PhyslibAlpha.CondensedMatter.SpinLattice.ProductStates

For a finite set of sites $X$ and a configuration $c: X \to \sigma$ that assigns a spin state $c(x)$ to each site $x \in X$, the function calculates the joint probability of the configuration as the product of the individual site probabilities: $$\prod_{x \in X} p(c(x))$$ where $p$ is the probability distribution for a single spin.

Let $p$ be a probability distribution on the spin space $\sigma$ (specifically, an element of the standard simplex). For any finite set of lattice sites $X$ and any spin configuration $c : X \to \sigma$, the product probability of the configuration is non-negative: $$0 \le \prod_{x \in X} p(c(x))$$

Let $S$ be a set of sites and $\sigma$ be a finite set of spin states. Let $p$ be a probability distribution on $\sigma$ (i.e., $p$ is in the standard simplex). For any finite subset of sites $X \subseteq S$, let $c: X \to \sigma$ be a configuration that assigns a spin state to each site in $X$. The product probability of the configuration $c$ is defined as $\prod_{x \in X} p(c(x))$. Then, the sum of these product probabilities over all possible configurations $c$ is equal to $1$: $$\sum_{c: X \to \sigma} \prod_{x \in X} p(c(x)) = 1$$

The finite-volume state for a system of independent spins on a finite set of sites $X \subseteq S$ with a one-site probability distribution $p$ on the spin space $\sigma$. This state is a linear functional on the space of classical observables $\text{Obs}_X$ (the set of functions $f: \sigma^X \to \mathbb{R}$) that assigns to each observable $f$ its expectation value: $$\sum_{c: X \to \sigma} \left( \prod_{x \in X} p(c(x)) \right) f(c)$$ where $c$ is a configuration that assigns a spin state $c(x) \in \sigma$ to each site $x \in X$.

Let $S$ be a lattice and $\sigma$ be a finite set of spin states. Let $p$ be a probability distribution on $\sigma$ (an element of the standard simplex). For any finite subset of sites $X \subseteq S$ and any classical observable $f: \sigma^X \to \mathbb{R}$, the value of the product state (the expectation value) applied to $f$ is given by the sum over all configurations $c: X \to \sigma$: $$\text{productState}(p, X)(f) = \sum_{c: X \to \sigma} \left( \prod_{x \in X} p(c(x)) \right) f(c)$$ where $\prod_{x \in X} p(c(x))$ is the product probability of the configuration $c$.

Let $S$ be a lattice and $\sigma$ be a set of spin states. Let $X$ and $Y$ be finite subsets of $S$ such that $X \subseteq Y$. For any spin configurations $c: X \to \sigma$ and $d: Y \to \sigma$, the indicator function for the condition that the restriction of $d$ to $X$ is equal to $c$ factors into constraints on individual sites: $$\mathbb{I}(d|_X = c) = \prod_{y \in Y} \begin{cases} \mathbb{I}(d(y) = c(y)) & \text{if } y \in X \\ 1 & \text{if } y \notin X \end{cases}$$ where $d|_X$ denotes the restriction of $d$ to $X$, and $\mathbb{I}(P)$ is the indicator function which is $1$ if the proposition $P$ is true and $0$ otherwise.

Let $S$ be a lattice and $\sigma$ a set of spin states. Let $p$ be a probability distribution on $\sigma$. For any finite subsets of sites $X$ and $Y$ such that $X \subseteq Y$, and for any spin configuration $c: X \to \sigma$, the sum of the product probabilities of configurations $d: Y \to \sigma$ that restrict to $c$ on $X$ is equal to the product probability of $c$ on $X$. Specifically, $$\sum_{d: Y \to \sigma} \left( \prod_{y \in Y} p(d(y)) \right) \cdot \mathbb{I}(d|_X = c) = \prod_{x \in X} p(c(x))$$ where $d|_X$ denotes the restriction of the configuration $d$ to the subregion $X$, and $\mathbb{I}(d|_X = c)$ is the indicator function which is $1$ if $d|_X = c$ and $0$ otherwise.

Let $S$ be a lattice and $\sigma$ be a set of spin states. Given a probability distribution $p$ on $\sigma$, let $\omega_Z$ denote the product state on a finite region $Z \subseteq S$, which is the state that assigns to any local observable $f: \sigma^Z \to \mathbb{R}$ the expectation value $$\omega_Z(f) = \sum_{c: Z \to \sigma} \left( \prod_{z \in Z} p(c(z)) \right) f(c).$$ For any finite subsets $X, Y \subseteq S$ such that $X \subseteq Y$, the restriction of the product state on $Y$ to the subregion $X$ is equal to the product state on $X$. That is, $$\omega_Y|_X = \omega_X.$$

The product state on the local net of observables $(\text{net } S \sigma)$ is a collection of compatible local states $\{\omega_X\}$ indexed by finite subsets $X \subseteq S$. For a given one-site probability distribution $p$ on the spin space $\sigma$, each local state $\omega_X$ is defined as: $$\omega_X(f) = \sum_{c: X \to \sigma} \left( \prod_{x \in X} p(c(x)) \right) f(c)$$ for any local observable $f: \sigma^X \to \mathbb{R}$. These local states are compatible under restriction, meaning for any finite subsets $X \subseteq Y \subseteq S$, the restriction of the state $\omega_Y$ to the subregion $X$ is equal to $\omega_X$.

Let $S$ be a lattice and $\sigma$ be a set of spin states. Given a one-site probability distribution $p$ on $\sigma$ (an element of the standard simplex), the **product global state** is the state $\omega$ on the global system of observables $\varinjlim \text{Obs}(X)$ obtained from the compatible family of product local states. For any local observable $f \in \text{Obs}(X)$ on a finite region $X \subseteq S$, the expectation value under this global state is given by: $$\omega(\iota_X f) = \sum_{c: X \to \sigma} \left( \prod_{x \in X} p(c(x)) \right) f(c)$$ where $\iota_X: \text{Obs}(X) \to \varinjlim \text{Obs}(Y)$ is the canonical inclusion of the local observable space into the global system.

Let $S$ be a lattice with a group $G$ acting on it, and let $\sigma$ be the set of possible spin values. Let $p$ be a probability distribution on $\sigma$. For any finite subset of sites $X \subseteq S$, any group element $g \in G$, and any spin configuration $c: g \cdot X \to \sigma$ on the translated region, the product probability of $c$ on $g \cdot X$ is equal to the product probability of its equivalent configuration on $X$. That is, $$\prod_{y \in g \cdot X} p(c(y)) = \prod_{x \in X} p(c(g \cdot x))$$ where the term on the right uses the equivalence mapping configurations on $g \cdot X$ to configurations on $X$ via the group action.

Let $S$ be a lattice and $\sigma$ be a set of spin states. Given a probability distribution $p$ on the spin space $\sigma$ (an element of the standard simplex), the product net state $\omega$ is the collection of compatible local states $\{\omega_X\}$ indexed by finite regions $X \subseteq S$, where for any local observable $f: \sigma^X \to \mathbb{R}$, the expectation value is defined by: $$\omega_X(f) = \sum_{c: X \to \sigma} \left( \prod_{x \in X} p(c(x)) \right) f(c).$$ If a group $G$ acts on the lattice $S$, then the product net state $\omega$ is invariant under the induced symmetry action on the net of local observables. That is, for every group element $g \in G$ and every finite region $X \subseteq S$, the local states satisfy: $$\omega_{gX} \circ \tau_{g,X} = \omega_X$$ where $\tau_{g,X}$ is the symmetry transformation mapping observables in region $X$ to those in the transformed region $g \cdot X$.