High q-State Clock Spin Glasses in Three Dimensions and the Lyapunov Exponents of Chaotic Phases and Chaotic Phase Boundaries
Spin-glass phases and phase transitions for q-state clock models and their q →∞ limit the XY model, in spatial dimension d = 3, are studied by a detailed renormalization-group study that is exact for the d=3 hierarchical lattice and approximate for the cubic lattice. In addition to the now well-established chaotic rescaling behavior of the spin-glass phase, each of the two types of spin-glass phase boundaries displays, under renormalization-group trajectories, their own distinctive chaotic behavior. These chaotic renormalization-group trajectories subdivide into two categories, namely as strong-coupling chaos (in the spin-glass phase and, distinctly, on the spinglass-ferromagnetic phase boundary) and as intermediate-coupling chaos (on the spinglass-paramagnetic phase boundary). We thus characterize each different phase and phase boundary exhibiting chaos by its distinct Lyapunov exponent, which we calculate. We show that, under renormalization-group, chaotic trajectories and fixed distributions are mechanistically and quantitatively equivalent. The phase diagrams of arbitrary even q-state clock spin-glass models in d=3 are calculated. These models, for all non-infinite q, have a finite-temperature spin-glass phase. Furthermore, the spin-glass phases exhibit a universal ordering behavior, independent of q. The spin-glass phases and the spinglass-paramagnetic phase boundaries exhibit universal fixed distributions, chaotic trajectories and Lyapunov exponents. In the XY model limit, our calculations indicate a zero-temperature spin-glass phase.
 E. Ilker and A.N. Berker, Phys. Rev. E 87, 032124 (2013).