Tight-binding model

Example

The simplest tight-binding model describes an electron hopping between the sites of the chain $\Z$. Let $\varepsilon_0 \in \R$ and $t > 0$, and define the operator $H$ on $\ell^2(\Z)$ by $$(H\psi)(n) = \varepsilon_0 \psi(n) - t\big(\psi(n+1) + \psi(n-1)\big).$$ It is a bounded self-adjoint operator. The Fourier series map $\psi \mapsto \hat\psi(k) = \sum_n \psi(n) e^{-ikn}$ from $\ell^2(\Z)$ onto $L^2([-\pi, \pi])$ turns $H$ into multiplication by $$E(k) = \varepsilon_0 - 2t\cos k, \qquad k \in [-\pi, \pi].$$ Check: the shift $\psi(n) \mapsto \psi(n+1)$ becomes multiplication by $e^{ik}$, and the shift $\psi(n) \mapsto \psi(n-1)$ becomes multiplication by $e^{-ik}$, whose sum is $2\cos k$. So the band structure is the single band $E(k)$ over the Brillouin zone $[-\pi, \pi]$, and the spectrum of $H$ is the interval $[\varepsilon_0 - 2t, \varepsilon_0 + 2t]$.

On the square lattice $\Z^2$ with hopping $t$ to the four neighbours, the same computation gives $E(k) = \varepsilon_0 - 2t(\cos k_1 + \cos k_2)$ on $[-\pi, \pi]^2$. Adding a quasi-periodic potential to the chain gives the almost Mathieu operator.