Morphology of ground states of two-dimensional frustration model

Barahona, Francisco; Maynard, R.; Rammal, R.; Uhry, J P

doi:10.1088/0305-4470/15/2/033

Cited by 156 publications

(121 citation statements)

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“…Assuming that, because of frustration, the defect energy is the sum of many correlated terms of different signs, Fisher [4,5,6] that, for each realization of the disorder in two dimensions (2D), there are bonds which are either always satisfied or always frustrated in all the GSs. These bonds define the Rigid Lattice (RL), or backbone of the system.…”

Section: Pacs Numbersmentioning

confidence: 99%

“…It is usually calculated by using the concept of defect energy, ∆E = E a − E p , which is the difference between the ground-state (GS) energies for antiperiodic (E a ) and periodic (E p ) boundary conditions, in one of the directions of a d-dimensional system of linear size L. In ferromagnetic systems, ∆E ∼ L θ , with θ = d s = d − 1, because the induced defect is a (d − 1)-dimensional domain wall with all its bonds frustrated. For spin glasses, the average over the distribution of bonds (denoted by [...]) must be taken and the scaling ansatz becomesAssuming that, because of frustration, the defect energy is the sum of many correlated terms of different signs, Fisher [4,5,6] that, for each realization of the disorder in two dimensions (2D), there are bonds which are either always satisfied or always frustrated in all the GSs. These bonds define the Rigid Lattice (RL), or backbone of the system.…”

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confidence: 99%

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Influence of the ground-state topology on domain-wall energy in the Edwards-Anderson±Jspin glass model

et al. 2007

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We study the phase stability of the Edwards-Anderson spin-glass model by analyzing the domain-wall energy. For the bimodal ±J distribution of bonds, a topological analysis of the ground state allows us to separate the system into two regions: the backbone and its environment. We find that the distributions of domain-wall energies are very different in these two regions for the three dimensional (3D) case. Although the backbone turns out to have a very high phase stability, the combined effect of these excitations and correlations produces the low global stability displayed by the system as a whole. On the other hand, in two dimensions (2D) we find that the surface of the excitations avoids the backbone. Our results confirm that a narrow connection exists between the phase stability of the system and the internal structure of the ground-state. In addition, for both 3D and 2D we are able to obtain the fractal dimension of the domain wall by direct means. PACS numbers:The spin glass state has been studied extensively during the last thirty years, but the role played by low energy excitations is still a matter of debate. These excitations are crucial to understand the nature of the ordering of the spin glass phase. Most studies have focused on the predictions of two theories: the replica-symmetry breaking (RSB) picture [1] and the droplet picture [2]. RSB, rigorously true for the SherringtonKirkpatrick model of spin glasses, predicts that there are excitations which involve flipping a finite fraction of the spins and, in the thermodynamic limit, cost only a finite amount of energy. The fractal dimension of the surface of these excitations, d s , is expected to be equal to the space dimension, d. On the other hand, in the droplet picture the lowest energy excitations of length L have d s < d and typically cost an energy of order L θ (θ is known as stiffness exponent). Thus, contrary to RSB, the droplet picture predicts that excitations involving a finite fraction of spins cost an infinite amount of energy in the thermodynamic limit.The exponent θ plays a central role in this debate. It is usually calculated by using the concept of defect energy, ∆E = E a − E p , which is the difference between the ground-state (GS) energies for antiperiodic (E a ) and periodic (E p ) boundary conditions, in one of the directions of a d-dimensional system of linear size L. In ferromagnetic systems, ∆E ∼ L θ , with θ = d s = d − 1, because the induced defect is a (d − 1)-dimensional domain wall with all its bonds frustrated. For spin glasses, the average over the distribution of bonds (denoted by [...]) must be taken and the scaling ansatz becomesAssuming that, because of frustration, the defect energy is the sum of many correlated terms of different signs, Fisher [4,5,6] that, for each realization of the disorder in two dimensions (2D), there are bonds which are either always satisfied or always frustrated in all the GSs. These bonds define the Rigid Lattice (RL), or backbone of the system. Spins connected by it are called solidary s...

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Section: Pacs Numbersmentioning

confidence: 99%

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confidence: 99%

Influence of the ground-state topology on domain-wall energy in the Edwards-Anderson±Jspin glass model

et al. 2007

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“…In the EAB model the fundamental level is degenerate and the spatial heterogeneities are well characterized by the so-called rigid lattice (RL). 8 This structure is composed by the set of bonds which do not change its condition (satisfied or frustrated) in all the configurations of the GS. These bonds are called rigid bonds.…”

Section: Introductionmentioning

confidence: 99%

Ground-state topology of the Edwards-Anderson±Jspin glass model

et al. 2010

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In the Edwards-Anderson model of spin glasses with a bimodal distribution of bonds, the degeneracy of the ground state allows one to define a structure called backbone, which can be characterized by the rigid lattice (RL), consisting of the bonds that retain their frustration (or lack of it) in all ground states. In this work we have performed a detailed numerical study of the properties of the RL, both in two-dimensional (2D) and three-dimensional (3D) lattices. Whereas in 3D we find strong evidence for percolation in the thermodynamic limit, in 2D our results indicate that the most probable scenario is that the RL does not percolate. On the other hand, both in 2D and 3D we find that frustration is very unevenly distributed. Frustration is much lower in the RL than in its complement. Using equilibrium simulations we observe that this property can be found even above the critical temperature. This leads us to propose that the RL should share many properties of ferromagnetic models, an idea that recently has also been proposed in other contexts. We also suggest a preliminary generalization of the definition of backbone for systems with continuous distributions of bonds, and we argue that the study of this structure could be useful for a better understanding of the low temperature phase of those frustrated models.

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“…-Recent studies [1][2][3] of spin-glass models suggest the possibility of some connections between the computational complexity of the problem of finding a ground state and the existence of a finite 7~ phase transition in a given model (e.g. ± J Ising models).…”

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confidence: 99%

“…find the ground states) is NP-complete in dimensions larger than 2. Though it is polynomial [1,2] in 2D, it becomes NP-complete for two coupled planes [3] (i.e. at d = 2 + e).…”

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confidence: 99%