Odd $\textbf{q}$-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases

Distinctive orderings and phase diagram structures are found, from renormalization-group theory, for odd $q$-state clock spin-glass models in $d=3$ dimensions [1]. These models exhibit asymmetric phase diagrams, as is also the case for quantum Heisenberg spin-glass models. No finite-temperature spin-glass phase occurs. For all odd $q\geq 5$, algebraically ordered antiferromagnetic phases [2,3] occur. One such phase is dominant and occurs for all $q\geq 5$. Other such phases occupy small low-temperature portions of the phase diagrams and occur for $5 \leq q \leq15$. All algebraically ordered phases have the same structure, determined by an attractive finite-temperature sink fixed point where a dominant and a subdominant pair states have the only non-zero Boltzmann weights. The phase transition critical exponents quickly saturate to the high $q$ value as previously observed for even q-state clock models [4]. \\[4pt] [1] E. Ilker and A. N. Berker, Phys. Rev. E {\bf 90}, 062112 (2014) \\[4pt] [2] A. N. Berker and L. P. Kadanoff, J. Phys. A {\bf 13}, L259 (1980) \\[4pt] [3] A. N. Berker and L. P. Kadanoff, J. Phys. A {\bf 13}, 3786 (1980) \\[4pt] [4] E. Ilker and A. N. Berker, Phys. Rev. E {\bf 87}, 032124 (2013)