Random Field Ising Magnets Research Articles

Anisotropic avalanches and critical depinning of three-dimensional magnetic domain walls.

Simulations with more than 10^{12} spins are used to study the motion of a domain wall driven through a three-dimensional random-field Ising magnet (RFIM) by an external field H. The interface advances in a series of avalanches whose size diverges at a critical external field H_{c}. Finite-size scaling is applied to determine critical exponents and test scaling relations. Growth is intrinsically anisotropic with the height of an avalanche normal to the interface ℓ_{⊥} scaling as the width along the interface ℓ_{∥} to a power χ=0.85±0.01. The total interface roughness is consistent with self-affine scaling with a roughness exponent ζ≈χ that is much larger than values found previously for the RFIM and related models that explicitly break orientational symmetry by requiring the interface to be single-valued. Because the RFIM maintains orientational symmetry, the interface develops overhangs that may surround unfavorable regions to create uninvaded bubbles. Overhangs complicate measures of the roughness exponent but decrease in importance with increasing system size.

Glassy Dynamics of Passive and Active Network Materials: A Microscopic Theory

Signatures of glassy dynamics have been identified experimentally for a rich variety of materials in which molecular networks provide rigidity. Here we present a theoretical framework to study the glassy behavior of both passive and active network materials. We construct a general microscopic network model that incorporates nonlinear elasticity of individual filaments and steric constraints due to crowding. By building constructive analogies between structural glass forming liquids and random field Ising magnets implemented using a heterogeneous self-consistent phonon method, we provide a microscopic approach to determine the mismatch surface tension and the configurational entropy, which compete in determining the barrier for structural rearrangements. This model recapitulates the influence of crosslinking on the fragility of inorganic network glass formers. For active network materials, the mapping, which correlates the glassy characteristics to the network architecture and properties of nonequilibrium motor processes, is shown to capture several key experimental observations on the cytoskeleton of living cells. Our calculations also identify a spinodal point where simultaneously the mismatch penalty vanishes and the mechanical stability of amorphous packing disappears.

Barkhausen noise in the random field Ising magnetNd2Fe14B

With sintered needles aligned and a magnetic field applied transverse to its easy axis, the rare-earth ferromagnet Nd_2Fe_(14)B becomes a room-temperature realization of the random field Ising model. The transverse field tunes the pinning potential of the magnetic domains in a continuous fashion. We study the magnetic domain reversal and avalanche dynamics between liquid helium and room temperatures at a series of transverse fields using a Barkhausen noise technique. The avalanche size and energy distributions follow power-law behavior with a cutoff dependent on the pinning strength dialed in by the transverse field, consistent with theoretical predictions for Barkhausen avalanches in disordered materials. A scaling analysis reveals two regimes of behavior: one at low temperature and high transverse field, where the dynamics are governed by the randomness, and the second at high temperature and low transverse field, where thermal fluctuations dominate the dynamics.

Exact ground states of one-dimensional long-range random-field Ising magnets

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 $r$ 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 $L={2}^{19}$ 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 $h$ 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 ${h}_{c}>0$.

Microscopic theory of the glassy dynamics of passive and active network materials

Signatures of glassy dynamics have been identified experimentally for a rich variety of materials in which molecular networks provide rigidity. Here we present a theoretical framework to study the glassy behavior of both passive and active network materials. We construct a general microscopic network model that incorporates nonlinear elasticity of individual filaments and steric constraints due to crowding. Based on constructive analogies between structural glass forming liquids and random field Ising magnets implemented using a heterogeneous self-consistent phonon method, our scheme provides a microscopic approach to determine the mismatch surface tension and the configurational entropy, which compete in determining the barrier for structural rearrangements within the random first order transition theory of escape from a local energy minimum. The influence of crosslinking on the fragility of inorganic network glass formers is recapitulated by the model. For active network materials, the mapping, which correlates the glassy characteristics to the network architecture and properties of nonequilibrium motor processes, is shown to capture several key experimental observations on the cytoskeleton of living cells: Highly connected tense networks behave as strong glass formers; intense motor action promotes reconfiguration. The fact that our model assuming a negative motor susceptibility predicts the latter suggests that on average the motorized processes in living cells do resist the imposed mechanical load. Our calculations also identify a spinodal point where simultaneously the mismatch penalty vanishes and the mechanical stability of amorphous packing disappears.

Excitations in high-dimensional random-field Ising magnets

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.

Critical behavior of the random-field Ising magnet with long-range correlated disorder

We study the correlated-disorder driven zero-temperature phase transition of the Random-Field Ising Magnet using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportional to r^a, where r is the distance between two lattice sites and a<0. To obtain exact ground states, we use a well established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than two million spins. We use finite-size scaling analyses for values a={-1,-2,-3,-7} to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility and of the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with delta-correlated disorder, except for the finite-size exponent of the susceptibility and for the case a=-1, where the results are also compatible with a phase transition at infinitesimal disorder strength. A summary of this work can be found at the papercore database at www.papercore.org.

