Ground-state optimization and hysteretic demagnetization: The random-field Ising model

We compare the ground state of the random-field Ising model with Gaussian distributed random fields, with its nonequilibrium hysteretic counterpart, the demagnetized state. This is a low-energy state obtained by a sequence of slow magnetic-field oscillations with decreasing amplitude. The main concern is how optimized the demagnetized state is with respect to the best-possible ground state. Exact results for the energy in $d=1$ show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For $d⩾3$, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points $R_c^{\left(\mathrm{DS}\right)}$ and $R_c^{\left(\mathrm{GS}\right)}$ differ for the demagnetized state and ground state. We argue based on the numerics that in $d=3$ the scaling at the transition is the same in both states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the critical points are also different.

Figure 1

The hysteresis loop of the RFIM computed exactly in $d=1$. We also report three minor loops. A demagnetization procedure corresponds to a series of minor loops obtained reversing the field at $H_0,H_1,H_2$,… as detailed in the text.

Figure 2

The energy of the GS is compared with the one of DS. The values are computed exactly in $d=1$ as a function of the disorder width $R$.

Figure 3

The energy difference between the GS and the DS computed exactly in $d=1$ is compared with the numerical simulations.

Figure 4

An illustration of the possible mechanisms for the deviations between GS and DS.

Figure 5

The average change in the spin-spin overlap between the GS and the DS $\left(\Delta q\right)$ and the contribution to that from completely “destroyed” GS domains $\left(\Delta q_{\mathrm{destr}}\right)$, as a function of $R$.

Figure 6

(Color online) The average overlap of a DS domain of size $l_{\mathrm{DS}}$ with the GS domain spin state at the same locations for $R=0.5$, 0.6, 0.8, 0.9, 1.0. The overlap decreases with $l_{\mathrm{DS}}$.

Figure 7

(Color online) The Zeeman energy of DS domains of the size, $l_{\mathrm{DS}}$. The black circles mark the average DS domain size for a given $R$. The two lines above and below the data indicate optimal, linear (GS-like) scaling and the Imry-Ma-like $l^{1∕2}$ scaling, respectively.

Figure 8

The magnetization of the GS and the DS in $d=3$ for different system size $L$ and disorder $R$.

Figure 9

Numerical results in $d=3$: The magnetization can be collapsed using $R_c=2.28$ (GS) and $R_c=2.16$ (DS), $\nu =1.37$, and $\beta =0.03$. The scaling curve is the same for DS and GS indicating universal behavior. The values for the ratios of the nonuniversal constants are $A_{\mathrm{DS}}∕A_{\mathrm{GS}}=1$ and $B_{\mathrm{DS}}∕B_{\mathrm{GS}}=0.68$.

Figure 10

The overlap between the GS and the DS in $d=3$ for different system sizes.

Figure 11

The difference in magnetization between the GS and the DS in $d=3$ for different system sizes.

Figure 12

The magnetization of the GS and the DS computed exactly on the Bethe lattice with $z=4$ in the thermodynamic limit, showing the ordering of the critical point (see the inset). When the data are plotted against the reduced parameter $(R_c-R)∕R_c$ the curves superimpose. The result implies that for the Bethe lattice $A_{\mathrm{GS}}=A_{\mathrm{DS}}$.

Figure 13

The distributions of the magnetization in the DS and the GS at their respective critical points on the Bethe lattice, obtained numerically for different lattice sizes $N$, can be all collapsed together.

Figure 14

Fraction of reachable states, averages over 100 realizations of disorder. Full symbols: fraction $f_{\mathrm{RND}}$ of reachable locally stable states, generated by random sampling the set of local minima. Open symbols: fraction $f_{\mathrm{GS}}$ of ground states reachable considering different realizations. System sizes are $N=10000$ (squares) and $N=5000$ (circles).