PhyslibAlpha.CondensedMatter.IsingModel.ClassicalExamples

For a set of sites $S$ and a uniform magnetic field $h \in \mathbb{R}$, this definition constructs an Ising model on $S$ characterized by zero interaction between any two sites (i.e., the coupling constant $J_{i,j} = 0$ for all $i, j \in S$) and a constant external field $h$ applied to every site.

Given an inverse temperature $\beta \in \mathbb{R}$, an external magnetic field $h \in \mathbb{R}$, and a spin state represented by a boolean $b$, the function calculates the one-site Boltzmann weight defined as $e^{\beta h \cdot \text{sgn}(b)}$. Here, $\text{sgn}(b)$ denotes the numerical spin value (typically $+1$ or $-1$) associated with the boolean state $b$.

For a given inverse temperature $\beta \in \mathbb{R}$ and external magnetic field $h \in \mathbb{R}$, the one-site partition function is defined as the sum of the one-site Boltzmann weights over all possible spin states $b \in \{\text{true}, \text{false}\}$. It is given by the formula: $$Z_{\text{site}}(\beta, h) = \sum_{b \in \{\text{true}, \text{false}\}} e^{\beta h \cdot \text{sgn}(b)}$$ where $\text{sgn}(b)$ is the numerical value assigned to the spin state $b$.

For any inverse temperature $\beta \in \mathbb{R}$ and external magnetic field $h \in \mathbb{R}$, the one-site partition function $Z_{\text{site}}(\beta, h)$ is strictly positive: $$Z_{\text{site}}(\beta, h) > 0$$

For a given inverse temperature $\beta \in \mathbb{R}$, an external magnetic field $h \in \mathbb{R}$, and a spin state $b \in \{\text{true}, \text{false}\}$, the function calculates the probability of a single independent spin being in state $b$. This probability is defined as the ratio of the one-site Boltzmann weight to the one-site partition function: $$P(b; \beta, h) = \frac{e^{\beta h \cdot \text{sgn}(b)}}{Z_{\text{site}}(\beta, h)}$$ where $\text{sgn}(b)$ denotes the numerical spin value (typically $+1$ or $-1$) associated with the state $b$, and $Z_{\text{site}}(\beta, h) = \sum_{b' \in \{\text{true}, \text{false}\}} e^{\beta h \cdot \text{sgn}(b')}$ is the one-site partition function.

For any inverse temperature $\beta \in \mathbb{R}$ and external magnetic field $h \in \mathbb{R}$, the one-site spin probability function $P(\cdot; \beta, h)$ is an element of the standard simplex on the set of spin states $\{\text{true}, \text{false}\}$. This means that for each $b \in \{\text{true}, \text{false}\}$, $P(b; \beta, h) \ge 0$, and the sum of the probabilities over all possible states is $\sum_{b \in \{\text{true}, \text{false}\}} P(b; \beta, h) = 1$.

In an Ising model on a set of sites $S$ with uniform magnetic field $h \in \mathbb{R}$ and no pair interactions, the Boltzmann weight for a configuration $c: Y \to \text{Bool}$ on a finite subset of sites $Y \subseteq S$ at inverse temperature $\beta$ is the product of the individual site weights: $$\text{weight}(\beta, Y, c) = \prod_{y \in Y} e^{\beta h \cdot \text{sgn}(c(y))}$$ where $\text{sgn}(c(y))$ is the numerical spin value ($+1$ or $-1$) associated with the state at site $y$.

For an Ising model on a set of sites $S$ with a uniform magnetic field $h \in \mathbb{R}$ and no interactions between sites, the partition function $Z(\beta, Y)$ for any finite subset of sites $Y \subseteq S$ at inverse temperature $\beta$ is the product of the individual one-site partition functions $Z_{\text{site}}(\beta, h)$ for each site in $Y$: $$Z(\beta, Y) = \prod_{y \in Y} Z_{\text{site}}(\beta, h)$$ where the one-site partition function is defined as $Z_{\text{site}}(\beta, h) = \sum_{b \in \{\text{true}, \text{false}\}} e^{\beta h \cdot \text{sgn}(b)}$.

