Overfrustrated and underfrustrated spin glasses in $d=3$ and 2: Evolution of phase diagrams and chaos including spin-glass order in $d=2$

In spin-glass systems, frustration can be adjusted continuously and considerably, without changing the antiferromagnetic bond probability $p$, by using locally correlated quenched randomness, as we demonstrate here on hypercubic lattices and hierarchical lattices. Such overfrustrated and underfrustrated Ising systems on hierarchical lattices in $d=3$ and 2 are studied. With the removal of just 51% of frustration, a spin-glass phase occurs in $d=2$. With the addition of just 33% frustration, the spin-glass phase disappears in $d=3$. Sequences of 18 different phase diagrams for different levels of frustration are calculated in both dimensions. In general, frustration lowers the spin-glass ordering temperature. At low temperatures, increased frustration favors the spin-glass phase (before it disappears) over the ferromagnetic phase and symmetrically the antiferromagnetic phase. When any amount, including infinitesimal, frustration is introduced, the chaotic rescaling of local interactions occurs in the spin-glass phase. Chaos increases with increasing frustration, as can be seen from the increased positive value of the calculated Lyapunov exponent $\lambda$, starting from $\lambda =0$ when frustration is absent. The calculated runaway exponent $y_R$ of the renormalization-group flows decreases with increasing frustration to $y_R=0$ when the spin-glass phase disappears. From our calculations of entropy and specific-heat curves in $d=3$, it is shown that frustration lowers in temperature the onset of both long- and short-range order in spin-glass phases, but is more effective on the former. From calculations of the entropy as a function of antiferromagnetic bond concentration $p$, it is shown that the ground-state and low-temperature entropy already mostly sets in within the ferromagnetic and antiferromagnetic phases, before the spin-glass phase is reached.

Figure 1

Randomly distributed ferromagnetic (blue) and antiferromagnetic (red) interactions on a square plane. In all three cases, the antiferromagnetic bond concentration is $p=0.5$. The frustrated squares are shaded. In the case at the center, the bonds were distributed in an uncorrelated fashion, leading to the frustration of half of the squares (stochastic frustration). In the case at the left, 25% of the frustration was randomly removed without changing $p=0.5$ (underfrustration). In the case at the right, 25% frustration was randomly added without changing $p=0.5$ (overfrustration). Frustration can thus be set between zero and complete frustration. It is clear that frustration can thus be adjusted in all hypercubic lattices.

Figure 2

(a) Migdal-Kadanoff approximate renormalization-group transformation for the $d=3$ cubic lattice with the length-rescaling factor of $b=3$. Bond moving is followed by decimation. (b) Exact renormalization-group transformation for the equivalent $d=3$ hierarchical lattice with the length-rescaling factor of $b=3$. (c) Pairwise applications of the quenched probability convolution of Eq. (5), leading to the exact transformation in (b) and to the numerically equivalent approximate transformation in (a).

Figure 3

(a) Migdal-Kadanoff approximate renormalization-group transformation for the $d=2$ square lattice with the length-rescaling factor of $b=3$. Bond moving is followed by decimation. (b) Exact renormalization-group transformation for the equivalent $d=2$ hierarchical lattice with the length-rescaling factor of $b=3$. (c) Pairwise applications of the quenched probability convolution of Eq. (5), leading to the exact transformation in (b) and to the numerically equivalent approximate transformation in (a).

Figure 4

Plot of $p_{\mathrm{effective}}$ versus $p$ for the range of underfrustration and overfrustration used in our study [Eq. (6)]. The curves are, consecutively from the lower right, for $f=0,0.2,0.5;f=1=g$ (thicker line); and $g=0.8,0.6,0.3$.