URBAN SEGREGATION WITH CHEAP AND EXPENSIVE RESIDENCES

In this paper we study urban segregation of two different communities A and B, rich and poor, distributed randomly on finite samples, to check cheap and expensive residences. For this purpose we avoid the complications of the Schelling model which are not necessary and instead we use the Ising model on 500 × 500 square lattices, which gives similar results, with random magnetic field at lower and higher temperatures (kBT/J = 2.0, 99.0) in finite times equal to 40, 400, 4000 and 40 000. This random-field Ising magnet is a suitable model, where each site of the square lattice carries a magnetic field ±h which is randomly up (expensive) or down (cheap). The resulting addition to the energy prefers up-spins on the expensive and down-spins on the cheap sites. Our simulations were carried out using a 50-line FORTRAN program. We present at a lower temperature (2.0) a time series of pictures, separating growing from non-growing domains. A small random field (h = ±0.1) allows for large domains, while a large random field (h = ±0.9) allows only small clusters. At higher temperature (99.0) we could not obtain growing domains.

Surface criticality in random field magnets

The boundary-induced scaling of three-dimensional random field Ising magnets is investigated close to the bulk critical point by exact combinatorial optimization methods. We measure several exponents describing surface criticality: ${\ensuremath{\beta}}_{1}$ for the surface layer magnetization and the surface excess exponents for the magnetization and the specific heat, ${\ensuremath{\beta}}_{s}$ and ${\ensuremath{\alpha}}_{s}$. The latter are related to the bulk phase transition by the same scaling laws as in pure systems, but only with the same violation of the hyperscaling exponent $\ensuremath{\theta}$ as in the bulk. The boundary disorders faster than the bulk, and the experimental and theoretical implications are discussed.

Magnetization processes in nanostructured metals and small-angle neutron scattering

The magnetization process in nanostructured $(n\text{\ensuremath{-}})$ Fe and Co was investigated via small-angle neutron scattering (SANS). In a zero field, the magnetization exhibits correlations extending over several grains. In intermediate applied magnetic fields around $1\phantom{\rule{0.3em}{0ex}}\mathrm{kOe}$, $n\text{\ensuremath{-}}\mathrm{Fe}$ and $n\text{\ensuremath{-}}\mathrm{Co}$ samples with small grain sizes exhibit an anisotropic scattering profile with an unusual intensity enhancement for scattering vectors parallel to the field direction. Comparing the experimental data with a modeled granular microstructure containing magnetic domains of arbitrary size and orientation, we conclude that magnetic domains extending over several grains are tilted considerably out of the external field direction in intermediate fields. Since the domain size does not change significantly with the magnitude of the external field, we conclude that the magnetization process does not proceed via domain-wall motion. Together with theoretical arguments showing the existence of marginally stable domains within the random-anisotropy model, our SANS data suggests that the magnetization process proceeds by simultaneous reversal of a few adjacent domains, presumably in the form of small avalanches. This resembles the magnetization process predicted for random-field Ising magnets. Our theoretical analysis of SANS data is general and applies to other systems consisting of magnetic nanoclusters embedded in a nonmagnetic matrix.

Domain walls in random field Ising magnets: wetting

Domain walls in random-field Ising magnets can be investigated in groundstates into which walls are induced by prepared boundary conditions. We outline recent progress, and new results on (domain wall) wetting in random field systems. This is studied in fixed disorder configurations in the presence of an external field, which is varied.

Domain walls in random field Ising magnets: wetting

Domain walls in random-field Ising magnets can be investigated in groundstates into which walls are induced by prepared boundary conditions. We outline recent progress, and new results on (domain wall) wetting in random field systems. This is studied in fixed disorder configurations in the presence of an external field, which is varied.

Spin wave propagation in the domain state of a random field magnet

Inelastic neutron scattering with high wave-vector resolution has characterized the propagation of transverse spin wave modes near the antiferromagnetic zone center in the metastable domain state of a random field Ising magnet. A well-defined, long wavelength excitation is observed despite the absence of long-range magnetic order. Direct comparisons with the spin wave dispersion in the long-range ordered antiferromagnetic state reveal no measurable effects from the domain structure. This result recalls analogous behavior in thermally disordered anisotropic spin chains but contrasts sharply with that of the phonon modes in relaxor ferroelectrics.

