  A renormalization scheme is developed to study an anisotropic quantum XY spin
chain in a quasiperiodic transverse field. The critical phase of the
quasi-particle excitations of the model with fractal wave functions exists in a
finite parameter interval and is sandwiched between the extended and localized
phases. The scaling properties of the critical phase fall into four distinct
universality classes referred as spectral, subconformal, conformal and Harper.
The spectral and conformal classes respectively describe the onsets of extended
to critical and critical to localized transitions while the subconformal class
describes the part of the phase diagram sandwiched between the conformal and
spectral transitions. The Harper universality class describes the isotropic
limit of the XY model. A decimation scheme is developed to compute the infinite
sets of universal scaling ratios characterizing the wave functions in the four
universality classes. The renormalization flow equations exhibit a limit cycle
at the band center and at the band edges providing a new method for determining
these energies with extremely high precision.
