Frustration effects in antiferromagnets on planar random graphs

Weigel, Martin; Johnston, D.A.

doi:10.1103/physrevb.76.054408

Cited by 18 publications

(19 citation statements)

References 54 publications

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“…Also, we reach much larger system sizes than previous work ͑also a bit larger than the recent study in Ref. 18, which is anyway for a different lattice type͒, which also reduces systematic errors due to unknown corrections to scaling. For the ±J model it was found that the MEDW energy saturates at a nonzero value for large system sizes, 7,23 which means = 0.…”

Section: Introductionmentioning

confidence: 77%

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Fractal dimension of domain walls in two-dimensional Ising spin glasses

Melchert

Hartmann

2007

Phys. Rev. B

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We study domain walls in 2d Ising spin glasses in terms of a minimum-weight path problem. Using this approach, large systems can be treated exactly. Our focus is on the fractal dimension $d_f$ of domain walls, which describes via $<\ell >\simL^{d_f}$ the growth of the average domain-wall length with %% systems size $L\times L$. %% 20.07.07 OM %% Exploring systems up to L=320 we yield $d_f=1.274(2)$ for the case of Gaussian disorder, i.e. a much higher accuracy compared to previous studies. For the case of bimodal disorder, where many equivalent domain walls exist due to the degeneracy of this model, we obtain a true lower bound $d_f=1.095(2)$ and a (lower) estimate $d_f=1.395(3)$ as upper bound. Furthermore, we study the distributions of the domain-wall lengths. Their scaling with system size can be described also only by the exponent $d_f$, i.e. the distributions are monofractal. Finally, we investigate the growth of the domain-wall width with system size (``roughness'') and find a linear behavior.Comment: 8 pages, 8 figures, submitted to Phys. Rev. B; v2: shortened versio

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Section: Introductionmentioning

confidence: 77%

“…24͒ and d f = 1.283͑11͒. 18 Nevertheless, in these works there is still no systematics concerning the selection of a representative 27,28 suggested that the behavior of zero-energy DWs is not consistent with the droplet scaling picture. In this context it was proposed to treat MEDWs of zero and non zero energies as distinct classes.…”

Section: Introductionmentioning

confidence: 97%

Fractal dimension of domain walls in two-dimensional Ising spin glasses

Melchert

Hartmann

2007

Phys. Rev. B

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“…Such consistence together with further assumptions would suggest a relation between stiffness exponent and fractal dimension, d f = 1 + 3/[4(3 + θ)]. 38 For the bimodal model, on the other hand, the fractal dimension is possibly different [41][42][43] , but calculations are complicated by sampling problems since the ground-state algorithms do not produce the degenerate ground states with the correct weights. These subtle differences between results for different coupling distributions and excitation types call for high-precision studies to distinguish random from systematic coincidences.…”

Section: Introductionmentioning

confidence: 99%

Domain-wall excitations in the two-dimensional Ising spin glass

Khoshbakht

Weigel

2018

Phys. Rev. B

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The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to 10 000 × 10 000 spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being θ = −0.2793(3) and d f = 1.273 19(9) for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding d f = 1.279(2). Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.

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“…In the meantime, there have been many studies of Boolean variables on scale-free networks [2][3][4][5] and, more recently, even with competing interactions [6][7][8][9][10]. There is general consensus that stable ferromagnetic and spin-glass phases emerge in these complex systems [10] and that for particular choices of the decay exponent λ the critical temperature diverges, i.e., Boolean variables with competing interactions are extremely robust to local perturbations.…”

Section: Introductionmentioning

confidence: 99%

Boolean decision problems with competing interactions on scale-free networks: Equilibrium and nonequilibrium behavior in an external bias

et al. 2014

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We study the equilibrium and nonequilibrium properties of Boolean decision problems with competing interactions on scale-free networks in an external bias (magnetic field). Previous studies at zero field have shown a remarkable equilibrium stability of Boolean variables (Ising spins) with competing interactions (spin glasses) on scale-free networks. When the exponent that describes the power-law decay of the connectivity of the network is strictly larger than 3, the system undergoes a spin-glass transition. However, when the exponent is equal to or less than 3, the glass phase is stable for all temperatures. First, we perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and show that the system has a spin-glass phase in a field, i.e., exhibits a de Almeida-Thouless line. Furthermore, we study avalanche distributions when the system is driven by a field at zero temperature to test if the system displays self-organized criticality. Numerical results suggest that avalanches (damage) can spread across the whole system with nonzero probability when the decay exponent of the interaction degree is less than or equal to 2, i.e., that Boolean decision problems on scale-free networks with competing interactions can be fragile when not in thermal equilibrium.

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Frustration effects in antiferromagnets on planar random graphs

Cited by 18 publications

References 54 publications

Fractal dimension of domain walls in two-dimensional Ising spin glasses

Fractal dimension of domain walls in two-dimensional Ising spin glasses

Domain-wall excitations in the two-dimensional Ising spin glass

Boolean decision problems with competing interactions on scale-free networks: Equilibrium and nonequilibrium behavior in an external bias

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