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arXiv:cond-mat/0206376 · Physical review. B, Condensed matter

Percolation in three-dimensional random field Ising magnets

E. Seppälä, A. M. Pulkkinen, Mikko J. Alava

Published Oct 4, 2002• 22 citations
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Abstract

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 ${\ensuremath{\Delta}}_{c}<\ensuremath{\Delta}<~{\ensuremath{\Delta}}_{p}.$ 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 ${\ensuremath{\Delta}}_{c}<\ensuremath{\Delta}<~{\ensuremath{\Delta}}_{p}$ 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.

DOI
10.1103/physrevb.66.144403

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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 [\ensuremathΔ/J]c\ensuremath≃2.27,[\ensuremath{\Delta}/J{]}_{c}\ensuremath{\simeq}2.27,[\ensuremathΔ/J]c​\ensuremath≃2.27, where \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is the standard deviation of the Gaussian random fields (J=1).(J=1).(J=1). With an external field H there is a disorder-strength-dependent critical field \ifmmode±\else\textpm\fiHc(\ensuremathΔ)\ifmmode\pm\else\textpm\fi{}{H}_{c}(\ensuremath{\Delta})\ifmmode±\else\textpm\fiHc​(\ensuremathΔ) for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. Hc\ensuremath∼(\ensuremathΔ\ensuremath−\ensuremathΔp)\ensuremathδ,{H}_{c}\ensuremath{\sim}(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{p}{)}^{\ensuremath{\delta}},Hc​\ensuremath∼(\ensuremathΔ\ensuremath−\ensuremathΔp​)\ensuremathδ, where \ensuremathΔp=2.43\ifmmode±\else\textpm\fi0.01{\ensuremath{\Delta}}_{p}=2.43\ifmmode\pm\else\textpm\fi{}0.01\ensuremathΔp​=2.43\ifmmode±\else\textpm\fi0.01 and \ensuremathδ=1.31\ifmmode±\else\textpm\fi0.03,\ensuremath{\delta}=1.31\ifmmode\pm\else\textpm\fi{}0.03,\ensuremathδ=1.31\ifmmode±\else\textpm\fi0.03, implying a critical line for \ensuremathΔc<\ensuremathΔ< \ensuremathΔp.{\ensuremath{\Delta}}_{c}<\ensuremath{\Delta}<~{\ensuremath{\Delta}}_{p}.\ensuremathΔc​<\ensuremathΔ< \ensuremathΔp​. When, with zero external field, \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability \ensuremathΠ\ensuremath↑\ensuremath↓=1.00{\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=1.00\ensuremathΠ\ensuremath↑\ensuremath↓​=1.00 to \ensuremathΠ\ensuremath↑\ensuremath↓=0.{\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=0.\ensuremathΠ\ensuremath↑\ensuremath↓​=0. This is located at \ensuremathΔ=2.32\ifmmode±\else\textpm\fi0.01,\ensuremath{\Delta}=2.32\ifmmode\pm\else\textpm\fi{}0.01,\ensuremathΔ=2.32\ifmmode±\else\textpm\fi0.01, i.e., above \ensuremathΔc.{\ensuremath{\Delta}}_{c}.\ensuremathΔc​. The spanning cluster has the fractal dimension of standard percolation, Df=2.53{D}_{f}=2.53Df​=2.53 at H=Hc(\ensuremathΔ).{H=H}_{c}(\ensuremath{\Delta}).H=Hc​(\ensuremathΔ). We provide evidence that this is asymptotically true even at H=0H=0H=0 for \ensuremathΔc<\ensuremathΔ< \ensuremathΔp{\ensuremath{\Delta}}_{c}<\ensuremath{\Delta}<~{\ensuremath{\Delta}}_{p}\ensuremathΔc​<\ensuremathΔ< \ensuremathΔp​ beyond a crossover scale that diverges as \ensuremathΔc{\ensuremath{\Delta}}_{c}\ensuremathΔc​ is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.
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.
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.
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.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to 603.{60}^{3}.603. For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.
Abstract In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent ν ∼ 1.3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent α ∼ − 1.9 . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.