PhyslibAlpha.CondensedMatter.Crystal.MomentumSectors

Given a representation $\beta : G \to \text{End}_K(V)$ of a translation group $G$ on a vector space $V$ over a field $K$, and a character $\chi : G \to K$, the **momentum sector** is the submodule of $V$ consisting of all vectors $v$ that satisfy $\beta(g)v = \chi(g)v$ for all $g \in G$. These vectors correspond to the Bloch waves associated with the crystal momentum defined by $\chi$.

Let $G$ be a group representing translations and $V$ be a $K$-module (typically a vector space) with a representation $\beta : G \to \text{End}_K(V)$. For a character or phase function $\chi : G \to K$, a vector $v \in V$ belongs to the crystal momentum sector $\text{sector}(\beta, \chi)$ if and only if $\beta(g)v = \chi(g) \cdot v$ holds for all $g \in G$.

Given a group $G$ (representing translations), a $K$-module $V$ (the space of states), a representation $\beta: G \to \text{End}_K(V)$, and a function $\chi: G \to K$ (representing a crystal momentum or phase), the coboundary submodule is the $K$-submodule of $V$ spanned by all vectors of the form $\beta(g)v - \chi(g) \cdot v$ for all $g \in G$ and $v \in V$. This submodule is the kernel of the projection onto the momentum fiber associated with $\chi$.

Let $G$ be a group representing translations and $V$ be a $K$-module representing the space of states. Given a representation $\beta: G \to \text{End}_K(V)$ and a function $\chi: G \to K$ representing a crystal momentum, for any $g \in G$ and $v \in V$, the vector $\beta(g)v - \chi(g) \cdot v$ is an element of the coboundary submodule $\text{cobound}(\beta, \chi)$.

Given a group $G$ (representing translations), a $K$-module $V$ (the space of states), a representation $\beta: G \to \text{End}_K(V)$, and a function $\chi: G \to K$ (representing a crystal momentum or character), the **momentum fiber** is the quotient space $V / \text{cobound}(\beta, \chi)$. It is defined as $V$ modulo the submodule generated by all vectors of the form $\beta(g)v - \chi(g) \cdot v$ for all $g \in G$ and $v \in V$. Physically, this quotient represents the space of states where the translation action $\beta(g)$ is identified with the phase $\chi(g)$.

Given a group $G$, a $K$-module $V$, a representation $\beta: G \to \text{End}_K(V)$, and a crystal momentum $\chi: G \to K$, this is the canonical $K$-linear projection from the space $V$ to the momentum fiber $V(\chi)$, where the fiber is the quotient space $V / \text{cobound}(\beta, \chi)$.

Let $G$ be a group (representing lattice translations), $V$ a $K$-module (the space of states), and $\beta: G \to \text{End}_K(V)$ a representation of $G$ on $V$. Let $\chi: G \to K$ be a function representing crystal momentum (a character). Let $V(\chi)$ denote the momentum fiber, defined as the quotient space $V / \text{span}\{\beta(g)v - \chi(g)v \mid g \in G, v \in V\}$, and let $\pi: V \to V(\chi)$ be the canonical projection. For any translation $g \in G$ and vector $v \in V$, it holds that: $$\pi(\beta(g)v) = \chi(g) \cdot \pi(v)$$ This shows that in the momentum fiber, the translation action $\beta(g)$ acts simply as scalar multiplication by the phase $\chi(g)$.

For a monoid $G$ and $K$-modules $V$ and $W$, let $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$ be representations (actions) of $G$. A $K$-linear map $T: V \to W$ is **equivariant** if it intertwines these actions, such that for all $g \in G$ and $v \in V$, the condition $T(\beta(g)v) = \beta'(g)(Tv)$ holds.

Let $V$ and $W$ be modules over a field $K$, and let $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$ be representations (actions) of a translation group $G$. If $T: V \to W$ is an equivariant linear map (satisfying $T(\beta(g)v) = \beta'(g)(Tv)$ for all $g \in G, v \in V$), then for any character $\chi: G \to K$, if a vector $v$ belongs to the momentum sector of $\chi$ in $V$ (meaning $\beta(g)v = \chi(g)v$ for all $g \in G$), then its image $T(v)$ belongs to the momentum sector of $\chi$ in $W$.

