Spin glasses in the limit of an infinite number of spin components

We consider spin glass models in which the number of spin components $m$ is infinite. In the formulation of the problem appropriate for numerical calculations proposed by several authors, we show that the order parameter defined by the long-distance limit of the correlation functions is actually zero and there is only “quasi-long-range order” below the transition temperature. Nonetheless, there can be a finite temperature phase transition where the decay of correlations changes from exponential to power law. We also show that the spin glass transition temperature is zero in three dimensions so power-law behavior only occurs at $T=0$ in this case. We also argue that the order of limits, $m\to \mathrm{\infty }$ and $N\to \mathrm{\infty }$ is important, where $N$ is the number of spins.

Figure 1

A plot of the average effective number of spin components, $m_0$, as a function of system size $N$ for a fixed, finite number of spin components $m$. For small $N$, $m_0\sim N^{\mu }$, but once $m_0$ hits the actual number of spin components $m$, it sticks at $m$ as $N$ is further increased.

Figure 2

The average number of nonzero spin components in the ground state $m_0$ as a function of $N$ for the infinite range model. We see that $m_0$ increases like $N^{\mu }$ with $\mu$ close to $2∕5$ as expected.

Figure 3

The square of the order parameter at $T=0$ for infinite range model. As expected, it decreases like $N^{-\mu }$ with $\mu =2∕5$.

Figure 4

The average number of nonzero spin components in the ground state $m_0$ as a function of $N$ for the short-range model in $d=3$. We see that $m_0$ increases like $N^{\mu }$ with $\mu \simeq 0.33$.

Figure 5

The spin glass susceptibility for the short-range model in $d=3$ for different system sizes. As expected it varies as $N^{1-\mu }$, where $\mu \simeq 0.33$ was also found in Fig. 4.

Figure 6

The average number of nonzero spin components in the ground state $m_0$ as a function of $N$ for the short-range model in $d=2$. We see that $m_0$ increases like $N^{\mu }$ with $\mu \simeq 0.29$.

Figure 7

The spin glass susceptibility for the short-range model in $d=2$ for different system sizes. As expected it varies as $N^{1-\mu }$, where $\mu \simeq 0.29$ was also found in Fig. 6.

Figure 8

The spin glass susceptibility as a function of temperature in three dimensions. The vertical axis has been divided by $L^{d(1-\mu )}$, in which we took $\mu =1∕3$ in order to collapse the data at $T=0$ according to the data in Figs. 4, 5.

Figure 9

A scaling plot of the spin glass susceptibility in Fig. 8 assuming a zero temperature transition.

Figure 10

The main figure is a plot of ${\xi }_L∕L$ against $T$ in three dimensions. The inset shows ${\xi }_L∕L$ at $T=0$ as a function of $L$. The dashed line is a guide to the eye.

Figure 11

A scaling plot of the data for ${\chi }_{\mathrm{SG}}$ in two dimensions, assuming a zero temperature transition. In the vertical axis, ${\chi }_{\mathrm{SG}}$ is divided by $L^{d(1-\mu )}\simeq L^{1.42}$ so that the data collapse at $T=0$.

Figure 12

Data for ${\xi }_L∕L$ as a function of $T$ in two dimensions. Clearly the data for larger sizes are merging at $T=0$ indicating a transition at $T_{\mathit{SG}}=0$. The inset shows data for ${\xi }_L∕L$ at $T=0$ confirming that the data become independent of size at $T=0$. The dashed line is a guide to the eye.

Figure 13

A scaling plot of the data from the largest sizes for ${\xi }_L∕L$ in two dimensions assuming $T_{\mathit{SG}}=0$.