Griffiths inequality explained

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.

Definitions

Let

style \sigma = \{\sigma _{j\}} _{j} 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 style \sigma _{A} = \prod _{j} \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 ⋅ \rangle = (1)/(Z) \sum _{\sigma} ⋅ (\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 ≠ 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} (J ^{k} _{B} \sigma ^{k} _{B})/(k!) = \sum _{\{k _{C\}} _{C}} \prod _{B} (J ^{k _{B}} _{B} \sigma ^{k _{B}} _{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} (J ^{k _{B}} _{B})/(k _{B!}) \int d\mu(\sigma) \sigma _{A} \sigma ^{k _{B}} _{B} \\ & = \sum _{\{k _{C\}} _{C}} \prod _{B} (J ^{k _{B}} _{B})/(k _{B!}) \int d\mu(\sigma) \prod _{j} \sigma ^{n _{A(j)} + k _{B} n _{B(j)}} _{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 ^{n(j)} _{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'~, \sigma'} _{j=\tau} _{j-\tau} _{j'~.}

The doubled system

\langle\langle ⋅ \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} \left[ 1+ ( -1 ) ^{|X|} \right] \tau _{A} \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} \tau _{A} \tau' _{X} ~, \\ \sigma _{B-\sigma'} _{B} & = \sum _{X\subset} \left[ 1- ( -1 ) ^{|X|} \right] \tau _{B} \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 ^[5] and.^[6]

Extension: Ginibre inequality

The Ginibre inequality is an extension, found by Jean Ginibre,^[4] 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) / \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 ^{n} _{j =1} (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 ^{2} _{h} \left( \langle fg\rangle _{h} - \langle f \rangle _{h} \langle g \rangle _{h} \right)\\ & = \iint d\mu(x) d\mu(y) f(x) (g(x) - g(y)) e ^{-h(x)-h(y)} \\ & = \sum ^{inf ty} _{k =0} \iint d\mu(x) d\mu(y) f(x) (g(x) - g(y)) (( - h(x) - h(y)) ^{k})/(k!) . \end{align}

Now the inequality follows from the assumption and from the identity

f(x) = (1)/(2) (f(x) + f(y)) + (1)/(2) (f(x) - f(y)).

Examples

• To recover the (second) Griffiths inequality, take Γ = ^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

(\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.^[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

J _{x,y} \sim |x - y| ^{-\alpha} if 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

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 s _{i ⋅} 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 ^{XY} _{c} \ge 2 \beta ^{\rm Is} _{c} ~;

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

\beta ^{XY} _{c} \ge ln ( 1 + \sqrt{ 2 }) ≈ 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]

Notes and References

1. Griffiths. R.B.. Robert Griffiths (physicist). Correlations in Ising Ferromagnets. I. J. Math. Phys.. 1967. 8. 3. 478 - 483. 10.1063/1.1705219. 1967JMP.....8..478G .
2. Kelly. D.J.. Sherman. S.. General Griffiths' inequalities on correlations in Ising ferromagnets. J. Math. Phys.. 1968. 9. 3. 466–484. 10.1063/1.1664600. 1968JMP.....9..466K .
3. Griffiths. R.B.. Robert Griffiths (physicist). Rigorous Results for Ising Ferromagnets of Arbitrary Spin. J. Math. Phys.. 1969. 10. 9. 1559–1565. 10.1063/1.1665005. 1969JMP....10.1559G .
4. Ginibre. J.. Jean Ginibre. General formulation of Griffiths' inequalities. Comm. Math. Phys.. 1970. 16. 4. 310 - 328. 10.1007/BF01646537. 1970CMaPh..16..310G . 120649586.
5. Book: Glimm. James Glimm. Jaffe. A.. Arthur Jaffe. Quantum Physics. A functional integral point of view. Springer-Verlag. 1987. 0-387-96476-2. New York.
6. Book: Friedli. Velenik. Y.. Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge University Press. Cambridge . 2017 . 9781107184824 .
7. Dyson. F.J.. Freeman Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys.. 1969. 12. 2. 91 - 107. 10.1007/BF01645907. 1969CMaPh..12...91D . 122117175.
8. Aizenman. M.. Michael Aizenman. Simon. B.. Barry Simon. A comparison of plane rotor and Ising models. Phys. Lett. A. 1980. 76. 3–4. 281–282. 10.1016/0375-9601(80)90493-4. 1980PhLA...76..281A .
9. Fröhlich. J.. Jürg Fröhlich. Park. Y.M.. Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems. Comm. Math. Phys.. 1978. 59. 3. 235 - 266. 10.1007/BF01611505. 1978CMaPh..59..235F . 119758048.
10. Book: Griffiths, R.B.. Robert Griffiths (physicist)

    . Phase Transitions and Critical Phenomena. 1. 1972. Academic Press. New York. 7. Robert Griffiths (physicist). C. Domb and M.S.Green. Rigorous results and theorems. Phase Transitions and Critical Phenomena.