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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?

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  • $\begingroup$ "obviously" there is a solution for every p integer, but for p>=0 this solution may not be algebraic/analytical. $\endgroup$
    – sintetico
    Jul 25, 2020 at 8:11
  • $\begingroup$ May I ask a question: Which physical system you have in mind? $\endgroup$
    – sintetico
    Jul 25, 2020 at 8:12
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    $\begingroup$ @sintetico correct, I will edit my wording. $\endgroup$ Jul 25, 2020 at 8:29
  • $\begingroup$ @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. $\endgroup$ Jul 25, 2020 at 8:35
  • $\begingroup$ In the Harper-Hofstadter model, one considers $\mu=\cos{(2\pi (p/q) n)}$ $\endgroup$
    – sintetico
    Jul 25, 2020 at 8:42

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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 $\sim e^{ikl}$.

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  • $\begingroup$ What do you mean by p states in every site, could you please elaborate? $\endgroup$ Jul 25, 2020 at 14:24
  • $\begingroup$ 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. $\endgroup$
    – Roger V.
    Jul 25, 2020 at 14:32
  • $\begingroup$ 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.. $\endgroup$ Jul 25, 2020 at 15:16
  • $\begingroup$ 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. $\endgroup$
    – Roger V.
    Jul 25, 2020 at 15:48
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    $\begingroup$ 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. $\endgroup$ Jul 25, 2020 at 15:55

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