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

My question is, can we diagonalize a general Hamiltonian , $$H=-t\sum_i^N (c_i^{\dagger}c_{i+1}+h.c.)+\sum_i \mu_i c_i^{\dagger}c_i$$ 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?

• "obviously" there is a solution for every p integer, but for p>=0 this solution may not be algebraic/analytical. Jul 25, 2020 at 8:11
• May I ask a question: Which physical system you have in mind? Jul 25, 2020 at 8:12
• @sintetico correct, I will edit my wording. Jul 25, 2020 at 8:29
• @sintetico I don't really have any particular system in mind, this was a general curiosity. maybe you can take a 1D lattice of electrons and introduce an external potential at certain lattice sites via laser? This is a long shot though. Jul 25, 2020 at 8:35
• In the Harper-Hofstadter model, one considers $\mu=\cos{(2\pi (p/q) n)}$ Jul 25, 2020 at 8:42

## 1 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 $$\sim e^{ikl}$$.

• What do you mean by p states in every site, could you please elaborate? Jul 25, 2020 at 14:24
• Your model as a single level for every $i$. In a more general model you have something like: $H = \sum_{i,\alpha,\beta}h_{\alpha\beta}c_{i,\alpha}^\dagger c_{i,\beta} - \sum_{i,\alpha,\beta}t_{\alpha\beta}c_{i,\alpha}^\dagger c_{i+1,\beta}$, which by Fourier transform is reduced to matrix diagonalization in $\alpha,\beta$ space. This is a standard tight-binding problem. So considering every $p$ sites in your as a cell in the Hamiltonian above, reduces your problem to this one. Jul 25, 2020 at 14:32
• Okay got it thanks, but that still does not supercede Abel's theorem right, I was actually thinking along the same lines because I expected p=5 would provide a 5th order characteristic equation, which probably does not have an analytical solution? At least matematica can't do it within a short period of time.. Jul 25, 2020 at 15:16
• You will have to diagonalize $p\times p$ matrix, whose characteristic equation is a polynomial of order $p$. So, generally speaking mathematically it may be not solvable. But you are dealing with a particularly simple problem, so the solution might exist for arbitrary $p$. From the physicists point of view the problem is exactly solvable, since one can find the roots if polynomial numerically... if one needs to get down to numbers. Jul 25, 2020 at 15:48
• corrected. Yes I agree. I have actually done it numerically and got all the results, but, if I could do it analytically for any p maybe it would have given me new insight to some features of the problem. But let's see, I will see if I can find any tricks to do it. Jul 25, 2020 at 15:55