No-passing rule in the ground state evolution of the random-field Ising model

We exactly prove the no-passing rule in the ground state evolution of the random-field Ising model with monotonically varying external field. In particular, we show that the application of the no-passing rule can speed up the calculation of the zero-temperature equilibrium $M\left(H\right)$ curve dramatically.

Figure 1

(Color online) An illustration of the algorithm to calculate the equilibrium $M\text{−}H$ curve. Calculate the energies $E_1$ and $E_2$ of the two simplest states $C_1=\left\{s_i=-1\right\}$ and $C_2=\left\{s_i=+1\right\}$, respectively, as a function of $H$. According to Proposition 1, $C_1$ (or $C_2$) would be the ground state for $H<-h_{\text{max}}$ (or $H>-h_{\text{min}}$). Calculate the crossing field $H^{\ast }\left(C_1,C_2\right)$ where $E_1=E_2$. Check whether there is a GS at $H^{\ast }$, which is different from $C_1$ and $C_2$. If no, the algorithm ends. If yes, denote the GS as $C$, calculate the crossing field $H^{\ast }\left(C,C_1\right)$ and $H^{\ast }\left(C,C_2\right)$. At the new crossing fields, check whether there is a GS which is different from the two intersected states. The algorithm will not end until all the crossing fields have been checked. An example of the calculated equilibrium $M\text{−}H$ curve is shown in Fig. 2.

Figure 2

(Color online) The equilibrium $M\text{−}H$ curve (the ground state evolution) for the Gaussian RFIM with $D=3$, $L=32$, and $R=2.837$. Here $R$ is the standard deviation of the Gaussian random-field distribution. The inset shows a detail of the $M\left(H\right)$ curve near $H=0$, where magnetization jumps are clearly seen.

Figure 3

States evolved from $C_1$, with only the change of the spin configuration, i.e., avalanches and reverse avalanches are explicitly shown. (a) State $C_2$: evolved from state $C_1$ with both avalanches and reverse avalanches. (Black dot) Spins flip up, forming avalanches ($A_1$, $A_2$, $A_3$, and $A_4$). (White dot) Spins flip down (reverse flip), forming reverse avalanches $\left(A_1^r,A_2^r\right)$. Note that there are three interacting bonds between avalanche $A_2$ and reverse avalanche $A_2^r$. (b) State $\tilde{C}$ evolved from state $C_1$ without reverse avalanches.

Figure 4

Running time difference $\left(\Delta t\right)$ between methods with and without using the earlier solution, i.e., the GS $C_1$ at field $H_1$, to calculate the GS $C_2$ at field $H_2$. The time difference $\Delta t$ (given in seconds) is plotted against the up-spin (or down-spin) density of the GS $C_1$. $\Delta t<0$ means using the earlier solution will save the running time. Calculations are done for 3D Gaussian RFIM (with disorder parameter $R=2.27$) for different system sizes. (Top) $H_2>H_1$. Up spins in $C_1$ at field $H_1$ will stay up at field $H_2$. (Bottom) $H_2<H_1$. Down spins in $C_1$ at field $H_1$ will stay down at field $H_2$.