PhyslibAlpha.CondensedMatter.SpinLattice.SpinWaves

The type $Gl$ represents the global system of classical observables for a lattice $\Lambda$ where each site has two possible spin states (binary spins, represented by `Bool`). It is constructed as the algebraic direct limit $\varinjlim_{X \subseteq \Lambda} \text{Obs}(X)$ over the directed system of finite regions $X$ of the lattice. For each finite subset $X \subseteq \Lambda$, the local space $\text{Obs}(X)$ consists of all real-valued functions $f: \{ \text{true, false} \}^X \to \mathbb{R}$ acting on the configuration space of that region. The direct limit identifies observables from a smaller region with their natural extensions to larger regions via inclusion maps.

Given a lattice $\Lambda$, a finite region $X \subseteq \Lambda$, and a site $x \in \Lambda$, the spin observable at $x$ on the region $X$ is a real-valued function defined on the space of configurations $c: X \to \text{Bool}$. For a configuration $c$, the observable value is given by: $$\text{spinObs}(X, x)(c) = \begin{cases} 1 & \text{if } x \in X \text{ and } c(x) = \text{true} \\ -1 & \text{if } x \in X \text{ and } c(x) = \text{false} \\ 0 & \text{if } x \notin X \end{cases}$$ Here, $c(x)$ represents the spin state (represented as a boolean) at site $x$.

For any site $x$ in the lattice $\Lambda$, this definition identifies the **spin observable** $s(x)$ as an element of the global system of observables $\text{Gl}$. It is constructed by taking the local spin observable at site $x$—which evaluates to $+1$ if the spin state is "true" and $-1$ if it is "false"—and embedding it into the global system via the canonical linear injection from the local observables on the singleton region $\{x\}$.

In a lattice $\Lambda$, for any site $x \in \Lambda$ and any finite region $X \subseteq \Lambda$ containing $x$, the canonical injection $\text{ofLoc}_X$ of the local spin observable $\text{spinObs}(X, x)$ into the global system of observables is equal to the global spin observable $s(x)$. That is, a spin observable at site $x$ represents the same global observable regardless of the local region $X$ used to define it, provided $x \in X$.

Let $\Lambda$ be a lattice and $Gl$ be the global system of classical observables on that lattice. The function `spinMap` is the linear map from the space of finitely supported real-valued coefficients $\psi: \Lambda \to \mathbb{R}$ to the global system $Gl$. For any such profile of coefficients $\psi$, the map returns the weighted sum of local spin observables: $$\text{spinMap}(\psi) = \sum_{x \in \Lambda} \psi(x) s(x)$$ where $s(x)$ denotes the spin observable at site $x$, which takes the value $+1$ or $-1$ depending on the spin state at that site.

Let $\Lambda$ be a lattice and $s(x)$ denote the spin observable at a site $x \in \Lambda$ within the global system of observables $Gl$. For any site $x \in \Lambda$ and any real coefficient $m \in \mathbb{R}$, let $\delta_{x, m}: \Lambda \to \mathbb{R}$ be the finitely supported function that takes the value $m$ at site $x$ and $0$ at all other sites. The linear map $\text{spinMap}$, which sends a coefficient profile $\psi$ to the weighted sum of observables $\sum_{y \in \Lambda} \psi(y) s(y)$, satisfies: $$\text{spinMap}(\delta_{x, m}) = m \cdot s(x)$$

Let $\Lambda$ be a lattice and $Y \subseteq \Lambda$ be a finite set of sites. Given a weighting function $\psi: \Lambda \to \mathbb{R}$ (represented as a finitely supported function `Finsupp`) and a spin configuration $c: Y \to \{\text{true}, \text{false}\}$, the value of the linear combination of spin observables $\sum_{x \in Y} \psi(x) \cdot \text{spinObs}(Y, x)$ evaluated at the configuration $c$ is given by the sum: $$\left( \sum_{x \in Y} \psi(x) \cdot \text{spinObs}(Y, x) \right)(c) = \sum_{x \in Y} \psi(x) \cdot \begin{cases} 1 & \text{if } c(x) = \text{true} \\ -1 & \text{if } c(x) = \text{false} \end{cases}$$ where $\text{spinObs}(Y, x)$ is the observable that returns $1$ or $-1$ based on the spin state at site $x$.

Let $\Lambda$ be a lattice and $Gl(\Lambda)$ be the global system of classical observables on that lattice. The linear map $\text{spinMap} : (\Lambda \to_0 \mathbb{R}) \to Gl(\Lambda)$, which sends a finitely supported profile of real coefficients $\psi$ to the weighted sum of local spin observables $\sum_{x \in \Lambda} \psi(x) s(x)$, is injective. This implies that the spin observables at distinct sites are linearly independent in the global system.

