Domain wall entropy of the bimodal two-dimensional Ising spin glass

Aromsawa, A.; Poulter, J.

doi:10.1103/physrevb.76.064427

Cited by 6 publications

(7 citation statements)

References 42 publications

(87 reference statements)

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“…7 and is compatible with both the value obtained for the case of fully periodic BCs and with the most recent estimate that was calculated by using the entropy ansatz, d f = 1.30͑3͒. 11 Even though a value of d f = 1.30͑1͒ was previously reported, 12 the small discrepancy with the value reported in this paper is due to the larger system sizes that are considered here ͑up to L = 100 in Ref. 12͒.…”

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confidence: 90%

Fractal dimension of domain walls in the Edwards-Anderson spin glass model

Risau-Gusmán

Romá

2008

Phys. Rev. B

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We directly study the length of the domain walls ͑DWs͒ obtained by comparing the ground states of the Edwards-Anderson spin glass model subject to periodic and antiperiodic boundary conditions. For the bimodal and the Gaussian bond distributions, we have isolated the DW and have directly calculated its fractal dimension d f . Our results show that, even though in three dimensions d f is the same for both distributions of bonds, this is clearly not the case for two-dimensional ͑2D͒ systems. In addition, contrary to what happens in the case of the 2D Edwards-Anderson spin glass with the Gaussian distribution of bonds, we find no evidence that the DW for the bimodal distribution of bonds can be described as Schramm-Loewner evolution processes.Whereas the properties of the long range Ising spin glasses are now well understood, after more than 20 years of work, the same cannot be said about short range spin glasses. This is true even for very simple models: the nature of the ordering of the low temperature phase of the twodimensional ͑2D͒ Edwards-Anderson ͑EA͒ spin glass model 1 is still being debated. Even though the fact that at T =0 EA models with Gaussian ͑EAG͒ and bimodal ͑EAB͒ bond distributions belong to two different universality classes seems well established, 2 new studies 3 of low energy excitations ͑fractal droplets͒ show that there is still room for discussion.It was recently suggested 4,5 that domain walls ͑DWs͒ can be described as Schramm-Loewner evolution ͑SLE͒ processes. These processes are the Brownian walks of diffusion constant and fractal dimension d f =1+ / 8. Furthermore, by using a conformal field theory, the stiffness exponent ͑which characterizes the scaling of the DW energy͒ can be related to the fractal dimension d f viaThis seems to be true for the 2D EAG, as for = −0.287͑4͒ ͑Ref. 6͒ Eq. ͑1͒ gives d f SLE = 1.2764͑4͒, which is compatible with the best numerical estimate d f = 1.274͑2͒. 7 It is not clear, however, whether such a relation should hold for the 2D EAB because of the high degeneracy of its ground state ͑GS͒. If it did, using the fact that the stiffness exponent seems to vanish, 8 Eq. ͑1͒ would yield d f SLE = 1.25. In the EAB model, the degeneracy of the GS precludes a clear-cut definition of the fractal dimension of the DW. For this reason, most of the estimates of d f are based on a scaling argument of Fisher and Huse, 9 which states that the entropy of droplets of size L should scale as S DW ϳ L d f /2 . It must be stressed that this was originally proposed for systems with only one GS. The estimates that were obtained by using this scaling range from d f Ϸ 1.0 to d f = 1.30͑3͒. 10,11 Even though very recently more direct measurements were attempted, 12,13 in those works, the sampling of the DWs was not controlled. 7 In Ref. 7, this problem is avoided and bounds are provided for the true d f : 1.095͑2͒ Ͻ d f Ͻ 1.395͑3͒.In this paper, we present the results of an extensive numerical study of the fractal dimension of DWs by using a direct measure of their length. We have studie...

