Griffiths inequality

Let $\backslash textstyle\; \backslash sigma=\backslash \{\backslash sigma\_j\backslash \}\_\{j\; \backslash in\; \backslash Lambda\}$ 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 $\backslash textstyle\; \backslash sigma\_A\; =\; \backslash prod\_\{j\; \backslash in\; A\}\; \backslash sigma\_j$ 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

:$H(\backslash sigma)=-\backslash sum\_\{A\}\; J\_A\; \backslash sigma\_A\; ~,$

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

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

be the partition function. As usual,

:$\backslash langle\; f\; \backslash rangle\; =\; \backslash frac\{1\}\{Z\}\; \backslash sum\_\backslash sigma\; f(\backslash sigma)\; e^\{-H(\backslash sigma)\}$

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

:$\backslash tau\_k\; =\; \backslash begin\{cases\}$

\sigma_k, &k\neq j, \\

- \sigma_k, &k = j.

\end{cases}

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

:$\backslash langle\; \backslash sigma\_A\backslash rangle\; \backslash geq\; 0$

for any list of spins A.

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

:$\backslash langle\; \backslash sigma\_A\backslash sigma\_B\backslash rangle\; \backslash geq$

\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

:$e^\{-H(\backslash sigma)\}\; =\; \backslash prod\_\{B\}\; \backslash sum\_\{k\; \backslash geq\; 0\}\; \backslash frac\{J\_B^k\; \backslash sigma\_B^k\}\{k!\}\; =\; \backslash sum\_\{\backslash \{k\_C\backslash \}\_C\}\; \backslash prod\_B\; \backslash frac\{J\_B^\{k\_B\}\; \backslash sigma\_B^\{k\_B\}\}\{k\_B!\}~,$

then

:$\backslash begin\{align\}Z\; \backslash langle\; \backslash sigma\_A\; \backslash rangle$

&= \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{align}

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

:$\backslash int\; d\backslash mu(\backslash sigma)\; \backslash prod\_j\; \backslash sigma\_j^\{n(j)\}\; =\; 0$

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, $\backslash sigma\text{'}$, with the same distribution of $\backslash sigma$. Then

:$\backslash langle\; \backslash sigma\_A\backslash sigma\_B\backslash rangle-$

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

:$\backslash begin\{align\}$

\sum_A J_A (\sigma_A+\sigma'_A) &= \sum_A J_A\sum_{X\subset A}

\left[1+(-1)^

\end{align}

Besides the measure on $\backslash tau,\backslash tau\text{'}$ is invariant under spin flipping because $d\backslash mu(\backslash sigma)d\backslash mu(\backslash sigma\text{'})$ is.

Finally the monomials $\backslash sigma\_A$, $\backslash sigma\_B-\backslash sigma\text{'}\_B$ are polynomials in $\backslash tau,\backslash tau\text{'}$ with positive coefficients

:$\backslash begin\{align\}$

\sigma_A &= \sum_{X \subset A} \tau_{A \setminus X} \tau'_{X}~, \\

\sigma_B-\sigma'_B &= \sum_{X\subset B}

\left[1-(-1)^

\end{align}

The first Griffiths inequality applied to $\backslash langle\backslash langle\backslash sigma\_A(\backslash sigma\_B-\backslash sigma\text{'}\_B)\backslash rangle\backslash rangle$ gives the result.

More details are in {{cite book|last1=Glimm|first=J.|author1-link=James Glimm|last2=Jaffe|first2=A.|author2-link=Arthur Jaffe|title=Quantum Physics. A functional integral point of view|publisher=Springer-Verlag|year=1987|isbn=0-387-96476-2|location=New York}} and.{{cite book|last1=Friedli|first=S.|last2=Velenik|first2=Y.|title=Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction|publisher=Cambridge University Press|location=Cambridge |year=2017 |isbn=9781107184824 |url=http://www.unige.ch/math/folks/velenik/smbook/index.html}}