Figure 5

Calculated phase diagrams of the overfrustrated, underfrustrated, and stochastically frustrated Ising spin-glass models on hierarchical lattices. The panels on the left side are for $d=3$ dimensions. In the left top panel, the outermost phase diagram, consisting of one horizontal and two vertical lines, is for no frustration, $f=0$. Starting from this outermost phase diagram, the three consecutive phase diagrams are for the underfrustrated cases (where frustration has been removed) of $f=0.1,0.2,0.5$. In the left middle panel, starting from the outermost phase diagram, the four consecutive phase diagrams are for the underfrustrated cases of $f=0.5,0.8$; the stochastic case (where frustration has been neither removed nor added) of $f=1=g$, drawn with the thicker lines; and the overfrustrated case (where frustration has been added) of $g=0.8$. In the left bottom panel, starting from the outermost phase diagram, the four consecutive phase diagrams are for the overfrustrated cases of $g=0.8,0.6,0.3,0.1$. In the latter three cases, $g=0.6,0.3,0.1$, no spin-glass phase occurs. Excessive overfrustration destroys the spin-glass phase. The panels on the right side are for $d=2$ dimensions. In the right top panel, the outermost phase diagram, consisting of one horizontal and two vertical lines, is for no frustration, $f=0$. Starting from this outermost phase diagram, the three consecutive phase diagrams are for the underfrustrated cases of $f=0.1,0.2,0.3$. In the right middle panel, starting from the outermost phase diagram, the three consecutive phase diagrams are for the underfrustrated cases of $f=0.3,0.4,0.5$. In the right bottom panel, starting from the outermost phase diagram, the three consecutive phase diagrams are the underfrustrated case of $f=0.5$; for the stochastic case of $f=1=g$, drawn with the thicker lines; and the overfrustrated case of $g=0.5$. In the latter three cases, $f=0.5,f=1=g,g=0.5$, no spin-glass phase occurs. However, in the underfrustrated cases of $f=0.1,0.2,0.3,0.4$, a spin-glass phase occurs in these $d=2$ dimensional systems with locally correlated randomness. All phase transitions in this figure are second order and, to the resolution of the figure, all multicritical points appear on the Nishimori symmetry line, shown with the dashed curves.

Figure 6

Interaction at a given position in the lattice at successive renormalization-group iterations, for $d=3$ systems with different frustrations. In all cases, the antiferromagnetic bond concentration is $p=0.5$ and the initial temperature is $1/J=0.2$, inside the spin-glass phase. For each frustration amount, a chaotic trajectory of the interaction at a given position is seen. The calculated Lyapunov exponent for each case is given in the upper right corner of each panel.

Figure 7

Interaction at a given position in the lattice at successive renormalization-group iterations, for $d=2$ systems with different frustrations. In all cases, the antiferromagnetic bond concentration is $p=0.5$ and the initial temperature is $1/J=0.2$, inside the spin-glass phase. For each frustration amount, a chaotic trajectory of the interaction at a given position is seen. The calculated Lyapunov exponent for each case is given in the upper right corner of each panel.

Figure 8

Chaotic visits of the consecutively renormalized interactions $J_{ij}$ at a given position of the system, in the spin-glass phase of overfrustrated, underfrustrated, and stochastically frustrated Ising models in $d=3$. These consecutively renormalized interactions at a given position of the system are shown here as scaled with the average interaction $\langle \left|J\right|\rangle$ across the system, which diverges as $b^{n{y}_R}$, where $n$ is the number of renormalization-group iterations and $y_R>0$ is the runaway exponent shown in Fig. 10. The number of visits into each interval of 0.1 on the horizontal axis has been scaled with the total number of renormalization-group iterations. Between 300 and 3500 renormalization-group iterations have been used for the different panels. The distributions of chaotic visits shown in the panels stabilize as the number of iterations is increased. The calculated Lyapunov exponent for each case is given in the upper right corner of each panel.

Figure 9

Chaotic visits of the consecutively renormalized interactions $J_{ij}$ at a given position of the system, in the spin-glass phase of underfrustrated Ising models in $d=2$. These consecutively renormalized interactions at a given position of the system are shown here as scaled with the average interaction $\langle \left|J\right|\rangle$ across the system, which diverges as $b^{n{y}_R}$, where $n$ is the number of renormalization-group iterations and $y_R>0$ is the runaway exponent shown in Fig. 10. The number of visits into each interval of 0.1 on the horizontal axis have been scaled with the total number of renormalization-group iterations. Between 700 and 5000 renormalization-group iterations have been used for the different panels. The distributions of chaotic visits shown in the panels stabilize as the number of iterations is increased. The calculated Lyapunov exponent for each case is given in the upper right corner of each panel. No spin-glass phase occurs for $f>0.49$, as seen in Figs. 5 and 10.

