Susceptibility and percolation in two-dimensional random field Ising magnets

Abstract

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.

Random-field Ising model: Insight from zero-temperature simulations

“…Hence, the critical behavior is the same everywhere along the phase boundary of figure 1, and we can predict it simply by staying at T = 0 and crossing the phase boundary at h = h c . This is a convenient approach, because we can determine the ground states of the system exactly using efficient optimization algorithms [20,21,25,65,66,[71][72][73][74][75][76] through an existing mapping of the ground state to the maximum-flow optimization problem [77]. A clear advantage of this approach is the ability to simulate large system sizes and disorder ensembles in rather moderate computational times.…”

Percolation and Schramm–Loewner evolution in the 2D random-field Ising model

“…Though there is no thermodynamic transition in 2D, the linear extent of the largest clusters diverges for sufficiently large H. In most aspects, this divergence appears to be consistent with standard site percolation [5,6,7]. It has been suggested that the divergence occurs even at H = 0 if ∆ is below a critical value [5,7].…”

“…However, it has been demonstrated that there exists a line of critical external fields H c (∆), as pictured schematically in Fig. 1, for which observables like the crossing probability and the fractal dimension of the spin clusters maintain the percolation values [6]. For very large disorder the line of critical external fields is found from the critical site percolation probability p c (p c ≈ 0.5927 for the square lattice)…”

“…Numerical ground-state calculations have shown, however, that even in the absence of a thermodynamic transition there exists a geometric transition at which the size of the spin clusters diverges in a manner bearing many similarities to classical site percolation [5,6]. While this transition is rather clearly established in the presence of an external field, it has been argued that a similar percolation phenomenon can also be observed in the absence of an external field for sufficiently small disorder [6,7]. Here, we re-investigate the zero-field behavior with large scale ground-state calculations, focusing on the possible percolation phenomenon.…”

Characterization of kinetic coarsening in a random-field Ising model

“…3. In this context, Seppälä and Alava [32] showed that below a critical random field strength, the largest domain spans the system and the two-dimensional (2D) RFIM shows a percolation transition. This was supported by some later studies [7,33].…”