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arXiv:cond-mat/0102318 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Extremal statistics in the energetics of domain walls

E. Seppälä, Mikko J. Alava, Phillip M. Duxbury

Published May 18, 2001• 10 citations
arXivHTMLPDF

Abstract

We study at T=0 the minimum energy of a domain wall and its gap to the first excited state, concentrating on two-dimensional random-bond Ising magnets. The average gap scales as deltaE1 approximately L(straight theta)f(N(z)), where f(y) approximately [ln y](-1/2), straight theta is the energy fluctuation exponent, L is the length scale, and N(z) is the number of energy valleys. The logarithmic scaling is due to extremal statistics, which is illustrated by mapping the problem into the Kardar-Parisi-Zhang roughening process. It follows that the susceptibility of domain walls also has a logarithmic dependence on the system size.

DOI
10.1103/physreve.63.066110

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We study at T=0 the minimum energy of a domain wall and its gap to the first excited state, concentrating on two-dimensional random-bond Ising magnets. The average gap scales as deltaE1 approximately L(straight theta)f(N(z)), where f(y) approximately ln y, straight theta is the energy fluctuation exponent, L is the length scale, and N(z) is the number of energy valleys. The logarithmic scaling is due to extremal statistics, which is illustrated by mapping the problem into the Kardar-Parisi-Zhang roughening process. It follows that the susceptibility of domain walls also has a logarithmic dependence on the system size.
The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds having probability ppp of ${J}_{ij}=\ensuremath{-}1$ and $(1\ensuremath{-}p)$ of Jij=+1{J}_{ij}=+1Jij​=+1 are studied. We are able to consider large samples of up to 3202{320}^{2}3202 spins by using sophisticated matching algorithms. We study $L\ifmmode\times\else\texttimes\fi{}L$ systems, but we also consider $L\ifmmode\times\else\texttimes\fi{}M$ samples, for different aspect ratios R=L∕MR=L∕MR=L∕M. We find that the scaling behavior of the ground-state energy and its sample-to-sample fluctuations inside the spin-glass region $({p}_{c}\ensuremath{\leqslant}p\ensuremath{\leqslant}1\ensuremath{-}{p}_{c})$ are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusplike singularity at pc{p}_{c}pc​. Inside the spin-glass region, the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding RRR fixed. Here, large finite-size effects are visible, which can be explained for all ppp by a single exponent $\ensuremath{\omega}\ensuremath{\approx}2∕3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R\ensuremath{\rightarrow}0$: the distribution of the domain-wall energies converges to a Gaussian for $R\ensuremath{\rightarrow}0$, although the domain walls of neighboring subsystems of size $L\ifmmode\times\else\texttimes\fi{}L$ are not independent.
Randomly placed impurities that alter the local exchange couplings, but do not generate random fields or destroy the long-range order, roughen domain walls in Ising systems for dimensionality 53<d<5\frac{5}{3}<d<535​<d<5. They also pin (localize) the walls in energetically favorable positions. This drastically slows down the kinetics of ordering. The pinned domain wall is a new critical phenomenon governed by a zero-temperature fixed point. For d=2d=2d=2, the critical exponents for domain-wall pinning energies and roughness as a function of length scale are estimated from numerically generated ground states.
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 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 statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for L×LL \times LL×L square lattices with L≤20L \le 20L≤20, and xxx = 0.25 and 0.5, where xxx is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy EdwE_{dw}Edw​ equal to 0 or 4. The probability distribution of the entropy, SdwS_{dw}Sdw​, is found to depend strongly on EdwE_{dw}Edw​. The results for SdwS_{dw}Sdw​ when Edw=4E_{dw} = 4Edw​=4 agree with the prediction of the droplet model. Our results for SdwS_{dw}Sdw​ when Edw=0E_{dw} = 0Edw​=0 agree with those of Saul and Kardar. In addition, we find that the distributions do not appear to be Gaussian in that case. The special role of Edw=0E_{dw} = 0Edw​=0 domain walls is discussed, and the discrepancy between the prediction of Amoruso, Hartmann, Hastings and Moore and the result of Saul and Kardar is explained.
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.