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arXiv:1010.5973 · Europhysics Letters (EPL)

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

J. D. Stevenson, M. Weigel

Published Jul 25, 2011• 26 citations
arXivHTMLPDF

Abstract

The concept of Schramm-Loewner evolution provides a unified description of domain boundaries of many lattice spin systems in two dimensions, possibly even including systems with quenched disorder. Here, we study domain walls in the random-field Ising model. Although, in two dimensions, this system does not show an ordering transition to a ferromagnetic state, in the presence of a uniform external field spin domains percolate beyond a critical field strength. Using exact ground-state calculations for very large systems, we examine ground-state domain walls near this percolation transition finding strong evidence that they are conformally invariant and satisfy the domain Markov property, implying compatibility with Schramm-Loewner evolution (SLE κ ) with parameter κ = 6. These results might pave the way for new field-theoretic treatments of systems with quenched disorder.

DOI
10.1209/0295-5075/95/40001

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The concept of Schramm-Loewner evolution provides a unified description of domain boundaries of many lattice spin systems in two dimensions, possibly even including systems with quenched disorder. Here, we study domain walls in the random-field Ising model. Although, in two dimensions, this system does not show an ordering transition to a ferromagnetic state, in the presence of a uniform external field spin domains percolate beyond a critical field strength. Using exact ground-state calculations for very large systems, we examine ground-state domain walls near this percolation transition finding strong evidence that they are conformally invariant and satisfy the domain Markov property, implying compatibility with Schramm-Loewner evolution (SLE κ ) with parameter κ = 6. These results might pave the way for new field-theoretic treatments of systems with quenched disorder.
Domain walls for spin glasses are believed to be scale invariant; a stronger symmetry, conformal invariance, has the potential to hold. The statistics of zero-temperature Ising spin glass domain walls in two dimensions are used to test the hypothesis that these domain walls are described by a Schramm-Loewner evolution ${\mathrm{SLE}}_{\ensuremath{\kappa}}$. Multiple tests are consistent with ${\mathrm{SLE}}_{\ensuremath{\kappa}}$, where $\ensuremath{\kappa}=2.32\ifmmode\pm\else\textpm\fi{}0.08$. Both conformal invariance and the domain Markov property are tested. The latter does not hold in small systems, but detailed numerical evidence suggests that it holds in the continuum limit.
We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length L, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows to tune the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic groundstate. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the the average domain-wall energy (~L^\theta) and the average domain-wall length (~L^df). Performing a finite-size scaling analysis for systems with a side length up to L=512, we find that both exponents remain constant in the spin-glass phase, i.e. \theta ~- 0.28 and df~1.275. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of Stochastic Loewner evolutions (SLEs) via df-1=3/[4(3+\theta)]. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.
We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with kappa approximately 2.1. An argument is given that their fractal dimension d(f) is related to their interface energy exponent theta by d(f) - 1 = 3/[4(3 + theta)], which is consistent with the commonly quoted values d(f) approximately 1.27 and theta approximately -0.28.
We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length LLL, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows tuning of the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic ground state. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the average domain-wall energy $(\ensuremath{\sim}{L}^{\ensuremath{\theta}})$ and the average domain-wall length $(\ensuremath{\sim}{L}^{{d}_{f}})$. Performing a finite-size scaling analysis for systems with a side length up to L=512L=512L=512, we find that both exponents remain constant in the spin-glass phase, i.e., $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.28$ and ${d}_{f}\ensuremath{\approx}1.275$. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of stochastic Loewner evolutions via ${d}_{f}\ensuremath{-}1=3/[4(3+\ensuremath{\theta})]$. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.
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.
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.