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

In summary, the conversation discusses solving the Harper equation for a single particle in a 2D square lattice with an Abelian magnetic field. This can be done through matrix multiplication and by plotting the Hofstadter butterfly using a Hamiltonian matrix. However, it is also possible to numerically solve the equation by expanding it into matrix form and diagonalizing it. MATLAB and Python are recommended tools for this process.
  • #1
tobix10
7
1
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

[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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  • #2
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:
for k = -pi:pi
       H = %some cell with k plugged in
       E(k,:) = eig(H)
end
 
  • #3
DeathbyGreen said:
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:
for k = -pi:pi
       H = %some cell with k plugged in
       E(k,:) = eig(H)
end
tobix10 said:
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

[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?
please if you found the numerical calculation code you can give me .
 
  • #4
Saidi said:
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. :smile:
 
  • #5
berkeman said:
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.
 

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

1. What is the Harper equation and why is it important?

The Harper equation is a mathematical model that describes the energy levels and wave functions of a particle in a periodic potential. It is important because it has applications in many areas of physics, including solid-state physics, condensed matter physics, and quantum mechanics.

2. How do scientists solve the Harper equation?

Scientists solve the Harper equation using various numerical and analytical techniques, such as the tight-binding approximation, Floquet theory, and perturbation theory. They also use computer simulations and mathematical models to study the behavior of particles in different potential landscapes.

3. What are the main challenges in solving the Harper equation?

One of the main challenges in solving the Harper equation is the complexity of the potential landscape, which can make it difficult to find exact solutions. Another challenge is the large number of energy levels and wave functions that need to be calculated, especially for systems with many particles.

4. What are some real-world applications of the Harper equation?

The Harper equation has been used to study the properties of electrons in crystalline solids, such as semiconductors and superconductors. It has also been applied to study the behavior of photons in photonic crystals and the dynamics of cold atoms in optical lattices. Additionally, the Harper equation has been used in the design of quantum computers and other quantum technologies.

5. Are there any unresolved issues or controversies surrounding the Harper equation?

There are ongoing debates and research efforts to improve the accuracy and efficiency of solutions to the Harper equation. Some scientists also question the validity of using the Harper equation in certain systems, such as disordered or non-periodic potentials. However, the Harper equation remains a widely accepted and useful model in the study of periodic systems in physics.

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