Conductivity of 2D lattice electrons in an incommensurate magnetic field

We consider conductivities of two-dimensional lattice electrons in a magnetic field. We focus on systems where the flux per plaquette $\phi$ is irrational (incommensurate flux). To realize the system with the incommensurate flux, we consider a series of systems with commensurate fluxes which converge to the irrational value. We have calculated a real part of the longitudinal conductivity $\sigma_{xx}(\omega)$. Using a scaling analysis, we have found $\Re\sigma_{xx}(\omega)$ behaves as $1/\omega ^{\gamma}$ \,$(\gamma =0.55)$ when $\phi =\tau,(\tau =\frac{\sqrt{5}-1}{2})$ and the Fermi energy is near zero. This behavior is closely related to the known scaling behavior of the spectrum.