The critical exponents of the two-dimensional Ising spin glass revisited: exact ground-state calculations and Monte Carlo simulations

Rieger, Heiko; Santen, Ludger; Blasum, Ulrich; Diehl, M.; Jünger, Michael; Rinaldi, Giovanni

doi:10.1088/0305-4470/29/14/018

Cited by 109 publications

(166 citation statements)

References 20 publications

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“…The straight line corresponds to the expected asymptotic power-law behavior with the exponent −x given in Eq. (8). To show more explicitly the quality of the collapse, the inset includes only the data points used in the fitting procedure and the polynomial fitting function (black curve).…”

Section: Order Parametermentioning

confidence: 99%

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Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

Xu¹,

Wu²,

Rubin³

et al. 2017

Phys. Rev. E

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We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the two-dimensional Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature T = 0. From a scaling analysis when T → 0 at different annealing velocities v, we find power-law scaling in the system size for the velocity required in order to relax toward the ground state; v ∼ L −(z+1/ν) , the Kibble-Zurek ansatz where z is the dynamic critical exponent and ν the previously known correlation-length exponent, ν ≈ 3.6. We find z ≈ 13.6 for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [S. J. Rubin, N. Xu, and A. W. Sandvik, Phys. Rev. E 95, 052133 (2017)] where the dynamics is faster (z is smaller) and the above two quantities relax with different dynamic exponents (with that of the energy being larger). We argue that the different behaviors arise as a consequence of the different low-energy landscapes-for normal-distributed couplings the ground state is unique (up to a spin reflection) while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here also indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasi-degenerate states, and the scaling function takes a different form.

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Section: Order Parametermentioning

confidence: 99%

“…Here the value of the exponent a ≈ 0.073 in the power law (L z+1/ν ) a is very close to half of the value of the exponent x in Eq. (8).…”

Section: Order Parametermentioning

confidence: 99%

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

Xu¹,

Wu²,

Rubin³

et al. 2017

Phys. Rev. E

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“…Assuming that E a (L) and E b (L) can be determined exactly for each sample, then ∆E(L) is also known exactly for each sample and the errors in ∆E(L) We are forced to conclude that the accessible sizes L are limited by the necessity of finding essentially exact global energy minima of each of a number N of samples subject to certain, yet to be defined, BC. To our knowledge, there is no algorithm applicable to the systems of interest which will find exact minima in polynomial time, such as the branch and cut algorithm 32 for the 2D Ising spin glass or numerically exact combinatorial optimization algorithms 33 for gauge and vortex glass models in the infinite screening limit, so we have to live with the fact that our problem is NP complete and the required CPU time explodes as L increases. We use simulated annealing 34,35 to estimate the lowest energies, which seems considerably more efficient than simple quenching to T = 0, but we are unable to go beyond L = 7 in 3D and L = 10 in 2D.…”

Section: Strategymentioning

confidence: 99%

Domain wall renormalization group study of theXYmodel with quenched random phase shifts

Akino

Kosterlitz

2002

Phys. Rev. B

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The XY model with quenched random disorder is studied by a zero temperature domain wall renormalization group method in 2D and 3D. Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents θs ≈ −0.36 in 2D and θs ≈ +0.31 in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the ±J XY spin glass in 3D, we obtain a spin stiffness exponent θs ≈ +0.10 which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain θs < 0. Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a re-entrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.

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“…Both estimates suggest that θ E (0) is slightly more negative than the accurately measured θ DW = −0.28 ± 0.01. 29,30,31 x(T ) shows a shallow minimum at T ≈ 0.4, unrelated to any ordering temperature.…”

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confidence: 95%

Size dependence of the internal energy in Ising and vector spin glasses

Katzgraber

Campbell

2003

Phys. Rev. B

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We study numerically the scaling correction to the internal energy per spin as a function of system size and temperature in a variety of Ising and vector spin glasses. From a standard scaling analysis we estimate the effective size correction exponent x at each temperature. For each system with a finite ordering temperature, as temperature is increased from zero, x initially decreases regularly until it goes through a minimum at a temperature close to the critical temperature, and then increases strongly. The behavior of the exponent x at and below the critical temperature is more complex than suggested by the model for the size correction that relates x to the domain-wall stiffness exponent.PACS numbers: 75.50. Lk, 05.50.+q, 75.40.Mg Since Gibbs, thermodynamic transitions have been classified according to the critical behavior of the specific heat, or equivalently of the critical temperature dependence of the internal energy. One of the disturbing features of the spin-glass transition has always been that there appears to be no thermodynamic signature whatsoever of the critical temperature, except in the mean-field limit. In finite dimensions, the specific-heat exponent α is strongly negative and the energy changes perfectly smoothly as a function of temperature through the critical temperature T c . 1 The standard scaling form for the finite-size correction to the internal energy per spin e(L) iswhere L represents the system size. An exponent θ E can be defined by θ E = d − x (d represents the space dimension). In Ising spin glasses (ISG) it has been surmised from "droplet" arguments 2,3,4 that this correction is directly related to the energy associated with domain walls, for which independent numerical measurements can also be carried out. Thus it is expected that at zero temperature θ E is identical to θ DW , the domain-wall stiffness exponent. This conjecture is related to the controversial question of the form of the elementary excitations in spin glasses, and the identity should be valid if periodic boundary conditions simply introduce supplementary domain walls. It is exact for Migdal-Kadanoff spin glasses. 4 While the argument was introduced for zero temperature, it has also been invoked for finite T . 3 At a continuous transition the singular part of the free energy divided by the temperature scales as length −d . Because (T − T c ) ∼ length −1/ν (see Refs. 5 and 6), if T c > 0,If T c = 0, θ DW = −1/ν at T = 0 and x(0) = d + 1/ν. 5 We have carried out Monte Carlo measurements of the size dependence of the energy as a function of temperature in the mean-field Sherrington-Kirkpatrick (SK) ISG model, in the Edwards-Anderson ISG with Gaussian interactions in dimensions 2, 3, and 4, in the gauge glass (GG) in dimensions 2, 3, and 4, and in the XY spin glass (XY SG) with Gaussian interactions in dimension 4. In all systems with a nonzero ordering temperature, x(T ) is strongly temperature dependent below as well as above T c . The effective exponent initially decreases progressively as T increases from...

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The critical exponents of the two-dimensional Ising spin glass revisited: exact ground-state calculations and Monte Carlo simulations

Cited by 109 publications

References 20 publications

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

Domain wall renormalization group study of theXYmodel with quenched random phase shifts

Size dependence of the internal energy in Ising and vector spin glasses

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