arXiv:cond-mat/0402627 · Physical Review B

Energy size effects of two-dimensional Ising spin glasses

Ian A. Campbell, Alexander K. Hartmann, Helmut G. Katzgraber

Published Aug 31, 2004• 35 citations

Abstract

We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.

DOI: 10.1103/physrevb.70.054429

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arXiv:cond-mat/0402627 · Physical Review B

Energy size effects of two-dimensional Ising spin glasses

Ian A. Campbell, Alexander K. Hartmann, Helmut G. Katzgraber

Published Aug 31, 2004• 35 citations

arXiv HTML PDF

Abstract

\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282

describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.

DOI: 10.1103/physrevb.70.054429

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Stiffness exponent of two-dimensional Ising spin glasses for nonperiodic boundary conditions using aspect-ratio scaling

Alexander K. Hartmann, Alan J. Bray, Adam C. Carter +2 more

We study the scaling behavior of domain-wall energies in two-dimensional Ising spin glasses with Gaussian and bimodal distributions of the interactions and different types of boundary conditions. The domain walls are generated by changing the boundary conditions at $T=0$ in one direction. The ground states of the original and perturbed system are calculated numerically by applying an efficient matching algorithm. Systems of size $L\ifmmode\times\else\texttimes\fi{}M$ with different aspect-ratios $1/8<~L/M<~64$ are considered. For Gaussian interactions, using the aspect-ratio scaling approach, we find a stiffness exponent $\ensuremath{\theta}=\ensuremath{-}0.287(4),$ which is independent of the boundary conditions in contrast to earlier results. Furthermore, we find a scaling behavior of the domain-wall energy as predicted by the aspect-ratio approach. Finally, we show that this approach does not work for the bimodal distribution of interactions.We study the scaling behavior of domain-wall energies in two-dimensional Ising spin glasses with Gaussian and bimodal distributions of the interactions and different types of boundary conditions. The domain walls are generated by changing the boundary conditions at $T=0$ in one direction. The ground states of the original and perturbed system are calculated numerically by applying an efficient matching algorithm. Systems of size $L\ifmmode\times\else\texttimes\fi{}M$ with different aspect-ratios $1/8<~L/M<~64$ are considered. For Gaussian interactions, using the aspect-ratio scaling approach, we find a stiffness exponent $\ensuremath{\theta}=\ensuremath{-}0.287(4),$ which is independent of the boundary conditions in contrast to earlier results. Furthermore, we find a scaling behavior of the domain-wall energy as predicted by the aspect-ratio approach. Finally, we show that this approach does not work for the bimodal distribution of interactions.

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Lower critical dimension of Ising spin glasses

Alexander K. Hartmann, A. P. Young

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Finite-size scaling of the Domain Wall Entropy for the 2D \pm J Ising Spin Glass

Ronald Fisch

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for $L \times L$ square lattices with $L \le 20$ , and $x$ = 0.25 and 0.5, where $x$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy $E_{dw}$ equal to 0 or 4. The probability distribution of the entropy, $S_{dw}$ , is found to depend strongly on $E_{dw}$ . The results for $S_{dw}$ when $E_{dw} = 4$ agree with the prediction of the droplet model. Our results for $S_{dw}$ when $E_{dw} = 0$ 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 $E_{dw} = 0$ 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.The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for $L \times L$ square lattices with $L \le 20$, and $x$ = 0.25 and 0.5, where $x$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy $E_{dw}$ equal to 0 or 4. The probability distribution of the entropy, $S_{dw}$, is found to depend strongly on $E_{dw}$. The results for $S_{dw}$ when $E_{dw} = 4$ agree with the prediction of the droplet model. Our results for $S_{dw}$ when $E_{dw} = 0$ 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 $E_{dw} = 0$ 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.