Figure 10

Lyapunov exponent $\lambda$ and runaway exponent $y_R$ of the spin-glass phases of overfrustrated, underfrustrated, and stochastically frustrated Ising models in $d=3$ (upper curves) and $d=2$ (lower curves). The horizontal scale shows, to the left of the dashed line, the $f$ values of the underfrustrated cases and, to the right of the dashed line, the $g$ values of the overfrustrated cases. The dashed line marks the stochastic frustration $(f=1=g)$. As can be seen in this figure and in Figs. 8 and 9, as soon as frustration is introduced $(f>0)$, the Lyapunov exponent becomes positive and chaotic behavior occurs inside the spin-glass phase. The average interaction $\langle \left|J\right|\rangle$ across the system diverges as $b^{n{y}_R}$, where $n$ is the number of renormalization-group iterations and $y_R>0$ is the runaway exponent. The Lyapunov exponent $\lambda$ monotonically increases with frustration from $\lambda =0$ at zero frustration and the runaway exponent $y_R$ monotonically decreases with frustration from $y_R=d-1$ at zero frustration. The spin-glass phase disappears when $y_R$ reaches zero, for $g=0.67$ in $d=3$ and $f=0.49$ in $d=2$.

Figure 11

Calculated entropy per site $S/kN$ (upper panel) and specific heat per site $C/kN$ (lower panel) as a function of temperature $1/J$ at fixed antiferromagnetic bond concentration $p=0.5$, for $d=3$ systems with underfrustration $(f=0.02,0.2,0.5)$, the stochastic frustration $(f=1=g)$, and overfrustration $(g=0.7)$. The tick mark shows the phase transition point between the spin-glass phase and the paramagnetic phase for each frustration case. It is seen that frustration lowers this transition temperature. Thus, for stochastic frustration ($f=1=g$), the specific-heat peak occurs outside the spin-glass phase, indicating that considerable short-range ordering occurs at higher temperatures before the onset of spin-glass long-range order. In contrast, for the more underfrustrated cases $(f=0.02,0.2)$, the specific-heat peak occurs inside the spin-glass phase, indicating that considerable short-range disorder persists into the higher temperatures of the spin-glass phase. This conclusion is also reached from the entropy curves in the upper panel. The changeover between these two regimes occurs at the underfrustrated system of $f=0.5$. Overfrustrated systems show understandably specific-heat behavior similar to $f=1$, with frustration lowering the long-range-order temperature and short-range order setting at higher temperatures with a specific-heat peak.

Figure 12

The top panel shows the calculated entropy per site $S/kN$ as a function of the antiferromagnetic bond concentration $p$ at fixed temperature $1/J=0.5$, for systems with no frustration $(f=0)$, underfrustration $(f=0.5,0.8)$, the stochastic frustration $(f=1=g)$, and overfrustration $(g=0.8)$. The tick mark shows the phase transition point between the ferromagnetic phase and the spin-glass phase for each frustration case. It is seen that frustration favors the spin-glass phase over the ferromagnetic phase. It is also seen that, as soon as frustration is introduced, the major portion of the entropy is created within the ferromagnetic phase as opposed to the spin-glass phase. The lower panel shows the calculated derivative of the entropy per site $(1/kN)(\partial S/\partial p)$ as a function of the antiferromagnetic bond concentration $p$ at temperature $1/J=0.5$, for the stochastic frustration system $(f=1)$ in $d=3$. The tick mark shows the phase transition point between the ferromagnetic phase and the spin-glass phase. The peak being inside the ferromagnetic phase shows that short-range disorder sets inside the ferromagnetic phase.