Consider an Ising model on a set of sites $S$ with a uniform magnetic field $h \in \mathbb{R}$ and no interactions between sites. For any finite subset of sites $Y \subseteq S$, the Gibbs probability $P_\beta(c)$ of a spin configuration $c: Y \to \{\text{true}, \text{false}\}$ at inverse temperature $\beta$ is equal to the product of the individual one-site probabilities: $$P_\beta(c) = \prod_{y \in Y} P(c(y); \beta, h)$$ where $P(b; \beta, h)$ is the probability of a single independent spin being in state $b \in \{\text{true}, \text{false}\}$.

In an Ising model on a set of sites $S$ with zero interaction between sites and a uniform external magnetic field $h$, the classical Gibbs state $\langle \cdot \rangle_{Y, \beta}$ for any finite region $Y \subseteq S$ at inverse temperature $\beta$ is equal to the product state formed by the independent one-site spin probability distributions $P(\cdot; \beta, h)$.

For an Ising model on a set of sites $S$ with a uniform magnetic field $h \in \mathbb{R}$ and no interaction between sites, the thermodynamic limit state at inverse temperature $\beta$ is equal to the infinite-volume product net state. This product state is defined by the independent one-site spin probability distribution $P(b; \beta, h)$, where for each site, the probability of being in spin state $b \in \{\text{true}, \text{false}\}$ is given by $$P(b; \beta, h) = \frac{e^{\beta h \cdot \text{sgn}(b)}}{Z_{\text{site}}(\beta, h)}$$ and $Z_{\text{site}}(\beta, h)$ is the single-site partition function.

For real numbers $J$ and $h$, this defines the uniform nearest-neighbor Ising model on the integer lattice $\mathbb{Z}$. The model is characterized by an interaction strength $J_{ij}$ between sites $i, j \in \mathbb{Z}$ that equals $J$ if $|i - j| = 1$ and $0$ otherwise, along with a constant external magnetic field $h$ applied at every site $i \in \mathbb{Z}$.

For a uniform nearest-neighbor Ising chain on the integer lattice $\mathbb{Z}$ with coupling constant $J \in \mathbb{R}$ and external field $h \in \mathbb{R}$, the exchange coupling $J_{i, j}$ between any two sites $i, j \in \mathbb{Z}$ is invariant under translation. Specifically, for any integer $a \in \mathbb{Z}$, it holds that $J_{a+i, a+j} = J_{i, j}$.

For the uniform nearest-neighbor Ising model on the integer lattice $\mathbb{Z}$ with coupling constant $J \in \mathbb{R}$ and magnetic field $h \in \mathbb{R}$, the pair interaction $J_{ij}$ between any two sites $i, j \in \mathbb{Z}$ is zero if they are not nearest neighbors, i.e., if $j \neq i - 1$ and $j \neq i + 1$.

For an exchange coupling $J \in \mathbb{R}$, an external magnetic field $h \in \mathbb{R}$, and inverse temperature $\beta \in \mathbb{R}$, this defines the thermodynamic limit state of the uniform nearest-neighbor Ising chain on the integer lattice $\mathbb{Z}$. The state is a unital positive linear map $\omega: \varinjlim A(i) \to \mathbb{R}$ on the global system of local observables, constructed as the unique extension of the family of compatible local states obtained from the thermodynamic limit of finite-volume Gibbs states.

For the uniform nearest-neighbor Ising model on the integer lattice $\mathbb{Z}$ with coupling constant $J \in \mathbb{R}$, external magnetic field $h \in \mathbb{R}$, and inverse temperature $\beta \in \mathbb{R}$, the global thermodynamic limit state corresponds exactly to the compatible family of local states obtained from the thermodynamic limit of finite-volume Gibbs states. Specifically, the image of the global state $\omega \in \mathcal{S}(\varinjlim A(i))$ under the canonical isomorphism $\text{stateEquiv}$ is equal to the net state $\sigma$, where $\sigma$ is the collection of local states defined by the thermodynamic limit of the classical Ising model.