# The quantum hall effect and Hofstadter's butterfly spectrum

What is the connection between the quantum Hall effect and the Hofstadter's butterfly spectrum? I mean, can I understand something about the quantum Hall effect in the Hofstadter's butterfly spectrum?

When you study the Quantum Hall effect and more precisely the Hall conductivity you learn that the latter is related to certain quantity called the Chern Number $$\; \sigma_{xy}= \frac{-e^{2}}{2\pi\hbar} C \;$$ which is also related to Berry Phase. When you consider particles on a lattice in a magnetic field, the resulting Schrodinger equation (under certain conditions) yields to the Harper equation which gives the Hofstadter's spectrum (for rational values of the flux). For this lattice model we can compute the Chern Number which leads to the Hall Conductivity as mentioned above. This is the only link that I see, hope it helps you a little bit.