Lattice Bloch transform and band projectors

Let $d,r\ge 1$ be integers, let $S\subset {\mathbb{Z}}^d$ be finite with $S=-S$, and let $T_{\delta }\in M_r(\mathbb{C})$ satisfy $T_{-\delta }=T_{\delta }^{\ast }$ for each $\delta \in S$. Give ${\mathbb{C}}^r$ the inner product $⟨v,w{⟩}_{{\mathbb{C}}^r}={\sum }_{a=1}^r\overline{v_a}{w}_a$, conjugate-linear in the first variable, and set $\mathcal{H}={\mathrm{\ell }}^2({\mathbb{Z}}^d;{\mathbb{C}}^r)$ with

$⟨\psi ,\varphi {⟩}_{\mathcal{H}}:={\sum }_{n\in {\mathbb{Z}}^d}⟨{\psi }_n,{\varphi }_n{⟩}_{{\mathbb{C}}^r}\mathrm{.}$

Let ${\mathbb{T}}_{2\pi }^d:={\mathbb{R}}^d\mathrm{/}(2\pi \mathbb{Z}{)}^d$ with normalized Haar measure $(2\pi {)}^{-d}dk$ and orientation $d{k}_1\wedge \cdots \wedge d{k}_d$, and set $\mathcal{K}=L^2({\mathbb{T}}_{2\pi }^d;{\mathbb{C}}^r)$ with

$⟨f,g{⟩}_{\mathcal{K}}:=(2\pi {)}^{-d}{\int }_{{\mathbb{T}}_{2\pi }^d}{\sum }_{a=1}^r\overline{f_a(k)}{g}_a(k)dk\mathrm{.}$

Thus the target pairing, like the sequence-space pairing, is conjugate-linear in its first variable. On finitely supported $\psi \in \mathcal{H}$ define

$(H_0\psi {)}_n={\sum }_{\delta \in S}{T}_{\delta }{\psi }_{n-\delta }\mathrm{.}$

On finitely supported sequences define

$({\mathcal{F}}_0\psi )(k)={\sum }_{n\in {\mathbb{Z}}^d}{\psi }_n{e}^{-ik\cdot n},h(k)={\sum }_{\delta \in S}{T}_{\delta }{e}^{-ik\cdot \delta }\mathrm{.}$

Assume Countable Choice for the unitary-extension assertion. The finite-range lattice Bloch model is the bounded extension $H$ of $H_0$, together with the unitary extension $\mathcal{F}:\mathcal{H}\to \mathcal{K}$ of ${\mathcal{F}}_0$ and its finite-matrix fibers $h(k)$. An isolated band cluster of rank $m$ is a fixed nonempty consecutive block of $m$ eigenvalues, counted with multiplicity, of the Hermitian matrix $h(k)$ whose distance from the complementary eigenvalues is bounded below by a single $\gamma >0$ for all $k$; if the block is the full spectrum, its projector is $I_r$. The band projector $P(k)$ is the orthogonal projection onto the sum of the eigenspaces in that block. For $d=2$, ${\mathbb{T}}_{2\pi }^2$ is the oriented Brillouin torus used on this page.

This definition concerns finite-range hopping on a discrete lattice. It does not identify that model with a continuum periodic Schrödinger operator.

The Fourier basis and Parseval's identity on the finite torus

Parseval equivalences for an orthonormal family

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)