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

Abstract

The presence of random fields is well known to destroy ferromagnetic order in Ising systems in two dimensions. When the system is placed in a sufficiently strong external field, however, the size of clusters of like spins diverges. There is evidence that this percolation transition is in the universality class of standard site percolation. It has been claimed that, for small disorder, a similar percolation phenomenon also occurs in zero external field. Using exact algorithms, we study ground states of large samples and find little evidence for a transition at zero external field. Nevertheless, for sufficiently small random-field strengths, there is an extended region of the phase diagram, where finite samples are indistinguishable from a critical percolating system. In this regime we examine ground-state domain walls, finding strong evidence that they are conformally invariant and satisfy Schramm-Loewner evolution (SLE κ ) with parameter κ = 6. These results add support to the hope that at least some aspects of systems with quenched disorder might be ultimately studied with the techniques of SLE and conformal field theory.The random-field Ising model (RFIM) is one of the earliest studied and simplest disordered systems showing non-trivial and glassy behavior [1,2]. It has a number of important realizations in nature, including diluted antiferromagnets in a field and binary liquids in porous media [2]. Through its long history, researchers have managed to gain a reasonable understanding of the critical behavior, although this progress has been neither straight nor smooth, and many questions remain unanswered [2]. It is known, for example, that the RFIM in two dimensions (2D) lacks ferromagnetic order [3,4]. Even at zero temperature it remains in the paramagnetic state for non-zero disorder. 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.The observed relations to classical site percolation at this (nonzero or zero field) geometrical transition motivate further questions of how far the similarities go. Interfaces in two-dimensional percolation satisfy Schramm-Loewner evolution (SLE) [8,9], but

Approximate ground states of the random-field Potts model from graph cuts

“…It is a fortunate coincidence that the problem of finding the ground state for an RFIM sample can be mapped to a maximumflow problem that is in P [32], i.e., there exist algorithms that solve it in a time that grows as a polynomial in the size of the system, including the Ford-Fulkerson algorithm of augmenting paths [33], the Goldberg-Tarjan push-relabel method [34], or variants thereof [35]. In the last few years, we have acquired significant knowledge about the ground-state properties of the RFIM [36][37][38][39][40][41]. The situation is different, however, for the case of the RFPM with q > 2 which corresponds to a multi-terminal flow or, equivalently, graph cut (GC) problem that is known to be NP hard [32,42].…”

Domain walls and Schramm-Loewner evolution in the random-field Ising model

“…On the contrary, for d ≥ 3, L b diverges at the thermodynamic transition point, below which the system is ferromagnetic [13]. For non-zero average fields H, on the other hand, even in 2D the size of spin clusters diverges at a critical value H c = H c (∆) [19][20][21][22]. However, the weight of these clusters is sub-extensive, such that the free energy remains analytic and no thermodynamic phase transition occurs.…”

Numerical evaluation of the Gauss hypergeometric function: Implementation and application to Schramm-Loewner evolution

“…Since SLE implies several geometrical properties of the associated curves, it is not clear a priori whether a given distribution of fractal curves can in fact be described by SLE. Therefore there have been a number of computer simulation studies testing numerically the compatibility of the properties of different ensembles of curves with the predictions of SLE [12][13][14][15][16][17][18]. The present work is based on a note in Ref.…”