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Background:

Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set $\{\exp(2\pi i j/N);1\leq j\leq N\}$ with uniform a priori distribution.

Consider the formal Hamiltonian given on the lattice $\mathbb Z^d$ by $$H_{ \Lambda }({\sigma}) = -\sum_{\langle x,y\rangle} J_{xy} \vec\sigma_{x} \vec\sigma_{y}$$ where $J_{xy}$ are nonnegative constants and $\vec\sigma_x \vec\sigma_y$ is inner product in $\mathbb R^2$. The sum is taken over all pair of first neighbors $\langle x,y\rangle$ means that $|x-y|=1$. The Partition function on a finite
$\Lambda\subset \mathbb Z^d$ is given by $$Z_\Lambda = \int\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big)\;d\sigma$$ where the integral is taken over all sites of $\Lambda$. The two point correlations are given by $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda}= Z_{\Lambda}^{-1}\int\vec\sigma_x \vec\sigma_y\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big) d\sigma$$ Question:

For which values of $N$ is known that the Lieb-Simon Inequality is true or false ?

Lieb-Simon Inequality $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda} \leq \sum_{b \in \partial B} \bigl<\vec\sigma_x \vec\sigma_b \bigr>_B \bigl<\vec\sigma_b \vec\sigma_y \bigr>_\Lambda,$$ where $B\subset\Lambda\subset\mathbb Z^d$ are finite, $x,y\in\Lambda$, $\partial B=\{z\in B; d(z,B^c)=1\}$ and $\partial B$ separates $x$ and $y$ ($i.e.$ any path from $x$ to $y$ must intercept $\partial B$).

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# For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?

Background:

Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set $\{\exp(2\pi j/N);1\leq j\leq N\}$ with uniform a priori distribution.

Consider the formal Hamiltonian given on the lattice $\mathbb Z^d$ by $$H_{ \Lambda }({\sigma}) = -\sum_{\langle x,y\rangle} J_{xy} \vec\sigma_{x} \vec\sigma_{y}$$ where $J_{xy}$ are nonnegative constants and $\vec\sigma_x \vec\sigma_y$ is inner product in $\mathbb R^2$. The sum is taken over all pair of first neighbors $\langle x,y\rangle$ means that $|x-y|=1$. The Partition function on a finite
$\Lambda\subset \mathbb Z^d$ is given by $$Z_\Lambda = \int\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big)\;d\sigma$$ where the integral is taken over all sites of $\Lambda$. The two point correlations are given by $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda}= Z_{\Lambda}^{-1}\int\vec\sigma_x \vec\sigma_y\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big) d\sigma$$ Question:

For which values of $N$ is known that the Lieb-Simon Inequality is true or false ?

Lieb-Simon Inequality $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda} \leq \sum_{b \in \partial B} \bigl<\vec\sigma_x \vec\sigma_b \bigr>_B \bigl<\vec\sigma_b \vec\sigma_y \bigr>_\Lambda,$$ where $B\subset\Lambda\subset\mathbb Z^d$ are finite, $x,y\in\Lambda$, $\partial B=\{z\in B; d(z,B^c)=1\}$ and $\partial B$ separates $x$ and $y$ ($i.e.$ any path from $x$ to $y$ must intercept $\partial B$).