PhyslibAlpha.CondensedMatter.IsingModel.KMS

For any finite index set $n$ and any complex $n \times n$ matrix $A$, let $\text{Re}(A) = \frac{1}{2}(A + A^*)$ be its self-adjoint real part (where $A^*$ denotes the conjugate transpose). For any index $c \in n$, the real part of the diagonal entry of $\text{Re}(A)$ is equal to the real part of the diagonal entry of $A$, that is: $$\text{Re}((\text{Re}(A))_{cc}) = \text{Re}(A_{cc})$$

Let $n$ be a finite set and $A$ be an $n \times n$ complex matrix. Let $\text{Im}(A)$ denote the imaginary part of $A$ as a self-adjoint operator (defined as $\frac{A - A^*}{2i}$). For any index $c \in n$, the real part of the diagonal entry $(\text{Im}(A))_{cc}$ is equal to the imaginary part of the diagonal entry $A_{cc}$, i.e., $$\text{Re}((\text{Im } A)_{cc}) = \text{Im}(A_{cc}).$$

For a finite region $Y$ and an operator $M$ in the algebra $\text{Alg}(Y)$ of the Ising model, the expectation value of $M$ in the Gibbs state at inverse temperature $\beta$ is given by the sum over all configurations $c \in \text{Cfg}(Y)$ of the Gibbs probability $P_\beta(c)$ multiplied by the diagonal matrix elements $M(c, c)$: $$\langle M \rangle_{\beta, Y} = \sum_{c \in \text{Cfg}(Y)} P_\beta(c) M(c, c)$$ where $P_\beta(c)$ is the probability of configuration $c$ in the Gibbs distribution for the region $Y$.

Given a finite region $Y$, a complex time $z \in \mathbb{C}$, and an operator $M$ acting on the spin configurations of $Y$, the function computes the Heisenberg evolution $\alpha_z(M)$. The matrix elements of the evolved operator are given by $$(\alpha_z(M))_{c, c'} = e^{i z (E_Y(c) - E_Y(c'))} M_{c, c'}$$ where $c$ and $c'$ are spin configurations in $Y$, $E_Y(c)$ is the energy of configuration $c$ in region $Y$, and $i$ is the imaginary unit. For real $z = t$, this corresponds to the standard time evolution $e^{i t H_Y} M e^{-i t H_Y}$ under the Hamiltonian $H_Y$.

For a finite region $Y$, a complex time $z \in \mathbb{C}$, and an operator $M$ in the algebra of the Ising model $\text{Alg}(Y)$, the matrix elements of the Heisenberg-evolved operator $\alpha_z(M)$ are given by: $$(\alpha_z(M))_{c, c'} = e^{i z (E_Y(c) - E_Y(c'))} M_{c, c'}$$ where $c, c' \in \text{Cfg}(Y)$ are spin configurations in the region $Y$, $E_Y(c)$ is the energy of configuration $c$, and $i$ is the imaginary unit.

Let $X$ be a finite region of sites and $Q$ be an operator in the local algebra $\text{Alg}(X)$. For any finite region $Y$ such that $X \subseteq Y$, let $\langle Q \rangle_Y$ be the expectation value of $Q$ with respect to the Gibbs state of the Ising model at inverse temperature $\beta$. As $Y$ increases to the infinite lattice along an ultrafilter $\mathcal{U}$ (extending the filter of finite sets ordered by inclusion), the expectation values converge to the expectation value of $Q$ in the thermodynamic limit state $\omega_\infty$: $$\lim_{Y \in \mathcal{U}} \langle Q \rangle_Y = \omega_\infty(Q)$$

For any real numbers $e$, $e'$, and $t$, the following identity holds in the complex numbers: $$e^{-\beta e} \exp(i (t + i\beta) (e' - e)) = e^{-\beta e'} \exp(i t (e' - e))$$ where $\beta$ is the inverse temperature and $i$ is the imaginary unit. This identity provides the basis for the KMS (Kubo-Martin-Schwinger) condition in the Ising model by demonstrating that a complex time shift of $i\beta$ in the evolution factor corresponds to the ratio of the Gibbs weights $e^{-\beta e}$ and $e^{-\beta e'}$.

