PhyslibAlpha.CondensedMatter.IsingModel.GibbsStates

For a quantum Ising model at inverse temperature $\beta$, the Gibbs state in a finite region $Y \subset S$ is a unital positive linear map on the algebra of observables $\mathcal{A}_Y$ (self-adjoint operators on the spin space). For an observable $M \in \mathcal{A}_Y$, its expectation value is given by the weighted sum over all spin configurations $c \in \text{Cfg}(Y)$: $$\omega_{Y, \beta}(M) = \sum_{c \in \text{Cfg}(Y)} P_\beta(c) \text{Re}(\langle c | M | c \rangle)$$ where $P_\beta(c)$ is the Gibbs probability of configuration $c$ defined by $P_\beta(c) = \frac{e^{-\beta E_Y(c)}}{Z_\beta(Y)}$, and $\langle c | M | c \rangle$ denotes the diagonal element of the operator $M$ in the $\sigma^z$ basis. This corresponds to the quantum mechanical trace $\text{Tr}(\rho M)$ where the density matrix $\rho = \frac{e^{-\beta H_Y}}{Z_Y}$ is diagonal in the configuration basis.

For a finite region $Y$ and a quantum observable $M$ (a self-adjoint operator on the spin space of $Y$), the expectation value of $M$ in the Gibbs state at inverse temperature $\beta$ is equal to the sum over all spin configurations $c \in \text{Cfg}(Y)$ of the product of the Gibbs probability $P_\beta(c)$ and the real part of the diagonal matrix element of $M$ corresponding to configuration $c$: $$\omega_{Y, \beta}(M) = \sum_{c \in \text{Cfg}(Y)} P_\beta(c) \text{Re}(\langle c | M | c \rangle)$$

For an Ising model $I$ at inverse temperature $\beta$ on a lattice $S$, the thermodynamic family is a collection of Gibbs states $(\omega_{Y, \beta})_{Y \in \mathcal{F}(S)}$ indexed by the set of finite regions $\mathcal{F}(S)$. For each finite region $Y$, the associated state is the Gibbs state $\omega_{Y, \beta}$ of the quantum Ising model. This family is equipped with an ultrafilter $\mathcal{U}$ on $\mathcal{F}(S)$ that refines the filter of finite sets directed by inclusion (the "at top" filter), which is used to define the thermodynamic limit of local expectations as the regions exhaust the lattice.

For a quantum Ising model $I$ on a lattice $S$ at inverse temperature $\beta$, the thermodynamic limit $\omega_\beta$ is a state of the infinite lattice (represented as a `NetState`). This state is constructed by taking the limit of the expectations of the thermodynamic family of finite-volume quantum Gibbs states $\omega_{Y, \beta}$ along a chosen ultrafilter $\mathcal{U}$ on the finite subsets $Y \subseteq S$. For any local observable $M$, the expectation value in the thermodynamic limit state is given by $$\omega_\beta(M) = \lim_{Y \to \mathcal{U}} \omega_{Y, \beta}(M)$$ where $\omega_{Y, \beta}$ is the Gibbs state for the finite region $Y$.

For a quantum Ising model on a lattice $S$ at inverse temperature $\beta$, let $X \subseteq S$ be a finite region and $M$ be a local observable associated with $X$. Let $\omega_{Y, \beta}$ denote the Gibbs state of the system in a finite region $Y \subseteq S$, and let $\omega_\beta$ denote the thermodynamic limit state constructed via an ultrafilter $\mathcal{U}$ on the finite subsets of $S$. The expectation value of the local observable $M$ in the finite-volume Gibbs states converges along $\mathcal{U}$ to the expectation value of $M$ in the thermodynamic limit state: $$\lim_{Y \to \mathcal{U}} \omega_{Y, \beta}(M) = \omega_\beta(M)$$ where $\omega_{Y, \beta}(M)$ is the expectation value of $M$ (embedded into the algebra of region $Y$) when $X \subseteq Y$.