Exact Diagonalization of a tight-binding Hamiltonian with periodically alternating potential

Question

My question is, can we diagonalize a general Hamiltonian ,
$$H=-tsum_i^N (c_i^{dagger}c_{i+1}+h.c.)+sum_i mu_i c_i^{dagger}c_i$$ where,
$$mu_i=begin{cases}
mu_0, &text{if mod}(i,p)=0 
0, &text{otherwise}.
end{cases}$$
Obviously, $p$ is the periodicity of the lattice and $c$ is Fermionic annihilation operator. I know $p=2,3,4$  will have an analytic solution but from Abel-Ruffini's theorem $p=5$ onwards may or may not have an analytic solution. Now I am sure because of periodicity there should be a certain degree of symmetry present in solutions, from Bloch's theorem, but I just cannot find a method to analytically solve the problem to get the eigenvalues and eigenvectors.
Numerically, I have found the solution, but any suggestions to solve it analytically?

Vadim · Accepted Answer

I think this should be treated as a tight-binding model with period $p$ and $p$ states in every site. One could do it by first introducing operators:
$$a_{l, nu} = c_{pl +nu}, nu=0,...,p-1,$$
and then looking for plane wave solutions $sim e^{ikl}$.

Exact Diagonalization of a tight-binding Hamiltonian with periodically alternating potential

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