Renormalization approach to quasiperiodic quantum spin chains

A renormalization scheme which takes into account the natural frequency of the system is developed to study an anisotropic quantum XY spin chain in a quasiperiodic transverse field. The quasiparticle excitations of the model exhibit extended, localized as well as critical phase, with fractal characteristics, in a finite parameter interval. The scaling properties of the critical phase fall into four distinct universality classes. The isotropic limit of the model describes the extensively studied Harper equation. The renormalization approach provides a new method for determining energies and transition thresholds with extremely high precision.