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.
Definitionsedit
Let be a configuration of (continuous or discrete) spins on a latticeΛ. If A ⊂ Λ is a list of lattice sites, possibly with duplicates, let 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
The system is called ferromagnetic if, for any list of sites A, JA ≥ 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
Statement of inequalitiesedit
First Griffiths inequalityedit
In a ferromagnetic spin system which is invariant under spin flipping,
for any list of spins A.
Second Griffiths inequalityedit
In a ferromagnetic spin system which is invariant under spin flipping,
for any lists of spins A and B.
The first inequality is a special case of the second one, corresponding to B = ∅.
Proofedit
Observe that the partition function is non-negative by definition.
Proof of first inequality: Expand
then
where nA(j) stands for the number of times that j appears in A. Now, by invariance under spin flipping,
if at least one n(j) is odd, and the same expression is obviously non-negative for even values of n. Therefore, Z<σA>≥0, hence also <σA>≥0.
Proof of second inequality. For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin, , with the same distribution of . Then
Introduce the new variables
The doubled system is ferromagnetic in because is a polynomial in with positive coefficients
Besides the measure on is invariant under spin flipping because is.
Finally the monomials , are polynomials in with positive coefficients
The first Griffiths inequality applied to gives the result.
Let A be a set of real functions on Γ such that. for every f1,f2,...,fn in A, and for any choice of signs ±,
Then, for any f,g,−h in the convex cone generated by A,
Proofedit
Let
Then
Now the inequality follows from the assumption and from the identity
Examplesedit
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.
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 JB for a certain subset B. By the second Griffiths inequality
Hence is monotonically increasing with the volume; then it converges since it is bounded by 1.
The one-dimensional, ferromagnetic Ising model with interactions displays a phase transition if .
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 if .
Aizenman and Simon[8] used the Ginibre inequality to prove that the two point spin correlation of the ferromagnetic classical XY model in dimension , coupling and inverse temperature is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagneticIsing model in dimension , coupling , and inverse temperature
Hence the critical of the XY model cannot be smaller than the double of the critical temperature of the Ising model
in dimension D = 2 and coupling J = 1, this gives
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]
Referencesedit
^Griffiths, R.B. (1967). "Correlations in Ising Ferromagnets. I". J. Math. Phys. 8 (3): 478–483. Bibcode:1967JMP.....8..478G. doi:10.1063/1.1705219.
^Kelly, D.J.; Sherman, S. (1968). "General Griffiths' inequalities on correlations in Ising ferromagnets". J. Math. Phys. 9 (3): 466–484. Bibcode:1968JMP.....9..466K. doi:10.1063/1.1664600.
^Griffiths, R.B. (1969). "Rigorous Results for Ising Ferromagnets of Arbitrary Spin". J. Math. Phys. 10 (9): 1559–1565. Bibcode:1969JMP....10.1559G. doi:10.1063/1.1665005.
^ abcGinibre, J. (1970). "General formulation of Griffiths' inequalities". Comm. Math. Phys. 16 (4): 310–328. Bibcode:1970CMaPh..16..310G. doi:10.1007/BF01646537. S2CID 120649586.
^Glimm, J.; Jaffe, A. (1987). Quantum Physics. A functional integral point of view. New York: Springer-Verlag. ISBN 0-387-96476-2.
^Friedli, S.; Velenik, Y. (2017). Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge: Cambridge University Press. ISBN 9781107184824.
^Dyson, F.J. (1969). "Existence of a phase-transition in a one-dimensional Ising ferromagnet". Comm. Math. Phys. 12 (2): 91–107. Bibcode:1969CMaPh..12...91D. doi:10.1007/BF01645907. S2CID 122117175.
^Aizenman, M.; Simon, B. (1980). "A comparison of plane rotor and Ising models". Phys. Lett. A. 76 (3–4): 281–282. Bibcode:1980PhLA...76..281A. doi:10.1016/0375-9601(80)90493-4.
^Fröhlich, J.; Park, Y.M. (1978). "Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems". Comm. Math. Phys. 59 (3): 235–266. Bibcode:1978CMaPh..59..235F. doi:10.1007/BF01611505. S2CID 119758048.