Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions

Katzgraber, Helmut G.; Young, A. P.

doi:10.1103/physrevb.67.134410

Cited by 104 publications

(238 citation statements)

References 45 publications

(77 reference statements)

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“…Note that T c decreases continuously with increasing σ and is expected to drop to zero at σ = 1. 18 For the SK model (σ = 0), one has T c = 1, essentially the result we find for σ = 0.55, so it is possible that T c has little variation with σ for For all values of σ, the data cross indicating that there is a spin-glass transition at finite temperature. σ ≤ 0.55.…”

Section: Resultssupporting

confidence: 52%

“…As discussed in an earlier work (see Ref. 18 and references therein), for 1/2 < σ ≤ 1, the system has a finite-temperature transition into a spin-glass phase in a long-range (LR) universality class at zero field. For 1 < σ ≤ 2, the system has T c = 0 and the critical behavior is also determined by the LR universality class.…”

Section: Model Observables and Numerical Detailsmentioning

confidence: 88%

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Probing the Almeida-Thouless line away from the mean-field model

2005

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Results of Monte Carlo simulations of the one-dimensional long-range Ising spin glass with powerlaw interactions in the presence of a (random) field are presented. By tuning the exponent of the power-law interactions, we are able to scan the full range of possible behaviors from the infiniterange (Sherrington-Kirkpatrick) model to the short-range model. A finite-size scaling analysis of the correlation length indicates that the Almeida-Thouless line does not occur in the region with non-mean-field critical behavior in zero field. However, there is evidence that an Almeida-Thouless line does occur in the mean-field region.

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Section: Resultssupporting

confidence: 52%

Section: Model Observables and Numerical Detailsmentioning

confidence: 88%

Probing the Almeida-Thouless line away from the mean-field model

2005

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“…Note however that the numerical measures via Monte-Carlo on sizes L ≤ 256 (see Fig. 13 and Table III of [28]) are not a clear support of this theoretical expectation, in particular in the region σ → (1/2) + where the theoretical prediction of Eq. 6 corresponds to θ LR (d = 1, σ → (1/2) + ) → (1/2) − , whereas the numerical results of [28] display a saturation around θ ≃ 0.3.…”

Section: A Gaussian Distributionmentioning

confidence: 85%

“…The interpretation proposed in [28] is that Eq. 7 is nevertheless exact in the whole region 1 2 < σ < 2 as predicted by the theoretical derivations [23,25], and despite their numerical results [28] . Another interpretation could be that the saturation seen in the numerics is meaningful, and that Eq.…”

Section: A Gaussian Distributionmentioning

confidence: 99%

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

Monthus¹

2014

J. Stat. Mech.

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For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J(r) ∼ r −σ and distributed with the Lévy symmetric stable distribution of index 1 < µ ≤ 2 (including the usual Gaussian case µ = 2), we consider the region σ > 1/µ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings JL ∝ L θµ(σ) is found to be θµ(σ) = 2 µ −σ whenever the long-ranged couplings are relevant θµ(σ) ≥ −1. For the statistics of the ground state energy E GS L over disordered samples, we obtain that the droplet exponent θµ(σ) governs the leading correction to extensivity of the averaged value E GS L ≃ Le0 + L θµ (σ) e1. The characteristic scale of the fluctuations around this average is of order L 1 µ , and the rescaledGaussian distributed for µ = 2, or displays the negative power-law tail in 1/(−u) 1+µ for u → −∞ in the Lévy case 1 < µ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.

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“…Our results for low temperatures show that for this model the phase space has an UM signature and exhibits many phase-space components, the number growing with system size in the mean-field as well as non-mean-field case. This suggests that for large enough system sizes SR spin glasses at low enough temperatures might have an UM phase space structure.Model.-The Hamiltonian of the 1D Ising chain with long-range power-law interactions [16,17] is given bywhere S i ∈ {±1} are Ising spins and the sum ranges over all spins in the system. The L spins are placed on a ring and r ij = (L/π) sin(π|i − j|/L) is the distance between the spins.…”

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confidence: 99%

Ultrametricity and Clustering of States in Spin Glasses: A One-Dimensional View

Katzgraber

Hartmann

2009

Phys. Rev. Lett.

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We present results from Monte Carlo simulations to test for ultrametricity and clustering properties in spin-glass models. By using a one-dimensional Ising spin glass with random power-law interactions where the universality class of the model can be tuned by changing the power-law exponent, we find signatures of ultrametric behavior both in the mean-field and non-mean-field universality classes for large linear system sizes. Furthermore, we confirm the existence of nontrivial connected components in phase space via a clustering analysis of configurations. The nature of the spin-glass state is controversial and it is unclear if the mean-field replica symmetry breaking (RSB) picture [2], the droplet picture [7,8], or an intermediate phenomenological scenario dubbed as TNT [9,10] (for "trivial-nontrivial") describes the nature of the spin-glass state best. One way to settle the applicability of the RSB picture to short-range (SR) spin glasses is by testing if the phase space is UM. Unfortunately, the existence of an UM phase structure for SR spin glasses is controversial, mainly because only small linear system sizes have been accessible so far. Recent results [11] suggest that SR systems are not UM, whereas other opinions exist [12,13,14,15]. Thus it is of paramount importance to test if SR spin glasses have an UM phase space.In this work we approach the problem from a different angle: First, we use a one-dimensional (1D) Ising spin-glass with power-law interactions. The model has the advantage that large linear system sizes can be studied. Furthermore, by tuning the exponent of the power law, the universality class of the model can be tuned between a mean-field and a non-mean-field universality class. This allows us to test our analysis method on the mean-field SK model and then apply it to regions of phase space where the system is not mean-field like. We perform a clustering analysis of the data similar to the work of Hed et al.[11] to obtain nontrivial triangles in phase space and introduce a novel correlator which allows us to see an UM signature for low temperatures and delivers we expect SK-like infinite-range behaviour. For 1/2 < σ ≤ 2/3 we have mean-field (MF) behaviour corresponding to an effective space dimension d eff ≥ 6, whereas for 2/3 < σ ≤ 1 we have a long-range (non-MF) spin glass with a ordering temperature Tc > 0. Close to σ = 2/3 (vertical red line)no signal for high temperatures. Furthermore, we use a clustering analysis to search for connected components in phase space. The proposed method can be applied to any field of science to test for an UM structure of phase space, thus making the method generally applicable. Our results for low temperatures show that for this model the phase space has an UM signature and exhibits many phase-space components, the number growing with system size in the mean-field as well as non-mean-field case. This suggests that for large enough system sizes SR spin glasses at low enough temperatures might have an UM phase space structure.Model.-The Hamiltonian of the...

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Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions

Cited by 104 publications

References 45 publications

Probing the Almeida-Thouless line away from the mean-field model

Probing the Almeida-Thouless line away from the mean-field model

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

Ultrametricity and Clustering of States in Spin Glasses: A One-Dimensional View

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