Given an equivariant linear map $T: V \to W$ between two $K$-modules equipped with representations $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$ of a translation group $G$, and given a character $\chi: G \to K$, the **sector map** is the linear map obtained by restricting $T$ to the momentum sector of $\chi$. Because $T$ is equivariant, it maps any vector $v$ in the sector of $V$ (satisfying $\beta(g)v = \chi(g)v$ for all $g \in G$) to a vector $T(v)$ in the sector of $W$ (satisfying $\beta'(g)T(v) = \chi(g)T(v)$ for all $g \in G$).

Let $V$ and $W$ be modules over a field $K$, and let $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$ be representations of a group $G$. Given an equivariant linear map $T: V \to W$ (satisfying $T(\beta(g)v) = \beta'(g)T(v)$ for all $g \in G, v \in V$) and a character $\chi: G \to K$, let $T_\chi: \text{sector}(\beta, \chi) \to \text{sector}(\beta', \chi)$ be the induced map between momentum sectors. For any vector $v$ in the momentum sector $\text{sector}(\beta, \chi)$, the value of the sector map $T_\chi(v)$ is equal to $T(v)$.

Let $G$ be a group and $K$ a ring. Let $V$ and $W$ be $K$-modules equipped with representations $\beta : G \to \text{End}_K(V)$ and $\beta' : G \to \text{End}_K(W)$. If $T : V \to W$ is an equivariant linear map (i.e., $T(\beta(g)v) = \beta'(g)T(v)$ for all $g \in G$ and $v \in V$), then for any function $\chi : G \to K$, the coboundary submodule of $V$ associated with $\chi$ is contained in the preimage of the coboundary submodule of $W$ under $T$. That is, $$\text{cobound}(\beta, \chi) \le T^{-1}(\text{cobound}(\beta', \chi))$$ where $\text{cobound}(\beta, \chi)$ is the $K$-submodule spanned by all vectors of the form $\beta(g)v - \chi(g)v$ for $g \in G$ and $v \in V$.

Let $G$ be a group (or monoid), $K$ a ring, and $V, W$ be $K$-modules equipped with representations $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$. Given an equivariant linear map $T: V \to W$ (satisfying $T(\beta(g)v) = \beta'(g)T(v)$ for all $g \in G, v \in V$) and a function $\chi: G \to K$ representing crystal momentum, the map `fiberMap` is the induced $K$-linear map $T_\chi: \text{Fiber}(\beta, \chi) \to \text{Fiber}(\beta', \chi)$ between the momentum fibers. This map is well-defined because $T$ maps the coboundary submodule of $V$ into the coboundary submodule of $W$, such that $T_\chi([v]) = [T(v)]$ for any $v \in V$.

Let $G$ be a group (or monoid) and $K$ a ring. Let $V$ and $W$ be $K$-modules equipped with representations $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$. If $T: V \to W$ is an equivariant linear map such that $T(\beta(g)v) = \beta'(g)T(v)$ for all $g \in G, v \in V$, then for any crystal momentum $\chi: G \to K$ and any vector $v \in V$, the induced map $T_\chi$ between the momentum fibers satisfies: $$T_\chi([v]) = [T(v)]$$ where $[v]$ denotes the image of $v$ under the canonical projection $\pi_\chi: V \to \text{Fiber}(\beta, \chi)$ to the momentum fiber.

Let $V$ and $W$ be $K$-modules equipped with representations $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$ of a group $G$. Given an equivariant linear map $T: V \to W$ and a crystal momentum phase $\chi: G \to K$, let $T_\chi: \text{Fiber}(\beta, \chi) \to \text{Fiber}(\beta', \chi)$ be the induced linear map between the momentum fibers. Then, $T_\chi \circ \pi_V = \pi_W \circ T$, where $\pi_V: V \to \text{Fiber}(\beta, \chi)$ and $\pi_W: W \to \text{Fiber}(\beta', \chi)$ are the canonical projections to the respective momentum fibers.

Let $V$ be a $K$-module and $\beta: G \to \text{End}_K(V)$ be a representation of a monoid $G$ on $V$. The identity linear map $\text{id}_V: V \to V$ is equivariant with respect to the action $\beta$ on both the domain and the codomain. That is, for all $g \in G$ and $v \in V$, the condition $\text{id}_V(\beta(g)v) = \beta(g)(\text{id}_V v)$ is satisfied.

