Energy size effects of two-dimensional Ising spin glasses

Campbell, I. A.; Hartmann, Alexander K.; Katzgraber, Helmut G.

doi:10.1103/physrevb.70.054429

Cited by 34 publications

(78 citation statements)

References 39 publications

(65 reference statements)

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“…For simplicity of notation, we use ... to denote the combined MC expectation value and the average over disorder samples. It is known that the 2DISG with Gaussian couplings has a phase transition exactly at T = 0, and its critical behavior has been studied extensively [5][6][7]. Unlike the 2DISG with J = ±1 couplings, where there are many degenerate ground states, there is only a unique ground state (and the state with all spins reversed).…”

Section: A Equilibrium Finite-size Scalingmentioning

confidence: 99%

“…Here the energy per spin for infinite d = 2 dimensional system is E ∞ = −1.314788(4) [30] and the most precise value available for the critical exponent ν of the correlation length ξ (where in the case T c = 0 we have ξ ∼ T −ν ) is ν = 3.56(2) [6]. The prefactor a of the scaling in L was claimed to be exactly a = 1.…”

Section: A Equilibrium Finite-size Scalingmentioning

confidence: 99%

“…Thus, as T → 0 all the independent replicas will eventually fall into the same ground state configuration in the limit of a very slow SA process, and the EA order parameter q 2 must approach 1 without any finite-size corrections in the T = 0 value. However, according to the study in [6], the equilibrium ground-state energy density has a finite-size correction of the form…”

Section: A Equilibrium Finite-size Scalingmentioning

confidence: 99%

“…The system has long-range spinglass order (defined with the Edwards-Anderson, EA, order parameter) at T = 0, and the correlation length diverges as a power law, ξ ∼ T −ν when T → 0. Many works have been devoted to the nature of the critical behavior and to obtaining the critical exponents of 2DISG system with both normal-distributed (Gaussian) and bimodal couplings [5][6][7][8]. However, due to the considerable challenges with MC simulations, especially for large systems at low temperature, there are still significant issues under debate.…”

Section: Introductionmentioning

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: A Equilibrium Finite-size Scalingmentioning

confidence: 99%

Section: A Equilibrium Finite-size Scalingmentioning

confidence: 99%

Section: A Equilibrium Finite-size Scalingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…In recent years it has become possible to compute the free energy of the two-dimensional (2D) Ising spin glass with ±J bonds on L × L lattices with L of 100 or more. 2,3,4,5,6,7 From these calculations on large lattices we have learned that extrapolations of data from lattices with L < 30 are often misleading. 8,9,10,11,12 A better understanding of why this happens is clearly desirable.…”

Section: Introductionmentioning

confidence: 99%

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

Fisch¹

2006

J Stat Phys

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For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and -1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x = 0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L 1.78(2) . Anomalous scaling is not found for the characteristic energy, which essentially scales as L 2 . When x = 0.25, a crossover to L 2 scaling of the entropy is seen near L = 12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L.

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Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass

Fisch

2006

J Stat Phys

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The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When L is even, almost all domain walls have energy E dw = 0 or 4. When L is odd, most domain walls have E dw = 2. The probability distribution of the entropy, S dw , is found to depend strongly on E dw . When E dw = 0, the probability distribution of |S dw | is approximately exponential. The variance of this distribution is proportional to L, in agreement with the results of Saul and Kardar. For E dw = k > 0 the distribution of S dw is not symmetric about zero. In these cases the variance still appears to be linear in L, but the average of S dw grows faster than √ L. This suggests a one-parameter scaling form for the L-dependence of the distributions of S dw for k > 0.

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Energy size effects of two-dimensional Ising spin glasses

Cited by 34 publications

References 39 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

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass

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