Decimation studies of Bloch electrons in a magnetic field: Higher-order limit cycles underlying the phase diagram

A decimation method is applied to the tight binding model describing the two-dimensional electron gas with next-nearest-neighbor interaction in the presence of an inverse golden mean magnetic flux. The critical phase with fractal spectrum and wave function exists in a finite window in two-dimensional parameter space introducing universal features. Our decimation scheme identifies nine quantitative universality classes characterized by the limit cycles of the decimation equations. The limit cycles describe the self-similarity of the wave functions and divide them into three broader qualitative classes where the wave functions are either symmetric, asymmetric, or exhibit a type of shifted symmetry. We conjecture that the rest of the critical phase, where the fractal wave functions do not exhibit self-similarity, is characterized by strange attractors of the renormalization equations. The results are compared with those of Han et al. [Phys. Rev. B 50, 11 365 (1994)] on the same model.