PhyslibAlpha.CondensedMatter.SpinLattice.PeriodicWaves

The set of sites in a two-component periodic lattice is defined as the Cartesian product $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Each site is represented by a pair $(n, b)$, where $n \in \mathbb{Z}$ identifies the unit cell and $b \in \mathbb{Z}/2\mathbb{Z}$ (or $\{0, 1\}$) distinguishes between the two internal degrees of freedom or basis sites within that cell.

The space of complex amplitudes within a single unit cell of the lattice, defined as the set of finitely supported functions from the internal labels $\mathbb{Z}/2\mathbb{Z}$ to the complex numbers $\mathbb{C}$. This represents the two degrees of freedom (modes) per cell in the periodic system.

The space $\text{Sector}$ is the complex vector space of linear spin observables on a two-component periodic lattice. It is defined as the complexification of the real spin sector, $\mathbb{C} \otimes_{\mathbb{R}} \mathcal{V}$, where $\mathcal{V}$ is the real vector space of linear observables on the lattice sites $S = \mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. This space represents the set of all complex linear combinations of local spin observables across the lattice.

This definition provides the group homomorphism $\tau : \mathbb{Z} \to \text{End}_{\mathbb{C}}(\text{Sector})$ that describes how lattice translations act on the space of complexified linear spin observables $\text{Sector}$. For a translation $n \in \mathbb{Z}$, the linear operator $\tau(n)$ is obtained by complexifying the real action of site translations on the observable net, restricted to shifts of the unit cell index. Specifically, it represents the action induced by the spatial shift $(m, b) \mapsto (m + n, b)$ on the lattice sites $S = \mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, where $m$ is the cell index and $b$ is the internal basis label.

The unit-cell structure for the two-component periodic lattice system. It provides an equivariant isomorphism between the complexified sector of linear spin observables ($\text{Sector}$) and the space of finitely supported configurations $\psi: \mathbb{Z} \to \text{Cell}$, where $\text{Cell} \cong \mathbb{C}^2$ represents the internal degrees of freedom at each lattice site. This structure is constructed by complexifying the real unit-cell structure on the lattice sites $S = \mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and then regrouping the internal labels $b \in \{0, 1\}$ so that the global system is indexed by the cell index $n \in \mathbb{Z}$ with a two-dimensional local amplitude space.

Given a function $m: \mathbb{Z}/2\mathbb{Z} \to \mathbb{R}$ representing the on-site masses (or squared frequencies) for the two internal modes, this definition is the linear operator $M_{\text{cell}}: \text{Cell} \to \text{Cell}$ on the unit cell amplitude space. For any amplitude vector $\psi \in \text{Cell} \cong \mathbb{C}^2$, the operator acts as $(M_{\text{cell}} \psi)(b) = m(b) \psi(b)$ for each component $b \in \{0, 1\}$.

Let $\psi: \mathbb{Z} \to \mathbb{C}^2$ be a finitely supported configuration of complex amplitudes on a one-dimensional lattice, where each site $n \in \mathbb{Z}$ has two internal degrees of freedom. Given a mass function $m: \mathbb{Z}/2\mathbb{Z} \to \mathbb{R}$ for the internal modes and a stiffness constant $\kappa \in \mathbb{R}$, the wave operator $W$ is a linear operator acting on the space of configurations. For any site $n$, the operator is defined as: $$(W \psi)(n) = M \psi(n) + \kappa \left( 2 \psi(n) - \psi(n-1) - \psi(n+1) \right)$$ where $M = \text{diag}(m(0), m(1))$ is the diagonal on-site mass matrix. This operator represents the combination of on-site restoring forces and the discrete Laplacian acting on the lattice.

For any mass function $m: \mathbb{Z}/2\mathbb{Z} \to \mathbb{R}$ and stiffness constant $\kappa \in \mathbb{R}$, the wave operator $W$ for the two-component periodic lattice system is equivariant with respect to the translation action on the lattice $\mathbb{Z}$. That is, for any lattice vector $a \in \mathbb{Z}$ and any finitely supported configuration $\psi: \mathbb{Z} \to \mathbb{C}^2$, the operator $W$ commutes with the translation operator $T_a$, such that $W(T_a \psi) = T_a (W \psi)$, where $(T_a \psi)(n) = \psi(n - a)$.

