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arXiv:cond-mat/0407297 · Physical Review B

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

Mikko J. Alava, Vittorio Basso, Francesca Colaiori, Lorenzo Dante, Gianfranco Durin, Alessandro Magni, Stefano Zapperi

Published Feb 28, 2005• 19 citations
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Abstract

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 $d\ensuremath{\geqslant}3$, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points ${R}_{c}^{(\mathrm{DS})}$ and ${R}_{c}^{(\mathrm{GS})}$ 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.

DOI
10.1103/physrevb.71.064423

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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=1d=1d=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\ensuremath⩾3d\ensuremath{\geqslant}3d\ensuremath⩾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 Rc(DS){R}_{c}^{(\mathrm{DS})}Rc(DS)​ and Rc(GS){R}_{c}^{(\mathrm{GS})}Rc(GS)​ differ for the demagnetized state and ground state. We argue based on the numerics that in d=3d=3d=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.
The equilibrium and nonequilibrium disorder-induced phase transitions are compared in the random-field Ising model. We identify in the demagnetized state the correct nonequilibrium hysteretic counterpart of the T=0 ground state, and present evidence of universality. Numerical simulations in d=3 indicate that exponents and scaling functions coincide, while the location of the critical point differs, as corroborated by exact results for the Bethe lattice. These results are of relevance for optimization, and for the generic question of universality in the presence of disorder.
Phase transition in d=3d=3d=3 ferromagnetic Ising models with random fields is analyzed directly at the zero-temperature critical point. Critical behavior is extracted from correlation functions averaged over an ensemble of exact ground states obtained with a new integer optimization algorithm. For Gaussian distribution of random fields finite-size scaling demonstrates a continuous phase transition with effective disconnected susceptibility exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremath{\approx}\ensuremath{-}0.9$, correlation-length exponent $\ensuremath{\nu}\ensuremath{\approx}1.0$, and magnetization exponent $\ensuremath{\beta}\ensuremath{\approx}0.05$.
The ground state of the random-field Ising ferromagnet on a Bethe lattice is found. With decreasing random-field strength an infinite series of spin-flip transitions precedes the onset of ferromagnetism. The T=0T=0T=0 critical behavior is not mean-field-like. At finite temperatures a series of Griffiths singularities precedes the phase transition. As $T\ensuremath{\rightarrow}0$, the Griffiths singularities terminate at the spin-flip transitions. The $T\ensuremath{\ne}0$ critical behavior is argued to be mean-field-like.
The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in d=5,6,7d=5,6,7d=5,6,7 dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions d=6,7d=6,7d=6,7 which are larger or equal to the assumed upper critical dimension, du=6{d}_{u}=6du​=6, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.
We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where rrr is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to L=219L={2}^{19}L=219 via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength hhh of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength hc>0{h}_{c}>0hc​>0.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.