Flat bands and localization in real space When reading this paper I came across the following claim:

From a band-theory point of view, the flat bands should have localized wavefunction profiles in real space

Is there a rigorous proof of this statement? So far I've only seen a hand-waving argument that goes like this: a flat band means the effective mass of the particle is infinite, i.e. it's very heavy and can't move - hence the localization in real space.
Presumably, this can be shown with a Fourier transform. I know that if we start with a localized wavefunction, i.e. something like $\Psi(x,t)=\delta(x-a)$ (where $a$ is the site) we arrive at the following momentum representation:
$$
\Phi\left(p,t\right)=\frac{1}{\sqrt{2\pi\hbar}}e^{-\frac{ipa}{\hbar}}
$$
However I'm not sure how this representation leads to the dispersion relation of a flat band ($E(k)=\mathrm{const}$). Doesn't it depend on the Hamiltonian of the system?
 A: The quoted statement is not precise. A better formulation might be

From a band-theory point of view, flat bands lead to dispersionless evolution of wavefunctions.
In most cases, they admit localized wavefunction profiles in real space.

Flat bands are said to be dispersionless in the following sense: the group velocity for wave packets is $dE(k)/dk = 0$ and therefore each wave packet preserves its form under time evolution. If it is initially localized it stays localized forever. This can be easily shown using the Fourier transform:
$$
  \psi(t,\vec x) = \int dk\, e^{-i\vec k\vec x} \tilde \psi(t, \vec k) = \int dk\, e^{-i\vec k\vec x} e^{-i E(\vec k) t} \tilde \psi(0, \vec k)
$$
Now if $E(\vec k) = E_0$ then
$$
  \psi(t,\vec x) = e^{-i E_0 t} \int dk\, e^{-i\vec k\vec x} \tilde \psi(0, \vec k) = e^{-i E_0 t} \psi(0, \vec x)
$$
which means that the form of $|\psi(t,\vec x)|$ stays constant in time.
The flat band is a property of the Hamiltonian. But once the Hamiltonian admits a flat band then this flat band is always dispersionless in the same way, independently of further details of the Hamiltonian.
The question about the localization in real space goes further and is related to the fact whether well localized Wannier functions can be constructed for the given Hamiltonian. In most cases, Wannier functions are exponentially localized but the general criterion is not known. The papers arXiv:2104.01236 and    arXiv:2010.06782, however, offer several examples of systems with flat bands and localization.
A: If we look at this from the point of view of the tight-binding approximation, the band width is proportional to the hopping integral, i.e., in one dimension
$$
H=E_0\sum_n|n\rangle\langle n | - \Delta\sum_n\left(|n\rangle\langle n+1| + |n+1\rangle\langle n|\right),\\
E(k)=E_0-2\Delta\cos(ka)
$$
If the band is flat, it means that $\Delta=0$, i.e., there is no hopping, that is the wave functions are localized.
Remark
Note that this argument can be trivially generalized to a lattice in any number of dimensions: without hopping all the sites have the same energy - thus, we can construct the Bloch states which have the same energy for any $\mathbf{k}$. In case we have atoms with different site energies in a unit cell, we well have several flat bands.
