Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\beta =1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that $T_c=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $\nu$. We find $-1∕\nu =-0.295\left(40\right)$, which is compatible with the value for the domain-wall and droplet exponent $\theta \approx -0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

Figure 1

Sample equilibration test for $L=75$ and $\beta =5$ averaged over 200 samples, the curves [Eq. (2)] start to overlap around the equilibration time (here $\mathrm{25\hspace{0.17em}000}\mathrm{MC}$ steps). The dotted line is an exponential fit.

Figure 2

Binder cumulant $g$ as a function of inverse temperature $\beta$ for different system sizes $\mathrm{L}$. The inset shows an enlargement of the low-temperature region.

Figure 3

Scaling of the Binder cumulant in the critical region obtained by plotting $g$ as a function of the rescaled inverse temperature $\beta L^{-1∕\nu }$ using $1∕\nu =0.295$. The inset shows an enlargement of the low-temperature region.

Figure 4

Quality function $S\left(\nu \right)$ for the Binder-cumulant scaling. The minimum gives the estimation for the exponent $\nu$.

Figure 5

Scaling of the Binder cumulant out of the critical region, when choosing $1∕\nu =0.38$. The inset shows an enlargement of the low-temperature region.

Figure 6

The spin-glass susceptibility ${\chi }_{\mathrm{sg}}$ as a function of inverse temperature $\beta$ for different system sizes $L$.

Figure 7

Scaling of the spin-glass susceptibility ${\chi }_{\mathrm{sg}}$.

Figure 8

The distribution $P\left(q\right)$ of overlaps for different system sizes at inverse temperature $\beta =2$.

Figure 9

The overlap $q_{\mathrm{gs}}$ with the ground state as a function of the inverse temperature for different system sizes $L$.

Figure 10

Scaling plot of the overlap with the ground state: $q_{\mathrm{gs}}$ as a function of $\beta L^{-1∕\nu }$, with $1∕\nu =0.295$. The inset shows a similar scaling plot, $q_{\mathrm{gs}}$ as a function of $\beta L^{-\zeta }$ $(\zeta =0.83)$, corresponding to a rescaling of all lengths with the overlap length $L^{*}$ instead with the correlation length $\xi$.

Figure 11

The specific heat $c$ as a function of inverse temperature $\beta$ for different system sizes $L$. The straight line is a putative asymptotic behavior $c\sim T$ when $T\to 0$.