For any finite region $Y$ of sites in an Ising model, let $A$ and $B$ be local operators acting on the spins in $Y$. For any real time $t$ and inverse temperature $\beta$, the expectation value with respect to the Gibbs state of region $Y$ satisfies the KMS (Kubo-Martin-Schwinger) condition: $$\langle A \alpha_{t + i\beta}(B) \rangle_Y = \langle \alpha_t(B) A \rangle_Y$$ where $\alpha_z$ denotes the Heisenberg evolution by complex time $z$ (given by $\alpha_z(M) = e^{i z H_Y} M e^{-i z H_Y}$ for a Hamiltonian $H_Y$), and $i$ is the imaginary unit.

Let $X$ be a finite subset of a lattice $S$, and let $X^+$ denote its neighborhood (represented by `I.plus X`). For any operator $N \in \text{Alg}(X)$ and any complex time $z \in \mathbb{C}$, the function `evolveLocal` computes the Heisenberg evolution of $N$, denoted $\alpha_z(N)$, as an operator in the algebra $\text{Alg}(X^+)$. It is defined by embedding $N$ into the algebra of the neighborhood via the inclusion map $\text{inclMat}$ and then applying the evolution function $\alpha_z$ for the region $X^+$. This construction reflects the property that in the Ising model, the time evolution of an operator local to $X$ depends only on the spins within its immediate neighborhood $X^+$, regardless of the size of the total system.

Let $X$ and $Y$ be finite regions of a lattice $S$ such that the neighborhood $X^+$ of $X$ is contained in $Y$ (denoted $X^+ \subseteq Y$). For any local operator $N$ in the algebra $\text{Alg}(X)$ and any complex time $z \in \mathbb{C}$, the Heisenberg evolution of $N$ computed within the larger region $Y$ is equal to the evolution of $N$ computed within its neighborhood $X^+$ and then embedded into the algebra of $Y$. Mathematically, $$\iota_{X^+, Y}(\alpha_z^{\text{local}}(N)) = \alpha_z^{(Y)}(\iota_{X, Y}(N))$$ where $\alpha_z^{(Y)}$ is the Heisenberg evolution in region $Y$, $\alpha_z^{\text{local}}(N)$ is the evolution of $N$ computed in its neighborhood $X^+$, and $\iota_{A, B} : \text{Alg}(A) \to \text{Alg}(B)$ is the natural inclusion map for $A \subseteq B$. This identity demonstrates the **locality of the dynamics** in the Ising model, showing that the evolution of a local operator depends only on the spins in its immediate neighborhood.

For a finite region $Y$ of the lattice, let $L: \text{Alg}(Y) \to \mathbb{C}$ be a $\mathbb{C}$-linear functional on the algebra of operators. For any operator $Q \in \text{Alg}(Y)$, the value $L(Q)$ is given by the sum $$L(Q) = \sum_{a, b} Q_{ab} L(E_{ab})$$ where $Q_{ab}$ are the matrix elements of $Q$, and $E_{ab}$ (represented as `Matrix.single a b 1` in the formal text) are the matrix units. The summation indices $a$ and $b$ range over the configurations of the Ising model in the region $Y$.

Let $Y$ be a finite region of a lattice and $\text{Alg}(Y)$ be the algebra of operators acting on the spin configurations in $Y$. Given a $\mathbb{C}$-linear functional $L : \text{Alg}(Y) \to \mathbb{C}$ and two operators $P, N \in \text{Alg}(Y)$, the correlation function defined by $f(z) = L(P \alpha_z(N))$ is an entire function (differentiable for all $z \in \mathbb{C}$), where $\alpha_z(N)$ is the Heisenberg evolution of the operator $N$ by the complex time $z$. This evolution is defined by the matrix elements $(\alpha_z(N))_{c, c'} = e^{i z (E_Y(c) - E_Y(c'))} N_{c, c'}$, where $E_Y(c)$ is the energy of configuration $c$ in region $Y$.

