PhyslibAlpha.ProbabilisticTheory.LocalNet.Bloch

For a covariant representation $\rho$ of a group $G$ acting on a real vector space $B$, the **complex linear action** is the group representation $\pi_{\mathbb{C}} : G \to \text{End}_{\mathbb{C}}(\mathbb{C} \otimes_{\mathbb{R}} B)$ that maps each group element $g \in G$ to the complex linear endomorphism obtained by the base change of the real linear action $\rho.\text{linearAction}(g)$. Effectively, this extends the real-linear symmetry transformations of the system to the complexified space $\mathbb{C} \otimes_{\mathbb{R}} B$.

Let $B$ be a real state space (an order unit space) and $\rho$ be a covariant representation of a symmetry group $G$ on $B$. Let $D: B \to B$ be a channel. If $D$ commutes with the symmetry representation (i.e., $D(\beta(g)b) = \beta(g)(D(b))$ for all $g \in G$ and $b \in B$), then the complexified linear map $D_{\mathbb{C}} : \mathbb{C} \otimes_{\mathbb{R}} B \to \mathbb{C} \otimes_{\mathbb{R}} B$ (obtained by the base change of $D$ to $\mathbb{C}$) is equivariant with respect to the complexified linear action $\pi_{\mathbb{C}}$ of $G$ on the complexified space $\mathbb{C} \otimes_{\mathbb{R}} B$. That is, for all $g \in G$: $$D_{\mathbb{C}} \circ \pi_{\mathbb{C}}(g) = \pi_{\mathbb{C}}(g) \circ D_{\mathbb{C}}$$

Given a unit-cell structure on a vector space $V$ over a field $K$ that defines an isomorphism $\Phi : (\Lambda \to_0 M) \cong V$ between the space of lattice configurations and $V$, this function maps a linear operator $D : V \to V$ to its representation on the configuration space $(\Lambda \to_0 M)$. Here, $(\Lambda \to_0 M)$ denotes the space of finitely supported functions from the lattice $\Lambda$ to the unit-cell degrees of freedom $M$. The resulting operator is defined by the conjugation $\Phi^{-1} \circ D \circ \Phi$.

Let $V$ be a vector space over a field $K$ representing a system with lattice translations given by an abelian group $\Lambda$. Let $U$ be a unit-cell structure that identifies $V$ with the configuration space $\Lambda \to_0 M$ (finitely supported maps from the lattice to the unit-cell degrees of freedom $M$) via a linear isomorphism $\Phi : (\Lambda \to_0 M) \cong V$. For any lattice translation $a \in \Lambda$ and any state $v \in V$, the inverse identification map $\Phi^{-1}$ commutes with the translation action: $$\Phi^{-1}(\beta_a v) = T_a (\Phi^{-1} v)$$ where $\beta_a$ is the translation action on $V$ and $T_a$ is the standard translation operator on the configuration space $\Lambda \to_0 M$.

Let $V$ be a vector space over a field $K$ with a translation action $\beta$ of an abelian group (lattice) $\Lambda$. Let $U$ be a unit-cell structure that identifies $V$ with the configuration space $\Lambda \to_0 M$ (the space of finitely supported functions from the lattice to the unit-cell degrees of freedom $M$) via a linear isomorphism $\Phi: (\Lambda \to_0 M) \cong V$. If a linear operator $D: V \to V$ is translation-invariant (i.e., it is equivariant with respect to the action $\beta$), then its representation on the configuration space, $U.\text{conj}(D) = \Phi^{-1} \circ D \circ \Phi$, is equivariant with respect to the standard translation action on $\Lambda \to_0 M$.

Let $V$ be a vector space over a field $K$ with a unit-cell structure $U$ that identifies $V$ with the space of lattice configurations $\Lambda \to_0 M$, where $\Lambda$ is an abelian group of translations and $M$ represents the degrees of freedom within a unit cell. Given a linear operator $D: V \to V$ and a crystal momentum (character) $\chi: \Lambda \to K$, the **Bloch matrix** of $D$ at $\chi$ is the linear map $M \to M$ defined as the symbol of the conjugated operator $\Phi^{-1} \circ D \circ \Phi$ at $\chi$. For a translation-invariant operator, this matrix describes its action on the unit-cell degrees of freedom within the momentum sector $\chi$.