Given a mass function $m: \mathbb{Z}/2\mathbb{Z} \to \mathbb{R}$ and a stiffness constant $\kappa \in \mathbb{R}$, the sector wave operator is a linear operator on the space of complexified linear spin observables ($\text{Sector}$). It is defined by transporting the periodic wave operator $W$ from the space of finitely supported lattice configurations ($\mathbb{Z} \to_0 \mathbb{C}^2$) to the observable sector using the unit-cell isomorphism $\Phi$. Specifically, the operator is given by the composition $\Phi \circ W \circ \Phi^{-1}$, where $W$ is the operator representing the on-site restoring forces and the discrete Laplacian.

Consider a two-component periodic lattice with a mass function $m: \mathbb{Z}/2\mathbb{Z} \to \mathbb{R}$ and a stiffness constant $\kappa \in \mathbb{R}$. Let $W$ be the wave operator acting on the space of finitely supported configurations $\psi: \mathbb{Z} \to \mathbb{C}^2$, and let $\Phi$ be the unit-cell isomorphism that identifies the complexified linear spin observable sector with these configurations. For any observable $v \in \text{Sector}$ and its corresponding configuration $\psi = \Phi^{-1}(v)$, and for any additive character $\chi: \mathbb{Z} \to \mathbb{C}$ representing the crystal momentum, the Fourier evaluation $\text{ev}_\chi$ intertwines the wave operator with its Bloch matrix (symbol) $\hat{W}(\chi)$: $$\text{ev}_\chi (W \psi) = \hat{W}(\chi) (\text{ev}_\chi \psi)$$ where the Fourier evaluation is defined by the sum $\text{ev}_\chi \psi = \sum_{n \in \mathbb{Z}} \chi(n) \psi(n)$.

In a periodic lattice with two components per cell, given a mass function $m: \mathbb{Z}_2 \to \mathbb{R}$ representing on-site restoring terms, a stiffness coefficient $\kappa \in \mathbb{R}$, an additive character $\chi: \mathbb{Z} \to \mathbb{C}$ representing the crystal momentum, and an internal component index $b \in \mathbb{Z}_2$, the Bloch eigenvalue is defined as: $$m(b) + \kappa(2 - \chi(1) - \chi(-1))$$ This value is interpreted as the squared frequency $\omega^2$ for the wave dynamics of the component $b$.

In a periodic lattice with two components per cell, let $m: \mathbb{Z}_2 \to \mathbb{R}$ be the mass function and $\kappa \in \mathbb{R}$ be the stiffness coefficient. For any additive character $\chi: \mathbb{Z} \to \mathbb{C}$ and any internal label $b \in \mathbb{Z}_2$, the unit basis vector $\mathbf{e}_b$ in the unit cell amplitude space is an eigenvector of the Bloch matrix (symbol) $\hat{W}(\chi)$ of the wave operator. Specifically, $$\hat{W}(\chi) \mathbf{e}_b = \lambda_b(\chi) \mathbf{e}_b$$ where the eigenvalue $\lambda_b(\chi)$ is given by the dispersion relation $m(b) + \kappa(2 - \chi(1) - \chi(-1))$.

The momentum character at a real wavevector $k$ is the additive character from the integers $\mathbb{Z}$ to the complex numbers $\mathbb{C}$ defined by $n \mapsto e^{ikn}$ for each lattice cell index $n \in \mathbb{Z}$.

In a periodic lattice with two components per cell, let $m: \mathbb{Z}_2 \to \mathbb{R}$ be a mass function representing on-site restoring terms and $\kappa \in \mathbb{R}$ be a stiffness coefficient. For a real wavevector $k$, the Bloch eigenvalue (representing the dispersion branch) for the internal component index $b \in \mathbb{Z}_2$ is given by: $$m(b) + \kappa(2 - 2 \cos k)$$ This expression represents the squared frequency $\omega^2$ for the wave dynamics at momentum $k$.