Highq-state clock spin glasses in three dimensions and the Lyapunov exponents of chaotic phases and chaotic phase boundaries

Abstract

Spin-glass phases and phase transitions for $q$-state clock models and their $q\ensuremath{\rightarrow}\ensuremath{\infty}$ 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 spin-glass--ferromagnetic phase boundary) and as intermediate-coupling chaos (on the spin-glass--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 spin-glass--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.