Let $V$ be a vector space over a field $K$ with a unit-cell structure $U$ that identifies $V$ with the space of lattice configurations $\Lambda \to_0 M$ via a linear isomorphism $\Phi: (\Lambda \to_0 M) \cong V$, where $\Lambda$ is an abelian group of translations and $M$ represents the degrees of freedom within a unit cell. Let $D: V \to V$ be a linear operator that is translation-invariant (i.e., equivariant with respect to the lattice translation action $\beta$ on $V$). For any crystal momentum (character) $\chi: \Lambda \to K$ and any vector $v \in V$, the Fourier evaluation $\text{ev}_\chi$ (which maps a configuration $\psi \in \Lambda \to_0 M$ to $\sum_{n \in \Lambda} \chi(n) \cdot \psi(n)$) satisfies the relation: $$\text{ev}_\chi(\Phi^{-1}(D v)) = D(\chi)(\text{ev}_\chi(\Phi^{-1} v))$$ where $D(\chi)$ is the Bloch matrix of $D$ at momentum $\chi$. In other words, for a translation-invariant system, the Fourier evaluation intertwines the action of the operator $D$ with its corresponding Bloch matrix in the momentum sector.

Let $V$ be a vector space over a field $K$ equipped with a unit-cell structure $U$ that identifies $V$ with the space of lattice configurations $\Lambda \to_0 M$, where $\Lambda$ is an abelian group of translations and $M$ represents the degrees of freedom within a unit cell. For any linear operators $D_1, D_2: V \to V$ and any crystal momentum (additive character) $\chi: \Lambda \to K$, if $D_1$ is translation-invariant (equivariant with respect to the lattice translation action $\beta$), then the Bloch matrix of the composition $D_1 \circ D_2$ at momentum $\chi$ is the composition of their individual Bloch matrices: $$\text{blochMatrix}_U(\chi, D_1 \circ D_2) = \text{blochMatrix}_U(\chi, D_1) \circ \text{blochMatrix}_U(\chi, D_2)$$

Given a unit-cell structure $U$ that identifies a vector space $V$ with the space of lattice configurations $\Lambda \to_0 M$ via an isomorphism $\Phi$, this definition constructs a linear operator on $V$ for a given collection of hopping amplitudes $H$. Here $H$ is a finitely supported map where $H_n \in \text{End}_K(M)$ represents the hopping amplitude to a unit cell displaced by the lattice vector $n \in \Lambda$. The resulting operator on $V$ is defined by transporting the translation-invariant hopping dynamics from the configuration space to $V$ using $\Phi$.

Let $V$ be a vector space over a field $K$ equipped with a unit-cell structure $U$ relative to a lattice $\Lambda$ and a unit-cell space $M$. For any collection of finitely supported hopping amplitudes $H: \Lambda \to_0 \text{End}_K(M)$, the resulting hopping operator $U.\text{hopping } H: V \to V$ is equivariant with respect to the lattice translation action $\beta$ on $V$.

Let $V$ be a vector space with a unit-cell structure $U$ over a lattice $\Lambda$ and a unit-cell space $M$. For any finitely supported collection of hopping amplitudes $H: \Lambda \to \text{End}_K(M)$ and its associated hopping operator $\mathcal{H}$ on $V$, the Bloch matrix of $\mathcal{H}$ at a crystal momentum (represented by the character $\chi: \Lambda \to K$) is given by the Fourier transform of the hopping amplitudes: $$\text{blochMatrix}(U, \chi, \mathcal{H}) = \sum_{n \in \Lambda} \chi(n) H_n$$ where $H_n$ is the hopping matrix to a cell displaced by the lattice vector $n$.

