Critical point scaling of Ising spin glasses in a magnetic field

Critical point scaling in a field $H$ applies for the limits $t\to 0$ (where $t=T/T_c-1)$ and $H\to 0$ but with the ratio $R=t/H^{2/\Delta }$ finite. $\Delta$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\xi$ and from it the crossover scaling function $f\left(R\right)$ defined via $1/\left(\xi H^{4/(d+2-\eta )}\right)\sim f\left(R\right)$. We have calculated analytically $f\left(R\right)$ for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension $d=3$, we have determined the exponents and the critical scaling function $f\left(R\right)$ within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for $f\left(R\right)$ in $d=3$ in reasonable agreement with those of the Monte Carlo simulations at the values of $R$ for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\le 6$, it would appear as a zero of the function $f\left(R\right)$ at some negative value of $R$, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\le 6$, which we review.

Figure 1

Bond-moving scheme in the Migdal-Kadanoff approximation. In the first step two sets of four bonds are moved together. The renormalized couplings and fields are assigned in the second step according to the given scheme. Finally, a trace is taken over the middle spin.

Figure 2

The coupling strength $J\left(n\right)$ at iteration step $n$ at fixed external field $H=0.1$ in scheme I (top) and scheme II (bottom). The temperatures are $T=0.1$, 1.0, 1.6, 2.0, 2.6, and 3.0 from top to bottom.

Figure 3

The flow of the coupling $J\left(n\right)$ at fixed temperature and varying fields in schemes I and II. The upper panel for each scheme is at $T=1.0$ and the lower one is for $T=2.0$. From top to bottom, $H=0$, 0.02, 0.05, 0.1, 0.2, and 0.3.

Figure 4

The standard deviation of the fields as a function of iteration in scheme I (top) for fixed temperature $T=0.1$ and varying initial uniform field $H$. The solid line in the upper figure is the fit ${\sigma }_H\sim L^{1.47}$. The lower figure is for scheme II also for fixed temperature $T=0.1$ and varying standard deviation $H$ of the initial random field. The solid line is a fit to ${\sigma }_H\sim L^{1.46}$.

Figure 5

The upper panel is ${\xi }^{-1}{H}^{\frac{4}{d+2-\eta }}\equiv f\left(R\right)$ vs $t/H^{4/\left(\nu \right(d+2-\eta \left)\right)}=R$ for MK scheme I and Janus data [15]. The lower panel is the same but with the Janus data for $\xi$ rescaled by the factor 3.79, as described in the text.

Figure 6

The upper panel is ${\xi }^{-1}{H}^{\frac{4}{d+2-\eta }}\equiv f\left(R\right)$ vs $t/H^{4/\left(\nu \right(d+2-\eta \left)\right)}=R$ for MK scheme II and Janus data [15]. The lower panel is the same but with the Janus data for $\xi$ rescaled by the factor 7.13, as described in the text.

Figure 7

The correlation length $\xi (T_c,H)$ at zero-field transition temperature $T_c$ for small external field $H$. The dashed lines are fits, $\xi \sim H^{-0.83}$ for scheme I and $\xi \sim H^{-0.72}$ for scheme II.

Figure 8

The log-log plot of ${\xi }^{-1}{H}^{\frac{4}{d+2-\eta }}\equiv f\left(R\right)$ vs $-t/H^{4/\left(\nu \right(d+2-\eta \left)\right)}=-R$ for MK scheme I (upper panel) and scheme II (lower panel) to illustrate the power-law behavior $f\left(R\right)\sim 1/{\left|R\right|}^x$, which is predicted to set in at large values of $-R$. The straight line is the expected slope if $x$ is predicted by Eq. (18).