For a lattice $\Lambda$ with spin states $\sigma = \{\text{true}, \text{false}\}$, let $\mathcal{A}$ be the net of local classical observables. The definition `act` represents the covariant group action of the translation group (the additive group $\Lambda$ considered as a multiplicative group) on this net. For each translation $a \in \Lambda$, this action provides a family of isomorphisms $\tau_a: \text{Obs}(X) \cong \text{Obs}(a+X)$ for all finite regions $X \subseteq \Lambda$. For an observable $f$ defined on a region $X$, the translated observable $\tau_a f$ is defined such that for any configuration $c$ on $a+X$, $(\tau_a f)(c) = f(c \circ \tau_a)$, where $(c \circ \tau_a)(x) = c(a+x)$ for $x \in X$.

The `rep` is the covariant representation of the translation group $\Lambda$ on the global system of observables $Gl(\Lambda)$ for a spin lattice. It defines a group homomorphism $\beta$ from the translation group to the symmetries of the global system, such that for any translation $a \in \Lambda$ and any local observable $f$ defined on a finite region $X \subseteq \Lambda$, the inclusion of the translated observable $\tau_a f$ into the global system is equal to the action of the global symmetry $\beta_a$ on the image of $f$ in the global system. This construction identifies how local translations extend to the entire algebra of spin observables.

In a spin lattice $\Lambda$, let $s(x)$ denote the spin observable at site $x \in \Lambda$, and let $\beta_a$ be the symmetry of the global system of observables induced by a translation $a \in \Lambda$. Then, translating the lattice by $a$ moves the spin observable at $x$ to the spin observable at $a + x$: $$\beta_a(s(x)) = s(a + x)$$

Let $\Lambda$ be a lattice. For any finitely supported configuration of real coefficients $\psi: \Lambda \to_0 \mathbb{R}$ and any translation $a \in \Lambda$, let $T_a$ be the translation operator on the space of configurations defined by $(T_a \psi)(x) = \psi(x - a)$. Let $\pi_a$ be the linear representation of the translation action on the global system of spin observables $Gl(\Lambda)$. The linear map $\text{spinMap}$, which maps a coefficient profile $\psi$ to its corresponding spin observable $\sum_{x \in \Lambda} \psi(x) s(x)$, satisfies the relation: $$\text{spinMap}(T_a \psi) = \pi_a(\text{spinMap}(\psi))$$ This indicates that the map from coefficients to observables intertwines the action of translations.

Let $\Lambda$ be a lattice and $Gl(\Lambda)$ be the global system of classical observables on that lattice. The spin-wave sector is the $\mathbb{R}$-submodule of $Gl(\Lambda)$ defined as the range of the linear map $\text{spinMap}: (\Lambda \to_0 \mathbb{R}) \to Gl(\Lambda)$. Physically, this sector represents the subspace of all observables that are finite linear combinations of local spin observables, taking the form $\sum_{x \in \Lambda} \psi(x) s(x)$ for finitely supported coefficients $\psi(x)$.

Let $\Lambda$ be a lattice and $Gl(\Lambda)$ be the global system of observables on the spin lattice. Let $\mathcal{S} \subseteq Gl(\Lambda)$ denote the spin-wave sector, defined as the $\mathbb{R}$-submodule of observables consisting of finite linear combinations of local spin observables. For any translation $a \in \Lambda$ and any observable $w \in Gl(\Lambda)$, if $w$ belongs to the spin-wave sector $\mathcal{S}$, then its image under the translation action $\pi(a)w$ also belongs to $\mathcal{S}$, where $\pi$ is the linear representation of the translation group on the global system.

Let $\Lambda$ be a lattice and $\mathcal{S} \subseteq Gl(\Lambda)$ be the spin-wave sector, defined as the $\mathbb{R}$-submodule of global observables consisting of finite linear combinations of local spin observables. The map `actionW` is the group homomorphism from the translation group $\Lambda$ to the space of linear endomorphisms $\text{End}_{\mathbb{R}}(\mathcal{S})$ that represents the action of lattice translations on the spin-wave sector. For each translation $a \in \Lambda$, the corresponding linear operator is the restriction of the global translation action $\pi(a)$ to the subspace $\mathcal{S}$.

The unit-cell structure for the spin-wave sector $\mathcal{S}$ on a lattice $\Lambda$ identifies the space of linear spin observables with the space of finitely supported real-valued functions $\Lambda \to_0 \mathbb{R}$, representing one real mode per lattice cell. This structure is established via a linear equivalence $\Phi: (\Lambda \to_0 \mathbb{R}) \cong \mathcal{S}$ derived from the injective map $\text{spinMap}$, which sends a coefficient profile $\psi$ to the observable $\sum_{x \in \Lambda} \psi(x) s(x)$. Under this identification, the lattice translation action $\text{actionW}$ on the observables in $\mathcal{S}$ corresponds to the translation of the coefficient configurations on $\Lambda$.

