Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the two-dimensional Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature $T=0$. From a scaling analysis when $T\to 0$ at different annealing velocities $v$, we find power-law scaling in the system size for the velocity required in order to relax toward the ground state, $v\sim L^{-(z+1/\nu )}$, the Kibble-Zurek ansatz where $z$ is the dynamic critical exponent and $\nu$ the previously known correlation-length exponent, $\nu \approx 3.6$. We find $z\approx 13.6$ for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [Rubin et al., Phys. Rev. E 95, 052133 (2017)] where the dynamics is faster ($z$ is smaller) and the above two quantities relax with different dynamic exponents (with that of the energy being larger). We argue that the different behaviors arise as a consequence of the different low-energy landscapes: for normal-distributed couplings the ground state is unique (up to a spin reflection), while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here also indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasidegenerate states, and the scaling function takes a different form.

Figure 1

(a) Velocity scaling of the mean excess energy density, $\mathrm{\Delta }E=E(v,L)-E_{\infty }$. The data collapse for system sizes in the range $L=8$ to 64 is optimal for $z=13.6\left(4\right)$. The straight line indicates the power-law regime with the expected exponent $x^{\prime }$ given by Eq. (11). (b) The same data graphed according to the third line of Eq. (10). The line shows the expected power-law behavior with exponent $-x^{\prime }$. (c) The data graphed versus $ln(1/v)$.

Figure 2

(a) Velocity scaling of the minimum excess energy $\mathrm{\Delta }{E}_{\min }$ per spin, where the exponent $z+1/\nu =13.9$ is the same as in Fig. 1. (b) Scaling of both $\mathrm{\Delta }E$ and $\mathrm{\Delta }{E}_{\min }$, with the same exponent as in (a) and only including the well-collapsed data in order to make the scaling functions better visible. The straight line is the same as in Fig. 1. In both panels, the asymptotic value of the scaled quantities for small $v{L}^{z+1/\nu }$ is consistent with the coefficient $a=1$ in Eq. (4), as indicated by the dashed lines.

Figure 3

Velocity scaling of the EA order parameter. (a) The horizontal axis is rescaled according to the KZ ansatz with the dynamic exponent $z=13.6$ having the value extracted from $\mathrm{\Delta }E$ in Fig. 1. The straight line corresponds to the expected asymptotic power-law behavior with the exponent $-x$ given in Eq. (8). To show more explicitly the quality of the collapse, the inset includes only the data points used in the fitting procedure and the polynomial fitting function (black curve). (b) The goodness of the fit, ${\chi }^2$ per degree of freedom, versus the scaling exponent $z+1/\nu$. (c) The data are graphed according to the third line of Eq. (7), to show the nonuniversal high-velocity behavior and its crossover into the size-independent power-law behavior. The straight line has the same slope $x$ (up to the sign) as in (a).

Figure 4

(a) The deviation $1-\langle q^2\rangle$ from the asymptotic size-independent value 1 graphed against the KZ-scaled velocity. The collapsed low-velocity data are fitted to a power-law form (the line), $1-\langle q^2\rangle \propto {\left(v{L}^{z+1/\nu }\right)}^a$ with the exponent $a=0.073$. (b) The same data as in (a) graphed against $L/{\xi }_v$, where the velocity-dependent correlation length is ${\xi }_v$ defined in Eq. (13) with the same exponent $z+1/\nu =13.9$ as in (a). The straight line here has slope exactly 1.

Figure 5

Scaling of the deviation $1-\langle q^2\rangle$ of the EA order parameter from its size-independent equilibrium value 1, showing results only for small system sizes. (a) The velocity is scaled according to the standard KZ form; the same as in Fig. 4. (b) The scaling argument $v{L}^{z+1/\nu }(1-a{L}^{-b})$ contains a correction, with optimized parameter values $a=1.7$ and $b=0.39$. The line has the same slope as in Fig. 4.