Given a group $\Lambda$ (representing lattice translations) and a real vector space $V$, let $\beta: \Lambda \to \text{End}_{\mathbb{R}}(V)$ be a representation of $\Lambda$ on $V$. The complexified action is the representation of $\Lambda$ on the complexified vector space $\mathbb{C} \otimes_{\mathbb{R}} V$ where each element $a \in \Lambda$ acts as the complexification (base change) of the linear map $\beta(a)$.

Let $V$ be a real vector space and $\Lambda$ be a group with a representation $\beta: \Lambda \to \text{End}_{\mathbb{R}}(V)$. If $D: V \to V$ is an $\mathbb{R}$-linear map that is equivariant with respect to $\beta$ (i.e., it commutes with the action of $\Lambda$ such that $D \circ \beta(g) = \beta(g) \circ D$ for all $g \in \Lambda$), then its complexification $D_{\mathbb{C}}: \mathbb{C} \otimes_{\mathbb{R}} V \to \mathbb{C} \otimes_{\mathbb{R}} V$ is equivariant with respect to the complexified action $\beta_{\mathbb{C}}$ of $\Lambda$ on the complexified vector space $\mathbb{C} \otimes_{\mathbb{R}} V$.

Given a unit-cell structure $U$ over the real numbers $\mathbb{R}$ for a representation $\beta$ of a lattice group $\Lambda$ on a real vector space $V$, the `complexify` function constructs a corresponding unit-cell structure over the complex numbers $\mathbb{C}$. This structure identifies the complexified vector space $\mathbb{C} \otimes_{\mathbb{R}} V$ with the space of configurations $\Lambda \to_0 M_{\mathbb{C}}$, where $M_{\mathbb{C}} = \mathbb{C} \otimes_{\mathbb{R}} M$ represents the complexified degrees of freedom within a unit cell. This identification is equivariant with respect to the complexified translation action of $\Lambda$ on $\mathbb{C} \otimes_{\mathbb{R}} V$.

Let $\Lambda$ be a group representing lattice translations and $B$ be a group representing internal site indices within a unit cell. The group homomorphism `cellTranslations` maps a translation element $a \in \Lambda$ to the pair $(a, 0) \in \Lambda \times B$, where $0$ is the identity element of $B$. This map formalizes the notion that a translation by a lattice vector $a$ shifts the unit cell position but leaves the internal site label within the cell unchanged.

Let $\Lambda$ be an abelian group representing lattice translations, $B$ be a set of internal site indices within a unit cell, and $M$ be a module representing the degrees of freedom at each site. Let $\psi : \Lambda \to (B \to_0 M)$ be a configuration where each lattice point $n \in \Lambda$ is mapped to a local configuration of sites $B$. For any translation $a \in \Lambda$, let $\tau_a$ denote the translation operator acting on the lattice. The "uncurry" operation $\text{uncurry}(\psi) : (\Lambda \times B) \to M$ maps a pair $(n, b)$ to the value $\psi(n)(b)$. The theorem states that uncurrying a translated configuration is equivalent to translating the uncurried configuration by the lattice vector $(a, 0) \in \Lambda \times B$: $$\text{uncurry}(\tau_a \psi) = \tau_{(a, 0)} (\text{uncurry } \psi)$$ where the translation on the right-hand side is defined by the lattice translation homomorphism $a \mapsto (a, 0)$.

Given a unit-cell structure $U$ representing a system on a product lattice $\Lambda \times B$ with local degrees of freedom $M$ (where $\Lambda$ represents the lattice translations and $B$ the internal site indices), this definition constructs a "regrouped" unit-cell structure over the lattice $\Lambda$. In this new structure, the local degrees of freedom of a single unit cell are given by $B \to_0 M$, the space of all configurations across the internal sites $B$. The isomorphism $\Phi$ for the regrouped structure maps a configuration $\psi: \Lambda \to (B \to_0 M)$ to the state space $V$ by first uncurrying it to a configuration on the product space $\Lambda \times B$ and then applying the isomorphism from the original structure $U$. The translation action for an element $a \in \Lambda$ in this new structure is defined by the original representation $\beta$ evaluated at the lattice translation $(a, 0)$.