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(adsbygoogle = window.adsbygoogle || []).push({});

[tex]g(m+1) + g(m-1) = [E - 2 cos(2\pi m \alpha- \nu)]g(m)[/tex]

where

[tex]\psi(x,y)=\psi(ma,na) = e^{i\nu n} g(m) \\ \alpha= \frac{e a^2 B}{h c}[/tex]

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 [itex](E, \alpha) [/itex]. 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 [itex]\alpha[/itex] into equation. Do I need to assume how g(0) looks like? Any help?

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# A Solving the Harper equation

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