Random-field-induced disordering mechanism in a disordered ferromagnet: Between the Imry-Ma and the standard disordering mechanism

Random fields disorder Ising ferromagnets by aligning single spins in the direction of the random field in three space dimensions, or by flipping large ferromagnetic domains at dimensions two and below. While the former requires random fields of typical magnitude similar to the interaction strength, the latter Imry-Ma mechanism only requires infinitesimal random fields. Recently, it has been shown that for dilute anisotropic dipolar systems a third mechanism exists, where the ferromagnetic phase is disordered by finite-size glassy domains at a random field of finite magnitude that is considerably smaller than the typical interaction strength. Using large-scale Monte Carlo simulations and zero-temperature numerical approaches, we show that this mechanism applies to disordered ferromagnets with competing short-range ferromagnetic and antiferromagnetic interactions, suggesting its generality in ferromagnetic systems with competing interactions and an underlying spin-glass phase. A finite-size-scaling analysis of the magnetization distribution suggests that the transition might be first order.

Figure 1

Dimensionless temperature $T/J$ vs mean of the disorder distribution $J_0/J$ phase diagram for the model given in Eq. (1) with fixed $J_0=1$ and $H_r=0$. At $J\lesssim 1.65$ ($J_0/J\gtrsim 0.606$) and low temperatures the system is in a FM phase. For $J\gtrsim 1.65$ ($J_0/J\lesssim 0.606$) and low temperatures the system is in a SG phase. At high temperatures the system is in the PM phase independently of the value of $J_0/J$. Note that the model reduces to the three-dimensional Ising model for $J=0$ and to the EA spin glass for $J_0=0$.

Figure 2

Binder ratio $g$ as given by Eq. (2) for linear system sizes $L=4$, 6, 8, and 10 at a random field strength $H_r=0$ and $T=0$. The crossing of the curves for different sizes $L$ gives an estimate of the critical disorder $J_c$ denoting the transition between the FM phase at $J<J_c$ and the SG phase at $J>J_c$. The dashed lines are guides to the eye.

Figure 3

Ferromagnetic and spin-glass two-point correlation functions divided by the linear system size $L$ for $J=1.60$ (a) and 1.80 (b) and (c), respectively. (a) The crossing of the curves of the dimensionless quantity ${\xi }_L^{\mathrm{fm}}/L$ for linear system sizes $L=6–20$ indicates that a PM to FM transition occurs at $T_c\sim 3.09\left(4\right)$. (b) The dimensionless quantity ${\xi }_L^{\mathrm{fm}}/L$ does not cross in the studied temperature range, showing the lack of a FM transition. (c) Curves corresponding to different system sizes of the dimensionless quantity ${\xi }_L^{\mathrm{sg}}/L$ cross at $T_c\sim 1.90\left(14\right)$ indicating that a PM to SG transition occurs in this temperature range.

Figure 4

$J-H_r$ phase diagram at $T=0$. At $H_r=0$ there is a FM-SG transition at $J_c\left(0\right)=1.63$. At finite but small $H_r$ the disordered paramagnetic phase has finite size glassy domains, denoted here as a “quasi-SG” (QSG) phase. The crossover between the QSG phase and the PM phase with short range spin-spin correlations is speculative. In the inset, the solid black line is a fit $H_r\left(J^{-1}\right)=\alpha {(J^{-1}-J_c^{-1})}^{\beta }$ to the data at low random field strength. Here $J_c$ and $\beta$ are fixed parameters; we use the above estimated value $J_c\left(0\right)=1.63$, and the analytical result of Ref. [13] $\beta =0.466$. $\alpha =1.52\left(17\right)$ is a free fitting parameter.

Figure 5

$T-H_r$ phase diagram for $J=0.50$ (a), $J=1.50$ (b), and $J=1.60$ (c). For the slightly disordered system with $J=0.50$ in (a) the relation expected from mean-field theory $T\left(H_r\right)-T_c\sim H_r^2$ holds, except for a weak reentrance at the lowest temperatures. The data are consistent with the disordered phase being a standard PM. For the highly disordered systems with $J=1.50$ (b) and $J=1.60$ (c) where the FM phase is close to the SG phase at $H_r=0$ (see Fig. 4) we find a linear relation $T\left(H_r\right)-T_c\left(h^{*}\right)\propto H_r$ for $H_r>h^{*}$ consistent with the disordered phase being a pQSG. Furthermore, we find strong reentrance at low temperatures with critical $H_r\ll J_0$, in agreement with the disordered phase being a QSG. The inset in (c) shows ${\xi }_L^{\mathrm{fm}}$ for $L=8$, 10, and 12 for $H_r=0.40$ where the curves do not cross, indicating the lack of a phase transition to the FM phase for the studied temperature range. The dashed lines in all panels between the pQSG and PM phases and between the QSG and pQSG phases denote smooth crossovers, their functional form should be considered as a guide to the eye.

Figure 6

Magnetization histograms for $J=1.50$ with a random field $H_r=0.04$ and $L=40$ (below the critical temperature) (a), $H_r=0.40$ and $L=20$ (below the critical temperature) (b), and $H_r=0.40$ and $L=20$ (above the critical temperature) (c). The solid red lines are double-Gaussian fits, which are composed of the sum of the two modes (thin dashed black lines), the blue dashed lines are single Gaussian fits. For $H_r=0.04$, the double Gaussian fit does not differ much from the single Gaussian fit, i.e., the distribution is normally distributed as expected for all temperatures in a continuous phase transition. For $H_r=0.40$ a hump emerges for $T<T_c$, the single Gaussian fit (blue dashed line) cannot capture the hump, but the double Gaussian fit (solid red line) does. Close to the critical temperature the single Gaussian fails to fit the distribution and a bimodal double-Gaussian structure becomes evident suggesting a finite jump in the magnetization washed out by the disorder at $T_c$, as expected for a first-order phase transition.

Figure 7

Relative difference of the bootstrapped mean value of the two modes (with mean values ${\mu }_0^1$ and ${\mu }_0^2$) of the bimodal Gaussian fitted function as a function of temperature for (a) $H_r=0.30$, (b) $H_r=0.40$, and (c) $H_r=0.50$. The gray vertical line corresponds to the estimated critical temperature. The relative difference between the modes increases close to $T_c$, the discrepancy between the estimated value for $T_c$ and the maximal values of the curves can be attributed to the strong finite-size correction of this observable.