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arXiv:1201.0493 · Physical Review B

Excitations in high-dimensional random-field Ising magnets

Björn Ahrens, Alexander K. Hartmann

Published Jun 20, 2012• 4 citations
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Abstract

Domain walls and droplet-like excitation of the random-field Ising magnet are studied in $d={3,4,5,6,7}$ dimensions by means of exact numerical ground-state calculations. They are obtained using the established mapping to the graph-theoretical maximum-flow problem. This allows us to study large system sizes of more than 5 $\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins in exact thermal equilibrium. All simulations are carried out at the critical point for the strength $h$ of the random fields, $h={h}_{c}(d)$. Using finite-size scaling, energetic and geometric properties like stiffness exponents and fractal dimensions are calculated. Using these results, we test (hyper)scaling relations, which seem to be fulfilled below the upper critical dimension ${d}_{u}=6$. Also, for $d<{d}_{u}$, the stiffness exponent can be obtained from the scaling of the ground-state energy.

DOI
10.1103/physrevb.85.224421

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Domain walls and droplet-like excitation of the random-field Ising magnet are studied in d=3,4,5,6,7d={3,4,5,6,7}d=3,4,5,6,7 dimensions by means of exact numerical ground-state calculations. They are obtained using the established mapping to the graph-theoretical maximum-flow problem. This allows us to study large system sizes of more than 5 \ifmmode×\else\texttimes\fi106\ifmmode\times\else\texttimes\fi{}{10}^{6}\ifmmode×\else\texttimes\fi106 spins in exact thermal equilibrium. All simulations are carried out at the critical point for the strength hhh of the random fields, h=hc(d)h={h}_{c}(d)h=hc​(d). Using finite-size scaling, energetic and geometric properties like stiffness exponents and fractal dimensions are calculated. Using these results, we test (hyper)scaling relations, which seem to be fulfilled below the upper critical dimension du=6{d}_{u}=6du​=6. Also, for d<dud<{d}_{u}d<du​, the stiffness exponent can be obtained from the scaling of the ground-state energy.
We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds,{d}_{s},ds​, and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.
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The four dimensional Gaussian random field Ising magnet is investigated numerically at zero temperature, using samples up to size 64464^4644, to test scaling theories and to investigate the nature of domain walls and the thermodynamic limit. As the magnetization exponent βββ is more easily distinguishable from zero in four dimensions than in three dimensions, these results provide a useful test of conventional scaling theories. Results are presented for the critical behavior of the heat capacity, magnetization, and stiffness. The fractal dimensions of the domain walls at criticality are estimated. A notable difference from three dimensions is the structure of the spin domains: frozen spins of both signs percolate at a disorder magnitude less than the value at the ferromagnetic to paramagnetic transition. Hence, in the vicinity of the transition, there are two percolating clusters of opposite spins that are fixed under any boundary conditions. This structure changes the interpretation of the domain walls for the four dimensional case. The scaling of the effect of boundary conditions on the interior spin configuration is found to be consistent with the domain wall dimension. There is no evidence of a glassy phase: there appears to be a single transition from two ferromagnetic states to a single paramagnetic state, as in three dimensions. The slowing down of the ground state algorithm is also used to study this model and the links between combinatorial optimization and critical behavior.
With respect to the usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the random field Ising model (RFIM) has the peculiarity of involving some ‘droplet’ exponent θ that enters the generalized hyperscaling relation 2 − α = ν( d − θ). In the present paper, to better understand the meaning of this droplet exponent θ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches, i.e. the magnetization jumps that occur in the sequence of ground states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution dl / l 1 + θ of the linear size l of low-energy excitations transforms into the distribution ds / s 1 + θ/ d f for the size s (number of spins) of excitations of fractal dimension d f ( s ∼ l d f ). In the non-mean-field region d < d c , droplets are compact d f = d , whereas in the mean-field region d > d c , droplets have a fractal dimension d f = 2θ leading to the well-known mean-field result ds / s 3/2 . Zero-field equilibrium avalanches are expected to display the same distribution ds / s 1 + θ/ d f . We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent θ(σ) can be varied as a function of the effective long-range interaction J ( r ) ∼ 1/ r d + σ in d = 1.
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