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confidence: 90%

Fractal dimension of domain walls in the Edwards-Anderson spin glass model

Risau-Gusmán

Romá

2008

Phys. Rev. B

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“…Our data from L = 32 to L = 256 is very well fit by a power law, with θ S = 0.50 ± 0.01, (in contrast with Ref. 23). It is intriguing that this exponent is consistent with the simple rational number 1/2.…”

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confidence: 45%

Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

2011

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Recently extended precise numerical methods and droplet scaling arguments allow for a coherent picture of the glassy states of two-dimensional Ising spin glasses to be assembled. The length scale at which entropy becomes important and produces "chaos", the extreme sensitivity of the state to temperature, is found to depend on the type of randomness. For the ±J model this length scale dominates the low-temperature specific heat. Although there is a type of universality, some critical exponents do depend on the distribution of disorder.PACS numbers: 75.10. Nr, Glassy systems, characterized by extremely slow relaxation and resultant complex hysteresis and memory effects, are difficult to study because their dynamics encompass a great range of time scales [1]. Glassy materials include those without intrinsic disorder, such as silica glass, and those where quenched disorder influences the active degrees of freedom. An example of a model of the latter is the Edwards-Anderson spin glass model [2], which includes the disorder and frustration necessary to capture many of the complex behaviors seen in disordered magnetic materials. Though this prototypical model of glassy behavior was originally proposed well over 30 years ago, many aspects of it remain poorly understood. The droplet and replica-symmetry-breaking pictures of spin glasses provide distinct views of spin glass behavior [3]. Analytical results are rare, so numerical approaches are invaluable for both testing and motivating new ideas. But the numerics are also exceedingly difficult: computing spin glass ground states is in general a NP-hard problem [4]. It is believed that these classes of problems require exponential computational time to solve exactly [5].A fortunate special case which is not prone to this computational intractability is the two-dimensional Ising spin glass (2DISG). The Hamiltonian is H = ij J ij s i s j , where the couplings J = {J ij } are independent random variables coupling classical spins s = ±1 at sites i,j on a square toroidal grid with L 2 sites. The randomness of the sign of J ij leads to competing interactions, not all of which can be satisfied. In general, such models have complex (free) energy landscapes and very slow dynamics. The most commonly-used distributions for the J ij are the ±J distribution where each bond value is ±1 with equal probability, or the Gaussian distribution where the J ij are chosen from a univariate Gaussian distribution with zero mean. One apparent difficulty in using the 2DISG as a model system is that truly long-range spinglass order only occurs at zero temperature. However, at low enough temperature such that the correlation length ξ exceeds L one can study a regime of glassy behavior.In this glassy regime, the dominant spin configurations are very sensitive at large length scales to small changes in temperature or other global perturbations: this sensitivity is referred to as "chaos".Highly developed numerical algorithms [4,[6][7][8][9] can efficiently circumvent the complexity due to disorder and fr...

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“…If θ S = 0.5 as reported 32,36,37 then α = −3.0. However, other work [38][39][40][41] predicts values d f > 1 that imply α > −3.0. It seems unlikely that droplet theory can predict a value in agreement with α = −4.21 or α = −7.1.…”

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confidence: 92%

Elementary excitations and the phase transition in the bimodal Ising spin glass model

Jinuntuya¹,

Poulter²

2012

J. Stat. Mech.

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We show how the nature of the the phase transition in the two-dimensional bimodal Ising spin glass model can be understood in terms of elementary excitations. Although the energy gap with the ground state is expected to be 4J in the ferromagnetic phase, a gap 2J is in fact found if the finite lattice is wound around a cylinder of odd circumference L. This 2J gap is really a finite size effect that should not occur in the thermodynamic limit of the ferromagnet. The spatial influence of the frustration must be limited and not wrap around the system if L is large enough. In essence, the absence of 2J excitations defines the ferromagnetic phase without recourse to calculating magnetisation or investigating the system response to domain wall defects. This study directly investigates the response to temperature. We also estimate the defect concentration where the phase transition to the spin glass glass state occurs. The value pc = 0.1045(11) is in reasonable agreement with the literature.

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Domain wall entropy of the bimodal two-dimensional Ising spin glass

Cited by 6 publications

References 42 publications

Fractal dimension of domain walls in the Edwards-Anderson spin glass model

Fractal dimension of domain walls in the Edwards-Anderson spin glass model

Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

Elementary excitations and the phase transition in the bimodal Ising spin glass model

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