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We investigate the nature of the low-energy, large-scale excitations in the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings and free boundary conditions, by studying the response of the ground state to a coupling-dependent perturbation introduced previously. The ground states are determined exactly for system sizes up to ${12}^{3}$ spins using a branch-and-cut algorithm. The data are consistent with a picture where the surface of the excitations is not space filling, such as the droplet or the trivial-non-trivial (TNT) picture, with only minimal corrections to scaling. When allowing for very large corrections to scaling, the data are also consistent with a picture with space-filling surfaces, such as replica symmetry breaking. The energy of the excitations scales with their size with a small exponent ${\ensuremath{\theta}}^{\ensuremath{'}},$ which is compatible with zero if we allow moderate corrections to scaling. We compare the results with data for periodic boundary conditions obtained with a genetic algorithm, and discuss the effects of different boundary conditions on corrections to scaling. Finally, we analyze the performance of our branch-and-cut algorithm, finding that it is correlated with the existence of large-scale, low-energy excitations.We investigate the nature of the low-energy, large-scale excitations in the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings and free boundary conditions, by studying the response of the ground state to a coupling-dependent perturbation introduced previously. The ground states are determined exactly for system sizes up to ${12}^{3}$ spins using a branch-and-cut algorithm. The data are consistent with a picture where the surface of the excitations is not space filling, such as the droplet or the trivial-non-trivial (TNT) picture, with only minimal corrections to scaling. When allowing for very large corrections to scaling, the data are also consistent with a picture with space-filling surfaces, such as replica symmetry breaking. The energy of the excitations scales with their size with a small exponent ${\ensuremath{\theta}}^{\ensuremath{'}},$ which is compatible with zero if we allow moderate corrections to scaling. We compare the results with data for periodic boundary conditions obtained with a genetic algorithm, and discuss the effects of different boundary conditions on corrections to scaling. Finally, we analyze the performance of our branch-and-cut algorithm, finding that it is correlated with the existence of large-scale, low-energy excitations.

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\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282

T=0

in one direction. The ground states of the original and perturbed system are calculated numerically by applying an efficient matching algorithm. Systems of size $L\ifmmode\times\else\texttimes\fi{}M$ with different aspect-ratios

1/8<~L/M<~64

are considered. For Gaussian interactions, using the aspect-ratio scaling approach, we find a stiffness exponent $\ensuremath{\theta}=\ensuremath{-}0.287(4),$ which is independent of the boundary conditions in contrast to earlier results. Furthermore, we find a scaling behavior of the domain-wall energy as predicted by the aspect-ratio approach. Finally, we show that this approach does not work for the bimodal distribution of interactions.

{N=480}^{2}

spins to be investigated. We study domain walls induced by two rather different types of boundary condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy,'' which we denote by $\ensuremath{\Delta}E.$ For Gaussian disorder, we find a power-law behavior $\ensuremath{\Delta}E\ensuremath{\sim}{L}^{\ensuremath{\theta}},$ with $\ensuremath{\theta}=\ensuremath{-}0.266(2)$ and $\ensuremath{\theta}=\ensuremath{-}0.282(2)$ for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension

{d}_{c}.

For the $\ifmmode\pm\else\textpm\fi{}J$ model, we obtain a different result, namely that the domain-wall energy saturates at a nonzero value for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ so $\ensuremath{\theta}=0,$ indicating that the lower critical dimension for the $\ifmmode\pm\else\textpm\fi{}J$ model is exactly

{d}_{c}=2.

{d}_{s},

and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.

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

p

of ${J}_{ij}=\ensuremath{-}1$ and $(1\ensuremath{-}p)$ of

{J}_{ij}=+1

are studied. We are able to consider large samples of up to

{320}^{2}

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∕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

{p}_{c}

. Inside the spin-glass region, the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding

R

fixed. Here, large finite-size effects are visible, which can be explained for all

p

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.

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for

L \times L

square lattices with

L \le 20

, and

x

= 0.25 and 0.5, where

x

is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy

E_{dw}

equal to 0 or 4. The probability distribution of the entropy,

S_{dw}

, is found to depend strongly on

E_{dw}

. The results for

S_{dw}

when

E_{dw} = 4

agree with the prediction of the droplet model. Our results for

S_{dw}

when

E_{dw} = 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

E_{dw} = 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.

{12}^{3}

spins using a branch-and-cut algorithm. The data are consistent with a picture where the surface of the excitations is not space filling, such as the droplet or the trivial-non-trivial (TNT) picture, with only minimal corrections to scaling. When allowing for very large corrections to scaling, the data are also consistent with a picture with space-filling surfaces, such as replica symmetry breaking. The energy of the excitations scales with their size with a small exponent ${\ensuremath{\theta}}^{\ensuremath{'}},$ which is compatible with zero if we allow moderate corrections to scaling. We compare the results with data for periodic boundary conditions obtained with a genetic algorithm, and discuss the effects of different boundary conditions on corrections to scaling. Finally, we analyze the performance of our branch-and-cut algorithm, finding that it is correlated with the existence of large-scale, low-energy excitations.