Let $\Lambda$ be a lattice and $\mathcal{S}$ be the spin-wave sector with its associated unit-cell structure $U$. For any finitely supported hopping amplitudes $H: \Lambda \to_0 \text{End}_{\mathbb{R}}(\mathbb{R})$ and the corresponding hopping operator $D$ on $\mathcal{S}$, and for any additive character $\chi: \Lambda \to \mathbb{R}$, the Bloch matrix of $D$ at crystal momentum $\chi$ is given by the sum: $$\text{blochMatrix}(U, \chi, D) = \sum_{n \in \Lambda} \chi(n) \cdot H_n$$ where $H_n$ is the hopping amplitude associated with the lattice translation $n$.

Consider the complexified spin-wave sector of a lattice $\Lambda$, which is identified with the space of lattice configurations $\Lambda \to_0 \mathbb{C}$ via a unit-cell structure. Let $H: \Lambda \to \text{End}_{\mathbb{C}}(\mathbb{C})$ be a finitely supported map where $H_n$ represents the hopping amplitude to a site displaced by $n \in \Lambda$. For any complex character $\chi: \Lambda \to \mathbb{C}$, the Bloch matrix of the hopping operator associated with $H$ at momentum $\chi$ is given by the sum: $$\sum_{n \in \Lambda} \chi(n) H_n$$

Let $\Lambda$ be a lattice and $Gl(\Lambda)$ be the global system of classical observables on the spin lattice. Let $\mathcal{S} \subseteq Gl(\Lambda)$ be the spin-wave sector, defined as the subspace of observables that are finite linear combinations of local spin observables. Let $D: Gl(\Lambda) \to Gl(\Lambda)$ be a channel that commutes with the global translation action $\beta$ on $Gl(\Lambda)$, such that $D(\beta_a f) = \beta_a(D f)$ for all $a \in \Lambda$ and $f \in Gl(\Lambda)$. If $D$ preserves the spin-wave sector, i.e., $D(w) \in \mathcal{S}$ for all $w \in \mathcal{S}$, then the restricted linear map $D|_{\mathcal{S}}: \mathcal{S} \to \mathcal{S}$ is equivariant with respect to the translation action $\pi$ on the sector, satisfying $D|_{\mathcal{S}}(\pi_a w) = \pi_a(D|_{\mathcal{S}} w)$ for all $a \in \Lambda$ and $w \in \mathcal{S}$.

Let $\Lambda$ be a lattice and $Gl(\Lambda)$ be the global system of classical observables on that lattice. Let $\mathcal{S} \subseteq Gl(\Lambda)$ be the spin-wave sector, consisting of real linear combinations of local spin observables. Suppose $D: Gl(\Lambda) \to Gl(\Lambda)$ is a channel that commutes with the translation action and preserves the spin-wave sector (i.e., $D(\mathcal{S}) \subseteq \mathcal{S}$). Let $D_{\mathbb{C}}: \mathcal{S}_{\mathbb{C}} \to \mathcal{S}_{\mathbb{C}}$ be the complexification of the restriction of $D$ to $\mathcal{S}$. For any complex character (crystal momentum) $\chi: \Lambda \to \mathbb{C}$ and any element $v$ in the complexified spin-wave sector $\mathcal{S}_{\mathbb{C}}$, the Bloch relation states: $$\text{ev}_\chi \left( \Phi^{-1} (D_{\mathbb{C}} v) \right) = \text{Bloch}(D_{\mathbb{C}}, \chi) \left( \text{ev}_\chi (\Phi^{-1} v) \right)$$ where: - $\Phi: (\Lambda \to_0 \mathbb{C}) \cong \mathcal{S}_{\mathbb{C}}$ is the linear isomorphism provided by the complexified unit-cell structure, identifying a coefficient profile with an observable. - $\text{ev}_\chi$ is the Fourier evaluation map at character $\chi$. - $\text{Bloch}(D_{\mathbb{C}}, \chi)$ is the Bloch matrix of the operator $D_{\mathbb{C}}$ at momentum $\chi$.

On the 1D integer lattice $\mathbb{Z}$, let $U_{\mathbb{C}}$ be the complexified unit-cell structure that identifies the spin-wave sector (the space of linear spin observables) with the space of finitely supported complex-valued configurations $\mathbb{Z} \to_0 \mathbb{C}$. Consider a hopping operator $D$ defined by unit amplitudes to the two nearest neighbors, corresponding to the hopping configuration $H = \delta_1 \text{id} + \delta_{-1} \text{id}$, where $\delta_n$ is the Dirac delta at lattice site $n$ and $\text{id}$ is the identity map on the single-site degree of freedom. For any complex character $\chi: \mathbb{Z} \to \mathbb{C}$ representing a crystal momentum, the Bloch matrix of $D$ at $\chi$ is given by: $$\text{blochMatrix}(U_{\mathbb{C}}, \chi, D) = (\chi(1) + \chi(-1)) \cdot \text{id}$$ In terms of a momentum $k$ where $\chi(n) = e^{ikn}$, this result corresponds to the dispersion relation $2 \cos k$.