Critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model on a triangular lattice

We present a numerical study of the critical behavior of the nonequilibrium zero-temperature random field Ising model in two dimensions on a triangular lattice. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations of systems with linear sizes up to $L=65536$, show that the model is in a different universality class than the same model on a quadratic lattice, which is relevant for a better understanding of model universality and the analysis of experimental data.

Figure 1

Scaling collapse of the number $N_2(R,L)$ of 2D spanning avalanches per single run. The collapse is obtained for $R_c=0.86$ and $1/\nu =0.19$; the system sizes are given in legend. In the inset symbols show the $N_2(R,L)$ numbers as a function of disorder $R$ for the lattice size $L=65536$ and disorder $R=1.01$, while the solid curve shows the best fit of these data to the model function (5), $y=0.5\text{erfc}\left[A(x-R_c^{\text{eff}})\right]$, found for $A=30$ and $R_c^{\text{eff}}=1.01$.

Figure 2

Effective critical disorder $R_c^{\text{eff}}\left(L\right)$ versus system size $L$ (symbols), and the power-law prediction $[R_c^{\text{eff}}\left(L\right)-R_c]/R_c^{\text{eff}}\left(L\right)\sim L^{-1/\nu }$ (solid line) with $R_c=0.86$ and $\nu =5.26$.

Figure 3

Magnetization curves obtained for disorders $R=1.18–1.34$ and system size $L=65536$. The curves are averages over 100 random field configurations for each $R$.

Figure 4

Scaling collapse of magnetization curves obtained for disorders $R=1.18–1.34$ and system size $L=65536$. The curves are averages over 100 random field configurations for each $R$. The scaling collapse corresponds to the values of critical exponents and parameters from Tables 1 and 2, respectively.

Figure 5

Susceptibility $\chi =dM/dH$ versus external magnetic field $H$ for disorders $R=1.18–1.34$ and system size $L=65536$. The curves are averages over 100 random field configurations for each $R$.

Figure 6

Scaling collapse of susceptibility curves obtained for the same simulation data and the same values of scaling exponents as in Fig. 4.

Figure 7

Integrated avalanche size distributions $D_R^{\text{(int)}}\left(S\right)$ for disorders $R=1.18–1.34$. The system size is $L=65536$ and $R_c^{\text{eff}}=1.01$ for that size. The curves are averages over 100 random field configurations.

Figure 8

Scaling collapse of the integrated avalanche size distributions from Fig. 7. The solid line represents the phenomenological curve (11). The collapse is obtained for the values of critical exponents and critical parameters from Tables 1 and 2, respectively.

Figure 9

Windowed avalanche size distributions $D_R^{\text{(int)}}\left(S\right)$ collected in a window from the central part of magnetization curve in which magnetization varies from $-0.08\mathrm{to}+0.08$. The simulation data are the same as for the distributions shown in Figs. 7 and 8.

Figure 10

Scaling collapse of the windowed avalanche size distributions shown in Fig. 9. The collapse is obtained for the same values of scaling exponents and parameters as in Fig. 8. The solid curve represents the phenomenological fit ${\overline{\mathcal{D}}}_{+}^{\text{(w)}}\left(X\right)=e^{-2.4849X^{1/\sigma }}(0.021715-0.20277X+0.71027X^2-1.1094X^3+0.65448X^4)$ for the universal scaling function for windowed size distributions.

Figure 11

Integrated avalanche duration distributions $D_R^{\text{(int)}}\left(T\right)$ for disorders $R=1.18–1.34$. The system size is $L=65536$ and $R_c^{\text{eff}}=1.01$. The curves are averages over 100 random field configurations.

Figure 12

Scaling collapse of integrated avalanche duration distributions shown in Fig. 11. The collapse is presented for the values of critical exponents from Table 1. Phenomenological fit of the universal scaling function is shown with the solid curve ${\overline{\mathcal{D}}}_{+}^{\text{(int)}}\left(X\right)=e^{-0.685571X^{1/{\sigma }^{*}}}(0.0193-1.9386X+13.148X^2-30.652X^3+24.047X^4)$.

Figure 13

Windowed avalanche duration distributions $D_R^{\text{(int)}}\left(S\right)$ for disorders $R=1.18–1.34$. The system size is $L=65536$ and $R_c^{\text{eff}}=1.01$.

Figure 14

Scaling collapse of windowed avalanche duration distributions for the same data and same windows as in Fig. 10. The values of the critical parameters are the same as in Fig. 12. The solid line stands for the phenomenological fit of the universal scaling function ${\overline{\mathcal{D}}}_{+}^{\text{(w)}}\left(X\right)=e^{-0.40506X^{1/{\sigma }^{*}}}(0.14838-1.2543X+3.9672X^2-5.5779X^3+2.9513X^4)$.

Figure 15

Power spectra $P\left(f\right)$ versus frequency $f$ for disorders $R=1.18–1.34$ and the system with a triangular lattice of size $L=65536$. The spectra are averaged over 100 realizations of random magnetic field. The dashed line depicts the RFIM spectrum on a quadratic lattice. For all presented spectra $N=8192$ and the Welch window is used.

Figure 16

Magnetization $M$ versus magnetic field $H$ for four lattice of sizes $L$ shown in the legend. All magnetization curves correspond to the same disorder $R=1.30$. The inset shows a zoomed-in portion of the magnetization curves (shown in colors in the legend).