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Journal of Physics A Mathematical and General

The critical temperature of the two-dimensional JIsing spin glass

R Sungthong, J. Poulter

Published Jun 5, 2003• 7 citations
Publisher

Abstract

We have studied the two-dimensional ± J Ising model on the square lattice with a view to determine whether a spin glass can exist at small finite temperatures. By mapping the Ising model onto an ensemble of non-interacting lattice fermions, we have shown that the critical temperature for the spin glass phase transition appears to vanish. This result applies for all concentrations of negative bonds where the ground state is a spin glass.

DOI
10.1088/0305-4470/36/24/306

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We have studied the two-dimensional ± J Ising model on the square lattice with a view to determine whether a spin glass can exist at small finite temperatures. By mapping the Ising model onto an ensemble of non-interacting lattice fermions, we have shown that the critical temperature for the spin glass phase transition appears to vanish. This result applies for all concentrations of negative bonds where the ground state is a spin glass.
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.
We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, ΔF(T)‾\overline{\Delta F(T)}ΔF(T)​ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .
We study the time evolution of two configurations submitted to the same thermal noise for several two dimensional models (Ising ferromagnet, symmetric spin glass, non symmetric spin glass). For all these models, we find a non zero critical temperature above which the two configurations always meet. Using finite size scaling ideas, we determine for these three models this dynamical phase transition and some of the critical exponents. For the ferromagnet, the transition Tc ≃ 2.25 coincides with the Curie temperature whereas for the two spin glass models ( ± J distribution of bonds) we obtain Tc ≃ 1.5-1.7.
The Ising spin glass in two dimensions is a model system with extraordinarily rich behavior. Although it does not show a spin-glass phase at finite temperatures, it exhibits many of the salient features of fully fledged spin glass systems. The authors use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems with up to $10,000\ifmmode\times\else\texttimes\fi{}10,000$ spins. These and further results from new algorithms for fully periodic lattices and for systems with degeneracies allow for the determination of critical exponents with unprecedented accuracy.
An Ising (spin- 1 / 2 ) model with random positive, negative and zero nearest-neighbour exchange coefficients is considered at zero temperature. The authors define two 'spin glass' susceptibilities, through average sums over ((s i s j ) 2 ) av and ( mod (s i s j ) mod ) av , where ( ) means averaging over degenerate ground states and ( ) av means averaging over the random bond distribution. These susceptibilities are expanded in powers of the concentrations of the non-zero bonds, for the triangular (seven terms), square (nine terms) and d-dimensional hypercubic (eight terms) lattices. All the series show a singularity, which is associated with the paramagnetic to spin-glass phase transition. The phase boundary between these phases, and the corresponding critical exponents, are estimated.
A new method to numerically calculate the n th moment of the spin overlap of the two-dimensional ± J Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the n th moment of the spin overlap can be calculated as a simple average of the n th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, L , up to L = 23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about 10 5 bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of T c is strongly affected by the corrections to scaling within the present data ( L ⩽ 23). Thus, there still remains the possibility that T c = 0, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.