Multiplicity of algebraic order from fixed lines of potential surfaces: XY-Ashkin-Teller in spatial dimension d=2

A position-space renormalization-group study is done for the Ashkin-Tellerized XY model, as an exact solution on the d=2 hierarchical lattice and an approximate solution on the square lattice. A multiplicity of algebraic order is found in the phase diagram, in the form of renormalization-group fixed lines composed of a continuous sequence of interaction potential surfaces. In the Ashkin-Tellerized XY model, each site has two continuously varying spins, each spin being an XY spin, that is, having orientation continuously varying in 2π radians. Nearest-neighbor sites are coupled by two-spin and four-spin interactions. The phase diagram has algebraically ordered phases that are ferromagnetic and antiferromagnetic in each of the spins, and algebraically ordered phases that are ferromagnetic and antiferromagnetic in the combined spin variable. These phases are subtended by fixed lines of potential surfaces that are multiplicatively different Berezinskii-Kosterlitz-Thouless fixed potentials. The evolution of continuously varying criticality is traced within each of the four phases. The renormalization-group flows, the fixed lines, and the interaction surfaces are in terms of the doubly composite Fourier coefficients of the exponentiated energy of the four nearest-neighbor spins. The disordered phase is maintained along two semi-infinite a priori quasi-disorder lines.