Let $Y$ be a finite subset of a lattice $S$. For any operator $M$ in the algebra $\text{Alg}(Y)$ of the Ising model, its Heisenberg evolution $\alpha_z(M)$ at complex time $z = 0$ is equal to $M$ itself: $$\alpha_0(M) = M$$

For a finite region $Y$ in a lattice $S$, any complex times $s, t \in \mathbb{C}$, and any operator $M$ in the algebra of the spins in $Y$, the Heisenberg evolution $\alpha_z$ satisfies the composition property: $$\alpha_{s+t}(M) = \alpha_s(\alpha_t(M))$$ where $\alpha_z(M)$ represents the evolution of $M$ by the complex time $z$.

This definition establishes that the Ising model on a lattice $S$ constitutes a **local analytic evolution**. It specifies the following components for the dynamical system: - **Support:** The evolution of an operator local to a region $X$ is supported within its neighborhood $X^+$ (defined as `I.plus X`). - **Dynamics:** The evolution is given by the Heisenberg map $\alpha_z$ for complex time $z \in \mathbb{C}$. - **Locality:** The system satisfies the stabilization property, meaning the evolution of a local operator $N \in \text{Alg}(X)$ computed in a larger region $Y$ (where $X^+ \subseteq Y$) coincides with the local evolution computed in the neighborhood $X^+$. - **Analyticity:** The evolution is analytic (differentiable) with respect to the complex time $z$.

A state $\omega$ of the infinite Ising lattice satisfies the **KMS (Kubo-Martin-Schwinger) condition** at inverse temperature $\beta$ if for any two local operators $M$ and $N$, the correlation function $t \mapsto \omega(M \alpha_t(N))$ is the restriction to the real line of an entire function $F: \mathbb{C} \to \mathbb{C}$ that satisfies the periodic boundary condition $F(t + i\beta) = \omega(\alpha_t(N) M)$, where $\alpha_t$ denotes the local analytic Heisenberg evolution of the operator.

Let $I$ be a quantum Ising model on a lattice $S$ and let $\beta$ be the inverse temperature. Let $F$ be a thermodynamic family where, for each index $i$, the state $\omega_i$ is the Gibbs state of the model $I$ at inverse temperature $\beta$ for the finite region $\Lambda_i$. If $\omega$ is a thermodynamic limit of the family $F$ (meaning for every local observable $a$, the expectation values converge: $\lim_{i \to \mathcal{U}} \omega_i(a) = \omega(a)$), then the limit state $\omega$ satisfies the KMS condition at inverse temperature $\beta$.

Let $I$ be a quantum Ising model and $\beta$ be the inverse temperature. The thermodynamic limit of the Gibbs states of finite regions for the model $I$, denoted as $\omega_\beta$, satisfies the KMS (Kubo-Martin-Schwinger) condition at inverse temperature $\beta$. This means that for any two local operators $M$ and $N$, the correlation function $t \mapsto \omega_\beta(M \alpha_t(N))$ is the restriction to the real line of an entire function $F: \mathbb{C} \to \mathbb{C}$ that satisfies the periodic boundary condition $F(t + i\beta) = \omega_\beta(\alpha_t(N) M)$, where $\alpha_t$ is the Heisenberg evolution of the operator.

For any real numbers $J$ (interaction strength), $h$ (longitudinal magnetic field), and $\beta$ (inverse temperature), the state obtained by taking the thermodynamic limit of the local Gibbs states for the quantum longitudinal Ising chain satisfies the KMS (Kubo-Martin-Schwinger) condition at inverse temperature $\beta$.