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arXiv:0901.1538 · New Journal of Physics

Excitations of the bimodal Ising spin glass on the brickwork lattice

W Atisattapong, J Poulter

Published Jun 22, 2009• 3 citations
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Abstract

An exact algorithm is used to investigate the distributions of the degeneracies of low-energy excited states for the bimodal Ising spin glass on the brickwork lattice. Since the distributions are extreme and do not self-average, we base our conclusions on the most likely values of the degeneracies. Our main result is that the degeneracy of the first excited state per ground state and per spin is finite in the thermodynamic limit. This is very different from the same model on a square lattice where a divergence proportional to the linear lattice size is expected. The energy gap for the brickwork lattice is obviously 2 J on finite systems and predicted to be the same in the thermodynamic limit. Our results suggest that a 2 J gap is universal for planar bimodal Ising spin glasses. The distribution of the second contribution to the internal energy has a mode close to zero and we predict that the low-temperature specific heat is dominated by the leading term proportional to T −2 exp (−2 J / kT ).

DOI
10.1088/1367-2630/11/6/063039

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An exact algorithm is used to investigate the distributions of the degeneracies of low-energy excited states for the bimodal Ising spin glass on the brickwork lattice. Since the distributions are extreme and do not self-average, we base our conclusions on the most likely values of the degeneracies. Our main result is that the degeneracy of the first excited state per ground state and per spin is finite in the thermodynamic limit. This is very different from the same model on a square lattice where a divergence proportional to the linear lattice size is expected. The energy gap for the brickwork lattice is obviously 2 J on finite systems and predicted to be the same in the thermodynamic limit. Our results suggest that a 2 J gap is universal for planar bimodal Ising spin glasses. The distribution of the second contribution to the internal energy has a mode close to zero and we predict that the low-temperature specific heat is dominated by the leading term proportional to T −2 exp (−2 J / kT ).
An exact algorithm is used to compute the degeneracies of the excited states of the bimodal Ising spin glass in two dimensions. It is found that the specific heat at arbitrary low temperature is not a self-averaging quantity and has a distribution that is neither normal nor lognormal. Nevertheless, it is possible to estimate the most likely value and this is found to scale as L 3 T -2 exp (−4 J / kT ), for a L × L lattice. Our analysis also explains, for the first time, why a correlation length ξ∼exp (2 J / kT ) is consistent with an energy gap of 2 J . Our method allows us to obtain results for up to 10 5 disorder realizations with L ⩽96. Distributions of second and third excitations are also shown.
An analysis is given of numerical simulation data to size L=128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (±J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent η≡0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature T(*)(L) there is a regime of effectively continuous energy levels; below T(*)(L) there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. T(*)(L) tends to zero at very large L, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable t=T and with a scaling variable τ(b)=T(2)/(1+T(2)) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility χ(τ(b),L) and second moment correlation length ξ(τ(b),L) in the thermodynamic limit regime are extrapolated to the τ(b)=0 critical limit. The Gaussian critical exponent estimates from the simulations, η=0 and ν=3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are η=0.20(2) and ν=4.8(3), distinctly different from the Gaussian critical exponents.
We compute the exact partition function of 2d Ising spin glasses with binary couplings. In these systems, the ground state is highly degenerate and is separated from the first excited state by a gap of size 4J. Nevertheless, we find that the low temperature specific heat density scales as exp(-2J/T), corresponding to an "effective" gap of size 2J; in addition, an associated crossover length scale grows as exp(J/T). We justify these scalings via the degeneracy of the low lying excitations and by the way low energy domain walls proliferate in this model.
A detailed analysis of Monte Carlo data on the two-dimensional Ising spin glass with bimodal interactions shows that the free energy of the model has a nontrivial scaling. In particular, we show by studying the correlation length that much larger system sizes and lower temperatures are required to see the true critical behavior of the model in the thermodynamic limit. Our results agree with data by Lukic et al. in that the degenerate ground state is separated from the first excited state by an energy gap of 2J.
Monte Carlo data of the two-dimensional Ising spin glass with bimodal interactions are presented with the aim of understanding the low-temperature physics of the model. An analysis of the specific heat, spin-glass susceptibility, finite-size correlation length, and the Binder ratio is performed to try to verify a recent proposal in which for large system sizes and finite but low temperatures the effective critical exponents are identical to the critical exponents of the two-dimensional Ising spin glass with Gaussian interactions. Our results show that with present system sizes the recently proposed scenario, in which the two-dimensional Ising spin glass with bimodally distributed interactions is in the same universality class as the model with Gaussian-distributed disorder at low but finite temperatures, cannot be reliably proven.
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.