Aging in the three-dimensional random-field Ising model

We studied the nonequilibrium aging behavior of the random-field Ising model in three dimensions for various values of the disorder strength. This allowed us to investigate how the aging behavior changes across the ferromagnetic-paramagnetic phase transition. We investigated a large system size of $N={256}^3$ spins and up to ${10}^8$ Monte Carlo sweeps. To reach these necessary long simulation times, we employed an implementation running on Intel Xeon Phi coprocessors, reaching single-spin-flip times as short as 6 ps. We measured typical correlation functions in space and time to extract a growing length scale and corresponding exponents.

Figure 1

Sketch of the phase diagram. For low temperatures and low disorder, the system is in a ferromagnetic phase and otherwise in a paramagnetic phase.

Figure 2

Snapshots (two-dimensional slices) of a single RFIM sample for two disordered strengths $h_0=1.2$ (top, ferromagnetic phase) and $h_0=2.8$ (bottom, paramagnetic phase) at the different times $t_{\text{w}}={10}^2$ (left), $t_{\text{w}}={10}^4$ (middle), and $t_{\text{w}}={10}^6$ (right), respectively.

Figure 3

Distribution of flipping times $\tau$ (log) in the time interval $[t_{\text{w}},2t_{\text{w}}]$ with $t_{\text{w}}={10}^6$ for various values of the disorder strength $h_0$. Lines are guides to the eyes only.

Figure 4

Spatial correlation $C_2$ over the distance $r$ for different waiting times $t_{\text{w}}$ of a ${256}^3$ system with field strength set to $h_0=1.6$. Only data points at integer values of $r$ are shown to give a clearer picture.

Figure 5

Coherence length ${\xi }_{1,2}$ as a function of the waiting time $t_{\text{w}}$ for different field strengths $h_0$. The coherence length was extracted from the spatial correlation using integral estimators. Inset: section of the same data on a semilog scale.

Figure 6

Power law fits (dashed lines) to extract the exponent $\alpha$ from the integral estimators. The color gradient shows from which waiting time the values of ${\xi }_{1,2}$ and $I_1$ were extracted.

Figure 7

Scaling exponent $\alpha$ of the spatial correlation over the field strength $h_0$ for a ${256}^3$ system, extracted using different methods. The vertical line marks $h_0=h_{\text{c}}$. Inset: Associated scaling exponent $\beta$.

Figure 8

Autocorrelation $C$ as a function of time $t$ for different waiting times $t_{\text{w}}$ at field strength $h_0=1.2$.

Figure 9

Autocorrelation $C$ as a function of time $t$ for different waiting times $t_{\text{w}}$ at field strength $h_0=2.8$. Note that the data for $t_{\text{w}}={10}^4$, ${10}^5$, and ${10}^6$ falls on top of each other.

Figure 10

Equilibrium exponent $x$ for different values of disorder $h_0$. The vertical line marks $h_0=h_{\text{c}}$.

Figure 11

Collapse of autocorrelation $C$ for different waiting times $t_{\text{w}}$ at field strength $h_0=1.2$.

Figure 12

Collapse of autocorrelation $C$ with correction for the quasiequilibrated part $C_{\text{eq}}$ for different waiting times $t_{\text{w}}$ at field strength $h_0=2.8$. An additional additive constant ${\xi }_0\approx -1.333$ was added to the coherence length to give a better collapse.

Figure 13

Long-time behavior of the autocorrelation as a function of the coherence length for few example values of the disorder $h_0$. The lines show fits to tails of power laws according to (11). The inset shows the resulting exponents ${\lambda }_{\text{C}}$ as a function of the disorder strength $h_0$. The vertical line indicates the critical point $h_c$.

Figure 14

Space-time correlation $\widehat{C}$ [see Eq. (12)] as a function of the spin distance $r$ relative to the coherence length $\xi (t_{\text{w}}+t)$ ($h_0=2.2$ near the critical point). Two sets are shown, $\xi (t_{\text{w}}+t)/\xi \left(t_{\text{w}}\right)\approx 0.8$ (upper curve) and $\xi (t_{\text{w}}+t)/\xi \left(t_{\text{w}}\right)\approx 0.625$ (lower curve), for three different waiting times $t_{\text{w}}={10}^3$, ${10}^4$, and ${10}^5$.

Figure 15

Space-time correlation $\widehat{C}$ [see Eq. (12)] as a function of the spin distance $r$ relative to the coherence length $\xi (t_{\text{w}}+t)$ for a fixed ratio $\xi (t_{\text{w}}+t)/\xi \left(t_{\text{w}}\right)\approx 5/6$ and various values of the disorder strength $h_0$.

Figure 16

Example memory layout of a 2D system, before and after exchanging the odd (light color) spins. Each spin in one interleaved memory block can be updated independently because all neighbors are stored in the other memory block.