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

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

: $\backslash langle\; f\; \backslash rangle\_h\; =\; \backslash int\; f(x)\; e^\{-h(x)\}\; \backslash ,\; d\backslash mu(x)\; \backslash Big/\; \backslash int\; e^\{-h(x)\}\; \backslash ,\; d\backslash mu(x).$

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 ±,

:$\backslash iint\; d\backslash mu(x)\; \backslash ,\; d\backslash mu(y)\; \backslash prod\_\{j=1\}^n\; (f\_j(x)\; \backslash pm\; f\_j(y))\; \backslash geq\; 0.$

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

:$\backslash langle\; fg\backslash rangle\_h\; -\; \backslash langle\; f\; \backslash rangle\_h\; \backslash langle\; g\; \backslash rangle\_h\; \backslash geq\; 0.$

Let

: $Z\_h\; =\; \backslash int\; e^\{-h(x)\}\; \backslash ,\; d\backslash mu(x).$

Then

: $\backslash begin\{align\}$

&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

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

• 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

::$\backslash frac\{\backslash partial\}\{\backslash partial\; J\_B\}\backslash langle\; \backslash sigma\_A\backslash rangle=$

\langle \sigma_A\sigma_B\rangle-

\langle \sigma_A\rangle \langle \sigma_B\rangle\geq 0

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

• The one-dimensional, ferromagnetic Ising model with interactions $J\_\{x,y\}\backslash sim\; |x-y|^\{-\backslash alpha\}$ 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.{{cite journal|last=Dyson|first=F.J.|authorlink=Freeman Dyson|title=Existence of a phase-transition in a one-dimensional Ising ferromagnet|journal=Comm. Math. Phys.|year=1969|volume=12|issue=2|pages=91–107|doi=10.1007/BF01645907|bibcode=1969CMaPh..12...91D |s2cid=122117175|url=http://projecteuclid.org/euclid.cmp/1103841344 }}

• 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 $J\_\{x,y\}\backslash sim\; |x-y|^\{-\backslash alpha\}$ if .
• Aizenman and Simon{{cite journal|last1=Aizenman|first1=M.|author1-link=Michael Aizenman|last2=Simon|first2=B.|author2-link=Barry Simon|title=A comparison of plane rotor and Ising models|journal=Phys. Lett. A|year=1980|volume=76|issue=3–4|pages=281–282|doi=10.1016/0375-9601(80)90493-4|bibcode=1980PhLA...76..281A }} 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 $\backslash beta$ 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 $\backslash beta/2$

::$\backslash langle\; \backslash mathbf\{s\}\_i\backslash cdot\; \backslash mathbf\{s\}\_j\backslash rangle\_\{J,2\backslash beta\}$

\le \langle \sigma_i\sigma_j\rangle_{J,\beta}

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

::$\backslash beta\_c^\{XY\}\backslash ge\; 2\backslash beta\_c^\{\backslash rm\; Is\}~;$

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

::$\backslash beta\_c^\{XY\}\; \backslash ge\; \backslash ln(1\; +\; \backslash sqrt\{2\})\; \backslash approx\; 0.88~.$

• There exists a version of the Ginibre inequality for the Coulomb gas that implies the existence of thermodynamic limit of correlations.{{cite journal|last1=Fröhlich|first1=J.|author1-link=Jürg Fröhlich|last2=Park|first2=Y.M.|title=Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems|journal=Comm. Math. Phys.|year=1978|volume=59|issue=3|pages=235–266|doi=10.1007/BF01611505|bibcode=1978CMaPh..59..235F |s2cid=119758048|url=http://projecteuclid.org/euclid.cmp/1103901661 }}
• Other applications (phase transitions in spin systems, XY model, XYZ quantum chain) are reviewed in.{{cite book|last=Griffiths|first=R.B.|title=Phase Transitions and Critical Phenomena|volume=1|year=1972|publisher=Academic Press|location=New York|pages=7|authorlink=Robert Griffiths (physicist)|editor=C. Domb and M.S.Green|chapter=Rigorous results and theorems|title-link=Phase Transitions and Critical Phenomena}}

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Category:Inequalities (mathematics)

Category:Statistical mechanics