How Do You Solve Harper's Equation in Quantum Mechanics?

[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?

 

[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).

for k = -pi:pi
       H = %some cell with k plugged in
       E(k,:) = eig(H)
end

 

[Saidi]

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).

for k = -pi:pi
       H = %some cell with k plugged in
       E(k,:) = eig(H)
end

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?

please if you found the numerical calculation code you can give me .

 

[berkeman]

please if you found the numerical calculation code you can give me .

Welcome to PF.

This thread is old enough that the participants are no longer with us. Your best bet is to follow the hints and link in Post #2.

 

[jtbell]

This thread is old enough that the participants are no longer with us.

I can see (by clicking on their usernames, then on the numbers shown under "Messages") that tobix10 has not posted here since 2017, and DeathbyGreen has not posted since 2018.