A Solving the Harper equation

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1. Apr 18, 2017

tobix10

For a single particle in a 2D square lattice in the presence of an Abelian magnetic field Schroedinger's equation transforms into Harper's equation

$$g(m+1) + g(m-1) = [E - 2 cos(2\pi m \alpha- \nu)]g(m)$$
where
$$\psi(x,y)=\psi(ma,na) = e^{i\nu n} g(m) \\ \alpha= \frac{e a^2 B}{h c}$$

I am familiar with a solution that involves matrix multiplication and the condition about the trace of wilson loop.
I can also plot Hofstatder butterfly by constructing hamiltonian matrix in some gauge (e.g Landau) and diagonalizing it.

What I don't know is how to solve Harper equation in a different way or how to get butterfly by simply finding points $(E, \alpha)$. Solving this kind of equations is new to me. I don't know what to do if I put some particular values of E and $\alpha$ into equation. Do I need to assume how g(0) looks like? Any help?

2. Apr 23, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Jun 6, 2017

DeathbyGreen

Are you asking how to numerically solve this? Here is a nice paper going over matrix expansion of the Harper equation:
https://arxiv.org/pdf/cond-mat/9808328.pdf
But to numerically solve you need to write the equation in matrix form and loop over possible parameter values. You should just take the raw Hofstandter Hamiltonian, expand it in matrix form, and diagonalize it numerically (MATLAB and Python are good tools for this).
Code (Text):
for k = -pi:pi
H = %some cell with k plugged in
E(k,:) = eig(H)
end