Griffiths inequality

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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[edit]

Let  \textstyle \sigma=\{\sigma_j\}_{j \in \Lambda} be a configuration of (continuous or discrete) spins on a lattice Λ. If AΛ 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, 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

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

Statement of inequalities[edit]

First Griffiths inequality[edit]

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[edit]

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[edit]

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 nA(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<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, \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.[5]

Extension: Ginibre inequality[edit]

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

Formulation[edit]

Let (Γ, μ) be a probability space. For functions fh 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,...,fn 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[edit]

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[edit]

Applications[edit]

  • 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
\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 showed 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.[6]
  • 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[7] 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.[8]
  • Other applications (phase transitions in spin systems, XY model, XYZ quantum chain) are reviewed in.[9]

References[edit]

  1. ^ Griffiths, R.B. (1967). "Correlations in Ising Ferromagnets. I". J. Math. Phys. 8 (3): 478–483. doi:10.1063/1.1705219. 
  2. ^ Kelly, D.J.; Sherman, S. (1968). "General Griffiths' inequalities on correlations in Ising ferromagnets". J. Math. Phys. 9: 466. doi:10.1063/1.1664600. 
  3. ^ Griffiths, R.B. (1969). "Rigorous Results for Ising Ferromagnets of Arbitrary Spin". J. Math. Phys. 10: 1559. doi:10.1063/1.1665005. 
  4. ^ a b c Ginibre, J. (1970). "General formulation of Griffiths' inequalities". Comm. Math. Phys. 16 (4): 310–328. doi:10.1007/BF01646537. 
  5. ^ Glimm, J.; Jaffe, A. (1987). Quantum Physics. A functional integral point of view. New York: Springer-Verlag. ISBN 0-387-96476-2. 
  6. ^ Dyson, F.J. (1969). "Existence of a phase-transition in a one-dimensional Ising ferromagnet". Comm. Math. Phys. 12: 91–107. doi:10.1007/BF01645907. 
  7. ^ Aizenman, M.; Simon, B. (1980). "A comparison of plane rotor and Ising models". Phys. Lett. A 76. doi:10.1016/0375-9601(80)90493-4. 
  8. ^ 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. doi:10.1007/BF01611505. 
  9. ^ Griffiths, R.B. (1972). "Rigorous results and theorems". In C. Domb and M.S.Green. Phase Transitions and Critical Phenomena 1. New York: Academic Press. p. 7.