma_B^{k_B} \\ &= \sum_{\{k_C\}_C} \prod_B \frac{J_B^{k_B}}{k_B!} \int d\mu(\sigma) \prod_{j \in \Lambda} \sigma_j^{n_A(j) + n_B(j)}~,\end{align}$

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

$\int d\mu(\sigma) \prod_j \sigma_j^{n(j)} = 0$

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, $\sigma'$, with the same distribution of $\sigma$. Then

$\langle \sigma_A\sigma_B\rangle- \langle \sigma_A\rangle \langle \sigma_B\rangle= \langle\langle\sigma_A(\sigma_B-\sigma'_B)\rangle\rangle~.$

Introduce the new variables

$\sigma_j=\tau_j+\tau_j'~, \qquad \sigma'_j=\tau_j-\tau_j'~.$

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

$\begin{align} \sum_A J_A (\sigma_A+\sigma'_A) &= \sum_A J_A\sum_{X\subset A} \left[1+(-1)^{|X|}\right] \tau_{A \setminus X} \tau'_X \end{align}$

Besides the measure on $\tau,\tau'$ is invariant under spin flipping because $d\mu(\sigma)d\mu(\sigma')$ is. Finally the monomials $\sigma_A$, $\sigma_B-\sigma'_B$ are polynomials in $\tau,\tau'$ with positive coefficients

$\begin{align} \sigma_A &= \sum_{X \subset A} \tau_{A \setminus X} \tau'_{X}~, \\ \sigma_B-\sigma'_B &= \sum_{X\subset B} \left[1-(-1)^{|X|}\right] \tau_{B \setminus X} \tau'_X~. \end{align}$

The first Griffiths inequality applied to $\langle\langle\sigma_A(\sigma_B-\sigma'_B)\rangle\rangle$ gives the result.

More details are in.

Extension: Ginibre inequality

The Ginibre inequality is an extension, found by Jean Ginibre, of the Griffiths inequality.

Formulation

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

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

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 ±,

$\iint d\mu(x) \, d\mu(y) \prod_{j=1}^n (f_j(x) \pm f_j(y)) \geq 0.$

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

$\langle fg\rangle_h - \langle f \rangle_h \langle g \rangle_h \geq 0.$

Proof

Let

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

Then

$\begin{align} &Z_h^2 \left( \langle fg\rangle_h - \langle f \rangle_h \langle g \rangle_h \right)\\ &\qquad= \iint d\mu(x) \, d\mu(y) f(x) (g(x) - g(y)) e^{-h(x)-h(y)} \\ &\qquad= \sum_{k=0}^\infty \iint d\mu(x) \, d\mu(y) f(x) (g(x) - g(y)) \frac{(-h(x)-h(y))^k}{k!}. \end{align}$

Now the inequality follows from the assumption and from the identity

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

Examples

• 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.

Applications

• 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

$\frac{\partial}{\partial J_B}\langle \sigma_A\rangle= \langle \sigma_A\sigma_B\rangle- \langle \sigma_A\rangle \langle \sigma_B\rangle\geq 0$

Hence $\langle \sigma_A\rangle$ is monotonically increasing with the volume; then it converges since it is bounded by 1.

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

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.

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

$\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle_{J,2\beta} \le \langle \sigma_i\sigma_j\rangle_{J,\beta}$

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

$\beta_c^{XY}\ge 2\beta_c^{\rm Is}~;$

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

$\beta_c^{XY} \ge \ln(1 + \sqrt{2}) \approx 0.88~.$

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