Energy gap of the bimodal two-dimensional Ising spin glass

Poulter, J.

doi:10.1088/1367-2630/10/9/093012

Cited by 7 publications

(9 citation statements)

References 23 publications

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“…[23] is inconsistent with the data in Fig. 2 for all L and T (see also [24,25]) 25 as proposed in Ref. [15] or C v (β, L) ∼ T 4.2 as proposed in Ref.…”

Section: Specific Heatcontrasting

confidence: 55%

Bimodal and Gaussian Ising spin glasses in dimension two

Lundow

Campbell

2016

Phys. Rev. E

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An analysis is given of numerical simulation data to size L=128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (±J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent η≡0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature T(*)(L) there is a regime of effectively continuous energy levels; below T(*)(L) there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. T(*)(L) tends to zero at very large L, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable t=T and with a scaling variable τ(b)=T(2)/(1+T(2)) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility χ(τ(b),L) and second moment correlation length ξ(τ(b),L) in the thermodynamic limit regime are extrapolated to the τ(b)=0 critical limit. The Gaussian critical exponent estimates from the simulations, η=0 and ν=3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are η=0.20(2) and ν=4.8(3), distinctly different from the Gaussian critical exponents.

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“…[23] is inconsistent with the data in Fig. 2 for all L and T (see also [24,25]) 25 as proposed in Ref. [15] or C v (β, L) ∼ T 4.2 as proposed in Ref.…”

Section: Specific Heatcontrasting

confidence: 55%

Bimodal and Gaussian Ising spin glasses in dimension two

Lundow

Campbell

2016

Phys. Rev. E

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“…As a final note, we expect a correlation length ξ ∼ exp(2J/kT ) in probable agreement 2,7,8,23,24 with the square lattice. Our reasoning is based on the construction of correlation functions using reciprocal defects 17,18,20 and closed polygons.…”

Section: Discussionsupporting

confidence: 59%

“…To continue for higher orders, we require the Green's functions G r , as given in Ref. 7, obtained from previous orders; that is for states whose degeneracy has already been lifted. The general rule for D r (at rth order) can be expressed as…”

Section: Formalismmentioning

confidence: 99%

Excitations of the bimodal Ising spin glass on the brickwork lattice

Poulter

2009

New J. Phys.

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An exact algorithm is used to investigate the distributions of the degeneracies of low-energy excited states for the bimodal Ising spin glass on the brickwork lattice. Since the distributions are extreme and do not self-average, we base our conclusions on the most likely values of the degeneracies. Our main result is that the degeneracy of the first excited state per ground state and per spin is finite in the thermodynamic limit. This is very different from the same model on a square lattice where a divergence proportional to the linear lattice size is expected. The energy gap for the brickwork lattice is obviously 2J on finite systems and predicted to be the same in the thermodynamic limit. Our results suggest that a 2J gap is universal for planar bimodal Ising spin glasses. The distribution of the second contribution to the internal energy has a mode close to zero and we predict that the low-temperature specific heat is dominated by the leading term proportional to T −2 exp(−2J/kT ).

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“…This is absolutely non-trivial because the energy of any finite-size excitation for the square lattice is a multiple of 4, and this would lead instead to A = 4 as later claimed in [38]. Over the years the A = 2 result has been supported by many authors [27,[39][40][41]; recently another scenario has also been proposed in which c V behaves as a power law [10] with a universal exponent that is the same of models with Gaussian distributions of the couplings. The true nature of c V remains nevertheless unclear [42].…”

Section: Specific Heat At T =mentioning

confidence: 99%

Replica Cluster Variational Method

et al. 2010

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We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi's Cluster Variational Method (CVM). Using replicas and the message-passing formulation of CVM we obtain a variational expression of the replicated free energy of a system with quenched disorder, both averaged and on a single sample, and make the hierarchical ansatz using functionals of functions of fields to represent the messages. We obtain a set of integral equations for the message functionals. The main difference with the Bethe case is that the functionals appear in the equations in implicit form and are not positive definite, thus standard iterative population dynamic algorithms cannot be used to determine them. In the simplest cases the solution could be obtained iteratively using Fourier transforms. We begin to study the method considering the plaquette approximation to the averaged free energy of the Edwards-Anderson model in the paramagnetic Replica-Symmetric phase. In two dimensions we find that the spurious spin-glass phase transition of the Bethe approximation disappears and the paramagnetic phase is stable down to zero temperature on the square lattice for different random interactions. The quantitative estimates of the free energy and of various other quantities improve those of the Bethe approximation. The plaquette approximation fails to predict a second-order spin-glass phase transition on the cubic 3D lattice but yields good results in dimension four and higher. We provide the physical interpretation of the beliefs in the replica-symmetric phase as disorder distributions of the local Hamiltonian. The messages instead do not admit such an interpretation and indeed they cannot be represented as populations in the spin-glass phase at variance with the Bethe approximation. The approach can be used in principle to study the phase diagram of a wide range of disordered systems and it is also possible that it can be used to get quantitative predictions on single samples. These further developments present however great technical challenges

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Energy gap of the bimodal two-dimensional Ising spin glass

Cited by 7 publications

References 23 publications

Bimodal and Gaussian Ising spin glasses in dimension two

Bimodal and Gaussian Ising spin glasses in dimension two

Excitations of the bimodal Ising spin glass on the brickwork lattice

Replica Cluster Variational Method

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