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arXiv:cond-mat/0307583 · Physical review. B, Condensed matter

Geometry of large-scale low-energy excitations in the one-dimensional Ising spin glass with power-law interactions

Helmut G. Katzgraber, A. P. Young

Published Dec 8, 2003• 32 citations
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Abstract

Results are presented for the geometry of low-energy excitations in the one-dimensional Ising spin chain with power-law interactions, in which the model parameters are chosen to yield a finite spin-glass transition temperature. Both finite-temperature and ground-state studies are carried out. For the range of sizes studied the data cannot be fitted to any of the standard spin-glass scenarios without including corrections to scaling. Incorporating such corrections we find that the fractal dimension of the surface of the excitations, is either equal to the space dimension, consistent with replica symmetry breaking, or very slightly less than it. The latter case is consistent with the droplet and ``trivial-nontrivial'' pictures.

DOI
10.1103/physrevb.68.224408

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Results are presented for the geometry of low-energy excitations in the one-dimensional Ising spin chain with power-law interactions, in which the model parameters are chosen to yield a finite spin-glass transition temperature. Both finite-temperature and ground-state studies are carried out. For the range of sizes studied the data cannot be fitted to any of the standard spin-glass scenarios without including corrections to scaling. Incorporating such corrections we find that the fractal dimension of the surface of the excitations, is either equal to the space dimension, consistent with replica symmetry breaking, or very slightly less than it. The latter case is consistent with the droplet and ``trivial-nontrivial'' pictures.
The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.
We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds,{d}_{s},ds​, 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.
We report the results of Monte Carlo simulations on several spin glass models at low temperatures. By using the parallel tempering (Exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short range models are consistent with a picture proposed earlier that there are large scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite range Viana-Bray model, our results obtained for a similar number of spins are consistent with standard replica symmetry breaking.
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 123{12}^{3}123 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.
In order to overcome the limitations of small system sizes in spin-glass simulations, we investigate the one-dimensional Ising spin chain with power-law interactions.The model has the advantage over traditional higher-dimensional Hamiltonians in that a large range of system sizes can be studied.In addition, the universality class of the model can be changed by tuning the power law exponent, thus allowing us to scan from the mean-field to long-range and short-range universality classes.We illustrate the advantages of this model by studying the nature of the spin glass state where our results hint towards a replica symmetry breaking scenario.We also compute ground-state energy distributions and show that mean-field and non-mean-field models are intrinsically different. * )Recently Krzakala and Martin, as well as Palassini and Young, suggest an intermediate picture in which droplets have a fractal surface, and their energy is finite in the thermodynamic limit.See Ref. 5) and references therein for details.* * ) Note that we explicitly distinguish between θ , the stiffness exponent for droplet excitations, and θ, the exponent for domain walls, because DP predicts θ = θ , which we want to test here.
We study the low-energy excitations of the three dimensional spin glass through a large-scale Monte Carlo simulation on lattices up to L=18L=18L=18. We find smooth extrapolations down to zero temperature, which, in the case of the energy and of the link overlap, can be directly -- and favourably -- compared with previous investigations featuring ground states (i.e., at zero temperature). The best fit for the fractal dimension of the excitations is provided by Replica-Symmetry Breaking theory, but we also consider the alternative TNT description. The P(q)P(q)P(q) is found to verify the Parisi-Toulouse temperature scaling. Our data provides a spectacular confirmation of the overlap-equivalence hypothesis.