Let $V, W,$ and $U$ be $K$-modules, and let $\beta: G \to \text{End}_K(V)$, $\beta': G \to \text{End}_K(W)$, and $\beta'': G \to \text{End}_K(U)$ be representations of a monoid $G$. If $T: V \to W$ is an equivariant linear map with respect to $\beta$ and $\beta'$, and $S: W \to U$ is an equivariant linear map with respect to $\beta'$ and $\beta''$, then their composition $S \circ T: V \to U$ is also an equivariant linear map with respect to $\beta$ and $\beta''$.

Let $G$ be a group, $V$ a $K$-module, and $\beta: G \to \text{End}_K(V)$ a representation of $G$ on $V$. For any crystal momentum $\chi: G \to K$, the linear map on the momentum fiber $\text{Fiber}(\beta, \chi)$ induced by the identity map $\text{id}_V$ is the identity map on $\text{Fiber}(\beta, \chi)$.

Let $G$ be a group and $K$ be a ring. Let $V, W, U$ be $K$-modules equipped with representations $\beta: G \to \text{End}_K(V)$, $\beta': G \to \text{End}_K(W)$, and $\beta'': G \to \text{End}_K(U)$, respectively. For a given crystal momentum $\chi: G \to K$, let $T: V \to W$ and $S: W \to U$ be equivariant linear maps. Then the linear map induced on the momentum fibers by the composition $S \circ T$, denoted $(S \circ T)_\chi: \text{Fiber}(\beta, \chi) \to \text{Fiber}(\beta'', \chi)$, is equal to the composition of the induced linear maps $S_\chi: \text{Fiber}(\beta', \chi) \to \text{Fiber}(\beta'', \chi)$ and $T_\chi: \text{Fiber}(\beta, \chi) \to \text{Fiber}(\beta', \chi)$, such that $(S \circ T)_\chi = S_\chi \circ T_\chi$.

Let $V$ be a $K$-module and $\beta: G \to \text{End}_K(V)$ be a representation of a group $G$. Let $D: V \to V$ be a $K$-linear map that is equivariant with respect to $\beta$, meaning $D(\beta(g)v) = \beta(g)(Dv)$ for all $g \in G$ and $v \in V$. For any character $\chi: G \to K$, if a vector $v \in V$ belongs to the momentum sector of $\chi$ (i.e., $\beta(g)v = \chi(g)v$ for all $g \in G$), then $Dv$ also belongs to the same momentum sector.

Given a group $G$ representing translations, a $K$-module $V$, a representation $\beta: G \to \text{End}_K(V)$, and a crystal momentum $\chi: G \to K$, the map `sectorToFiber` is the canonical $K$-linear map from the momentum sector of $\chi$ to the momentum fiber of $\chi$. It is defined as the composition of the inclusion map $\iota: \text{sector}(\beta, \chi) \hookrightarrow V$ and the natural projection $\pi: V \to \text{Fiber}(\beta, \chi)$ onto the quotient space. This map associates each Bloch wave of momentum $\chi$ with its corresponding class in the momentum fiber.

Let $G$ be a group representing translations and $K$ be a ring. Let $V$ and $W$ be $K$-modules equipped with representations $\beta: G \to \text{End}_K(V)$ and $\beta': G \to \text{End}_K(W)$, respectively. Let $\chi: G \to K$ be a character representing a crystal momentum. Suppose $T: V \to W$ is an equivariant $K$-linear map (i.e., $T(\beta(g)v) = \beta'(g)T(v)$ for all $g \in G, v \in V$). Let: - $T_{\text{sector}, \chi} : \text{sector}(\beta, \chi) \to \text{sector}(\beta', \chi)$ be the restriction of $T$ to the momentum sectors; - $T_{\text{fiber}, \chi} : \text{Fiber}(\beta, \chi) \to \text{Fiber}(\beta', \chi)$ be the map induced by $T$ between the momentum fibers; - $\text{sectorToFiber}_{\beta, \chi}$ and $\text{sectorToFiber}_{\beta', \chi}$ be the canonical linear maps from the sectors to the fibers. Then the following identity holds: $$T_{\text{fiber}, \chi} \circ \text{sectorToFiber}_{\beta, \chi} = \text{sectorToFiber}_{\beta', \chi} \circ T_{\text{sector}, \chi}$$ This expresses the naturality of the canonical map from the crystal momentum sector to the fiber with respect to equivariant maps.