What is the procedure of "flattening of Hamiltonian"? Can someone explain what exactly is spectral flattening of a Hamiltonian? I hear about it in the course, but it was never explained and every article on the Internet also speaks about it as if it was obvious what it is. I would appreciate a rigorous answer.
 A: The basic idea of spectral flattening is that you can transform your single-particle Hamiltonian into one with a completely flat energy dispersion without changing the topological properties. Hamiltonians with flat dispersion are usually easier to study.
More rigorously, following 1, assume you have a Hamiltonian $H(k)$ with a gap for all $k$, e.g.
$$E_0(\mathbf{k})\leq E_1(\mathbf{k})\leq \dots \leq E_{p-1}(\mathbf{k}) < E_p(\mathbf{k})\leq E_{p+1}(\mathbf{k})\leq \dots \leq E_n(\mathbf{k}) \quad \forall \mathbf{k}$$
Then we can map this Hamiltonian to:
$H(\mathbf{k}) \rightarrow Q =  R\cdot[-I_p \oplus I_{n-p}]\cdot R^T$, where $R$ is the transformation which diagonalizes $H(\mathbf{k})$ and $I$ is the identity matrix.
This Hamiltonian has $p$ eigenvalues $-1$ and $n-p$ eigenvalues $+1$ for all $\mathbf{k}$, and is thus spectrally flat. By construction, we did this without closing a gap (e.g. adiabatically). Therefore, the strong topological invariants are still the same, and we can study $Q$ in lieu of $H(\mathbf{k})$, allowing us to ignore all the pesky details of what the exact energies are.
This is one of the most appealing things about topology! You can change the Hamiltonian as much as you want (as long as you do so adiabatically/without closing a gap), so you can study the simplest possible system.
Such a technique is sometimes also employed to study complicated many-body, if you can show that one can adiabatically map them to a flattened Hamiltonian.
Much more information on this is found in 1
1 Bouhon, A., Bzdušek, T., and Slager, R.-J. Geometric approach to fragile topology beyond symmetry indicators. Phys. Rev. B 102, 115135 (2020), available at arXiv:2005.02044
