Figure 1

(Color online) Constant-temperature cross sections of the global phase diagram for $K_z/K_{xy}=0.5$, as a function of $p_{xy}$ and $p_z$, which are the concentrations of antiferromagnetic $xy$ and $z$ bonds, respectively. At low temperatures (high $K_{xy}$), the central spin-glass (SG) phase separates the corner ferromagnetic (F), columnar (C), antiferromagnetic (A), and layered (L) phases. The diagrams are twofold symmetric along each axis, but not fourfold symmetric, due to the difference between longitudinal $\left(p_{xy}=0\right)$ and transverse $\left(p_z=0\right)$ spin glasses. As temperature increases, the paramagnetic (P) phase appears at the central point, first reaches the transverse spin-glass system and eliminates the spin-glass phase, then reaches the longitudinal spin-glass system and eliminates the spin-glass phase. In the latter system, the spin-glass and paramagnetic phases simultaneously occur for a very narrow range of temperatures, as also seen in the inset in the lower left panel of Fig. 3.Reuse & Permissions

Figure 2

Construction of the uniaxially anisotropic $\text{d}=3$ hierarchical model. Two graphs are mutually and repeatedly self-imbedded. Note that for $K_{xy}=0$, $K_z=0$, and $K_{xy}=K_z$, the system reduces, respectively, to the $d=1$, isotropic $d=2$, and isotropic $d=3$ systems.Reuse & Permissions

Figure 3

(Color online) Temperature-concentration phase diagrams for isotropically mixed (upper left), transverse (upper right), longitudinal (lower left), and $p_{xy}=0.5p_z$ spin-glass systems. In all cases, $K_z/K_{xy}=0.5$. The upper left and right phase diagrams are seen to be, respectively, reentrant and forward, namely, with a ferromagnetic phase that, respectively, recedes from or proceeds toward the spin-glass phase as temperature is lowered, as clearly seen in the insets. There are no points obeying Nishimori symmetry in the phase diagrams of this figure. Note the remarkably narrow spin-glass phase, reaching zero temperature, in the longitudinal spin-glass system, as also seen in the inset. All phase transitions in this figure are second order.Reuse & Permissions

Figure 4

(Color online) Zero-temperature phase diagrams of the longitudinal (left column) and transverse (right column) spin-glass systems. With the appropriate reversal in variables, the transverse and longitudinal spin-glass phase diagrams are seen here to be qualitatively similar, but quantitatively different. The spin-glass phase is more extensive in the transverse case. All phase transitions in this figure are second order.Reuse & Permissions

Figure 5

(Color online) Phase diagrams with Nishimori symmetry lines (dashed) for different anisotropy parameters: The ratio $K_z/K_{xy}$ is 2, 1, and 0.5 from top to bottom. In the left column, $p_{xy}$ satisfies the Nishimori condition. In the right column, $K_z$ satisfies the Nishimori condition. All phase transitions in this figure are second order.Reuse & Permissions

Figure 6

(Color online) The Nishimori condition for $K_z$ is held throughout the leftmost figure and for $K_{xy}$ throughout the center figure. The complimentary Nishimori condition, for $K_{xy}$ and $K_z$ respectively, is held along the dashed straight lines, which intersect the ordered (F, L, A, or C)–spin-glass–paramagnetic multicritical points. In the rightmost figure both conditions are satisfied throughout the figure. In this figure, the phase boundaries around the paramagnetic phase are actually lines of the multicritical points where the paramagnetic, ordered (F, L, A, or C), and spin-glass (not seen in this cross section) phases meet. In the side figures, first-order boundaries (dotted) occur between the ferromagnetic and layered phases, and between the antiferromagnetic and columnar phases, terminating at $d=2$ critical points. All other phase transitions (full lines) in this figure are second order.Reuse & Permissions

Figure 7

(Color online) Fixed distributions, with circles and crosses showing one renormalization-group transformation and thereby by their exact superposition attesting to the fixed nature of the distributions. The distributions have been binned for exhibition purposes. (a) For the ferromagnetic–spin-glass phase boundary, a runaway to infinite coupling; (b) for the paramagnetic–spin-glass phase boundary. Both of these fixed distributions are spatially isotropic, attracting isotropic and anisotropic boundaries, and do not obey Nishimori symmetry. (c) For the ferromagnetic–spin-glass–paramagnetic multicritical point. This fixed distribution is spatially isotropic and obeys Nishimori symmetry. This fixed distribution attracts the isotropic multicritical point, which obeys Nishimori symmetry, and anisotropic multicritical points, which obey and do not obey Nishimori symmetry. The fixed distributions for the antiferromagnetic–spin-glass, columnar–spin-glass, layered–spin-glass phase boundaries and for the antiferromagnetic–spin-glass–paramagnetic, columnar–spin-glass–paramagnetic, layered–spin-glass–paramagnetic multicritical points are as shown here in (a) and (c), respectively, but with the appropriate $K_{xy}\to -K_{xy}$ and/or $K_z\to -K_z$ reflections. (d) Fixed distribution for the spin-glass phase. This phase sink is an isotropic runaway, attracting both spatially isotropic and anisotropic spin-glass phase points, and does not obey Nishimori symmetry.Reuse & Permissions