Percolation in three-dimensional random field Ising magnets

The structure of the three-dimensional (3D) random field Ising magnet is studied by ground-state calculations. We investigate the percolation of the minority-spin orientation in the paramagnetic phase above the bulk phase transition, located at $[\ensuremath{\Delta}/J{]}_{c}\ensuremath{\simeq}2.27,$ where $\ensuremath{\Delta}$ is the standard deviation of the Gaussian random fields $(J=1).$ With an external field H there is a disorder-strength-dependent critical field $\ifmmode\pm\else\textpm\fi{}{H}_{c}(\ensuremath{\Delta})$ for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. ${H}_{c}\ensuremath{\sim}(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{p}{)}^{\ensuremath{\delta}},$ where ${\ensuremath{\Delta}}_{p}=2.43\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\delta}=1.31\ifmmode\pm\else\textpm\fi{}0.03,$ implying a critical line for ${\ensuremath{\Delta}}_{c}&lt;\ensuremath{\Delta}&lt;~{\ensuremath{\Delta}}_{p}.$ When, with zero external field, $\ensuremath{\Delta}$ is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability ${\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=1.00$ to ${\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=0.$ This is located at $\ensuremath{\Delta}=2.32\ifmmode\pm\else\textpm\fi{}0.01,$ i.e., above ${\ensuremath{\Delta}}_{c}.$ The spanning cluster has the fractal dimension of standard percolation, ${D}_{f}=2.53$ at ${H=H}_{c}(\ensuremath{\Delta}).$ We provide evidence that this is asymptotically true even at $H=0$ for ${\ensuremath{\Delta}}_{c}&lt;\ensuremath{\Delta}&lt;~{\ensuremath{\Delta}}_{p}$ beyond a crossover scale that diverges as ${\ensuremath{\Delta}}_{c}$ is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.

Three-dimensional random-field Ising magnet: Interfaces, scaling, and the nature of states

The nature of the zero temperature ordering transition in the 3D Gaussian random field Ising magnet is studied numerically, aided by scaling analyses. In the ferromagnetic phase the scaling of the roughness of the domain walls, $w\sim L^\zeta$, is consistent with the theoretical prediction $\zeta = 2/3$. As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales as for 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. These dimensions are argued to be related to combinations of the energy scaling exponent, $\theta$, which determines the violation of hyperscaling, the correlation length exponent $\nu$, and the magnetization exponent $\beta$. The value $\beta = 0.017\pm 0.005$ is derived from the magnetization: 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 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.

Specific-heat exponent of random-field systems via ground-state calculations

Exact ground states of three-dimensional random field Ising magnets (RFIM) with Gaussian distribution of the disorder are calculated using graph-theoretical algorithms. Systems for different strengths h of the random fields and sizes up to N=96^3 are considered. By numerically differentiating the bond-energy with respect to h a specific-heat like quantity is obtained, which does not appear to diverge at the critical point but rather exhibits a cusp. We also consider the effect of a small uniform magnetic field, which allows us to calculate the T=0 susceptibility. From a finite-size scaling analysis, we obtain the critical exponents \nu=1.32(7), \alpha=-0.63(7), \eta=0.50(3) and find that the critical strength of the random field is h_c=2.28(1). We discuss the significance of the result that \alpha appears to be strongly negative.

Susceptibility and percolation in two-dimensional random field Ising magnets.

The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength-dependent length scale. This breakup length scale L(b) scales exponentially with the squared random field, exp(A/delta(2)). By adding an external field H, we then study the susceptibility in the ground state. If L>L(b), domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as L-2. We define a random field strength-dependent critical external field value +/-H(c)(delta) for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing Delta and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) H(c) approximately (delta-delta(c))(delta), where delta(c)=1.65+/-0.05 and delta=2.05+/-0.10. Below Delta(c) the systems should percolate even when H=0. This implies that even for H=0 above L(b) the domains can be fractal at low random fields, such that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation D(f)=91/48. The structure of the spanning clusters is studied by defining red clusters, in analogy to the "red sites" of ordinary site percolation. The sizes of red clusters define an extra length scale, independent of L.

Combinatorial optimization methods in disordered systems

We give an overview of the applications of methods from combinatorial optimization to problems in disordered systems. The optimization methods are efficient, for example it is possible to find the ground state of a random field Ising magnet containing one million sites in a couple of minutes on a high end workstation. Combinatorial algorithms for rigidity percolation and minimal energy domain walls in random exchange magnets are even more efficient.

Disorder, order, and domain wall roughening in the two-dimensional random field Ising model

Ground states and domain walls are investigated with exact combinatorial optimization in two-dimensional random field Ising magnets. The ground states break into domains above a length scale that depends exponentially on the random field strength squared. For weak disorder, this paramagnetic structure has remnant long-range order of the percolation type. The domain walls are super-rough in ordered systems with a roughness exponent \ensuremath{\zeta} close to 6/5. The interfaces exhibit rare fluctuations and multiscaling reminiscent of some models of kinetic roughening and hydrodynamic turbulence.

Ground state structure of random magnets

Using exact optimization methods, we find all of the ground states of +/- h random-field Ising magnets (RFIM) and of dilute antiferromagnets in a field (DAFF). The degenerate ground states are usually composed of isolated clusters (two-level systems) embedded in a frozen background. We calculate the paramagnetic response (sublattice response) and the ground state entropy for the RFIM (DAFF) due to these clusters. In both two and three dimensions there is a broad regime in which these quantities are strictly positive, even at irrational values of h/J (J is the exchange constant).