Английская Википедия:Griffiths inequality

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Шаблон:Short description 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 <math> \textstyle \sigma=\{\sigma_j\}_{j \in \Lambda}</math> 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 <math> \textstyle \sigma_A = \prod_{j \in A} \sigma_j </math> 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

<math>H(\sigma)=-\sum_{A} J_A \sigma_A ~,</math>

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

<math> Z=\int d\mu(\sigma) e^{-H(\sigma)} </math>

be the partition function. As usual,

<math> \langle \cdot \rangle = \frac{1}{Z} \sum_\sigma \cdot(\sigma) e^{-H(\sigma)} </math>

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

<math> \tau_k = \begin{cases}

\sigma_k, &k\neq j, \\ - \sigma_k, &k = j. \end{cases} </math>

Statement of inequalities

First Griffiths inequality

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

<math> \langle \sigma_A\rangle \geq 0</math>

for any list of spins A.

Second Griffiths inequality

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

<math> \langle \sigma_A\sigma_B\rangle \geq

\langle \sigma_A\rangle \langle \sigma_B\rangle </math> 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

<math> 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!}~,</math>

then

<math>\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) + k_B n_B(j)}~,\end{align}</math>

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

<math>\int d\mu(\sigma) \prod_j \sigma_j^{n(j)} = 0 </math>

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, <math>\sigma'</math>, with the same distribution of <math>\sigma</math>. Then

<math> \langle \sigma_A\sigma_B\rangle-

\langle \sigma_A\rangle \langle \sigma_B\rangle= \langle\langle\sigma_A(\sigma_B-\sigma'_B)\rangle\rangle~. </math>

Introduce the new variables

<math>

\sigma_j=\tau_j+\tau_j'~, \qquad \sigma'_j=\tau_j-\tau_j'~. </math>

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

<math>\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}</math>

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

<math>\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}</math>

The first Griffiths inequality applied to <math>\langle\langle\sigma_A(\sigma_B-\sigma'_B)\rangle\rangle</math> 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

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

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 ±,

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

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

<math> \langle fg\rangle_h - \langle f \rangle_h \langle g \rangle_h \geq 0. </math>

Proof

Let

<math> Z_h = \int e^{-h(x)} \, d\mu(x).</math>

Then

<math>\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} </math>

Now the inequality follows from the assumption and from the identity

<math> f(x) = \frac{1}{2} (f(x)+f(y)) + \frac{1}{2} (f(x)-f(y)). </math>

Examples

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 JB for a certain subset B. By the second Griffiths inequality
<math>\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 </math>

Hence <math>\langle \sigma_A\rangle</math> is monotonically increasing with the volume; then it converges since it is bounded by 1.
  • The one-dimensional, ferromagnetic Ising model with interactions <math> J_{x,y}\sim |x-y|^{-\alpha} </math> displays a phase transition if <math> 1<\alpha <2 </math>.
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 <math> J_{x,y}\sim |x-y|^{-\alpha} </math> if <math> 2<\alpha < 4 </math>.
  • Aizenman and Simon[8] used the Ginibre inequality to prove that the two point spin correlation of the ferromagnetic classical XY model in dimension <math>D</math>, coupling <math>J>0</math> and inverse temperature <math>\beta</math> is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension <math>D</math>, coupling <math>J>0</math>, and inverse temperature <math>\beta/2</math>
<math>\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle_{J,2\beta}

\le \langle \sigma_i\sigma_j\rangle_{J,\beta}</math>

Hence the critical <math>\beta</math> of the XY model cannot be smaller than the double of the critical temperature of the Ising model
<math> \beta_c^{XY}\ge 2\beta_c^{\rm Is}~;</math>
in dimension D = 2 and coupling J = 1, this gives
<math> \beta_c^{XY} \ge \ln(1 + \sqrt{2}) \approx 0.88~.</math>
  • 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]

References

Шаблон:Reflist

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  1. ↑ Шаблон:Cite journal
  2. ↑ Шаблон:Cite journal
  3. ↑ Шаблон:Cite journal
  4. ↑ 4,0 4,1 4,2 Шаблон:Cite journal
  5. ↑ Шаблон:Cite book
  6. ↑ Шаблон:Cite book
  7. ↑ Шаблон:Cite journal
  8. ↑ Шаблон:Cite journal
  9. ↑ Шаблон:Cite journal
  10. ↑ Шаблон:Cite book
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