Low-temperature phase boundary of dilute-lattice spin glasses

The thermal-to-percolative crossover exponent $\varphi$, well known for ferromagnetic systems, is studied extensively for Edwards–Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold $p_c$ using an improved reduction algorithm. Simulations extend to system sizes above $N={10}^8$ in dimensions $d=2,\dots ,7$. The results can be related to the behavior of the transition temperature $T_g\sim {(p-p_c)}^{\varphi }$ between the paramagnetic and the glassy regime for $p\searrow p_c$. In three dimensions, where our simulations predict $\varphi =1.127\left(5\right)$, this scaling form for $T_g$ provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.

Figure 1

(Color online) Phase diagram for bond-diluted spin glasses $(d>d_l)$. In the spin glass phase (SG) for $T<T_g$ and $p>p_c$, $y$ in Eq. (2) is $>0$, while $y=y_P<0$ in Eq. (3) at $p=p_c$ and $T=T_g=0$. In the paramagnetic phase (PM) for $p<p_c$, defects decay exponentially for all $T$. The exponent $\varphi$ in Eq. (4) describes the boundary $T_g\left(p\right)$ for $p\searrow p_c$.

Figure 2

(Color online) Plot of $\sigma \left(\Delta E\right)$ in Eq. (3) as a function of system size $N=L^d$ in extrapolated form. Plotting $\ln \left(\sigma \right)∕\ln \left(N\right)$ vs $1∕\ln \left(N\right)$ and linearly extrapolating (dashed lines), we extract the asymptotic values for $y_P∕d$ in Table at $1∕\ln \left(N\right)=0$. Note that the fitted asymptotic regime here corresponds to orders of magnitude of scaling in a (less insightful) plot of $\ln \sigma$ vs $\ln N$. Note the increasing corrections to scaling for larger $d$ before asymptotic behavior is obtained.

Figure 3

(Color online) Comparison of the data for $d=3$ (top) and $d=4$ (bottom) for different bond distributions $P\left(J\right)$, plotted in the same way as in Fig. 2. There is little difference between the Gaussian and the Lorentzian bonds even in $d=3$, as both are similarly smooth near $J=0$. For our power-law bonds with vanishing support for $\mid J\mid <1$, our methods are very inefficient. Yet, at least in $d=4$, those bonds produce results far from but consistent with Gaussian bonds.

Figure 4

(Color online) Plot of the exponents $y_P∕d$ (top) and phi (bottom) in Table as a function of $1∕d$. For comparison, also plotted (top) are the stiffness exponents $y∕d$ inside the spin-glass regime (Ref. 9) $(p>p_c)$. Some of the large error bars for $\varphi =-\nu y_P$ originate with uncertainties in $\nu$ (Ref. 11).