At zero temperature, two-dimensional Ising spin glasses are known to fall into several universality classes. Here we consider the scaling at low but nonzero temperatures and provide numerical evidence that eta approximately equal 0 and nu approximately equal 3.5 in all cases, suggesting a unique universality class. This algebraic (as opposed to exponential) scaling holds, in particular, for the +/- J model, with or without dilutions, and for the plaquette diluted model. Such a picture, associated with an exceptional behavior at T = 0, is consistent with a real space renormalization group approach. We also explain how the scaling of the specific heat is compatible with the hyperscaling prediction.

We compute the exact partition function of 2d Ising spin glasses with binary couplings. In these systems, the ground state is highly degenerate and is separated from the first excited state by a gap of size 4J. Nevertheless, we find that the low temperature specific heat density scales as exp(−2J/T ), corresponding to an "effective" gap of size 2J; in addition, an associated cross-over length scale grows as exp(J/T ). We justify these scalings via the degeneracy of the low lying excitations and by the way low energy domain walls proliferate in this model. In this work we reconsider the nature of these singularities using recently developed methods [6,7] for computing the exact partition function of square lattices with periodic boundary conditions, focusing on the low T scaling properties of the model with binary couplings. We show that although the energy "quantum" of excitation above the ground state is 4J, such excitations behave as composite particles; in fact the specific heat near the critical point scales as if the elementary excitations were of energy 2J. We justify this picture using properties of excitations and domain walls in this model. Finally, the joint temperature and size dependence shows the presence of a characteristic temperature-dependent length that grows as exp(J/T ), in agreement with hyperscaling.The model and our measurements -The Hamiltonian of our two-dimensional (2d) spin glass iswhere the sum runs over all nearest neighbor pairs of Ising spins (σ i = ±1) on a square lattice of volume V = L × L with periodic boundary conditions. The quenched random couplings J ij take the value ±J with probability 1/2. The partition function at inverse temperature β ≡ T −1 is Z J = {σi} e −βHJ ({σi}) and can be written asHere P J (X) is the polynomial whose coefficient of X p is the number of spin configurations of energy E = (−2L 2 + 2p)J. Saul and Kardar [4,5] showed that determining P J can be reduced to computing determinants which they did using exact arithmetic of arbitrarily large integers. More recently a more powerful approach has been developed [6,7], based on the use of modular arithmetic to compute pfaffians. With this algorithm, one first finds the coefficients modulo a prime number, thereby avoiding costly arbitrary precision arithmetic. Then the computation is repeated for enough different primes to allow the reconstruction of the actual (huge) integer coefficients using the Chinese remainder theorem.The algorithm proposed and implemented in [6,7] is powerful enough to solve samples with L ≈ 100; the total CPU time needed to compute Z J grows approximately as L 5.5 . In our study we have determined Z J for a large number of disorder samples at different lattice sizes: for instance we have 400000 samples at L = 6, 100000 at L = 10, 10000 at L = 30, 1000 at L = 40 and 300 at L = 50. The total computation time used is equivalent to about 40 years of a 1.2 GHz Pentium processor. For each sample we derive from Z J various thermodynamic quantities such as the free energy F J (β) = −β −1 ln Z...

We study temperature chaos in a two-dimensional Ising spin glass with random quenched bimodal couplings, by an exact computation of the partition functions on large systems. We study two temperature correlators from the total free energy and from the domain wall free energy: in the second case we detect a chaotic behavior. We determine and discuss the chaos exponent and the fractal dimension of the domain walls.PACS numbers: 75.10. Nr, 05.50.+q, 75.40.Gb, 75.40.Mg Introduction -A characteristic feature of spin glasses is the presence of chaos under small changes in the quenched couplings, in the temperature, or in the magnetic field [1,2,3,4,5,6,7,8,9]. With the expression temperature chaos we refer to the fragility of the equilibrium states of a disordered system under small temperature changes. Let us consider two typical equilibrium configurations of such a system under the same realization of the quenched disorder: the first configuration is in equilibrium at temperature T , while the second one is in equilibrium at temperature T ′ = T + ∆T . One says that there is temperature chaos if for arbitrarily small (but non-zero) values of ∆T , the typical overlap of two configurations at T and T ′ goes to zero when the system size diverges. The spatial distance ℓ(T, ∆T ) over which such overlaps decay is called the chaos length, and, as we will discuss better in the following, it scales as ℓ ∼ ∆T

The ground state and low T behavior of two-dimensional spin systems with discrete binary couplings are subtle but can be analyzed using exact computations of finite volume partition functions. We first apply this approach to Villain's fully frustrated model, unveiling an unexpected finite size scaling law. Then we show that the introduction of even a small amount of disorder on the plaquettes dramatically changes the scaling laws associated with the T = 0 critical point.

We consider the random-anisotropy model on the square and on the cubic lattice in the strong-anisotropy limit. We compute exact ground-state configurations, and we use them to determine the stiffness exponent at zero temperature; we find theta=-0.275(5) and theta approximate to 0.2, respectively, in two and three dimensions. These results strongly suggest that the low-temperature phase of the model is the same as that of the usual Ising spin-glass model. We also show that no magnetic order occurs in two dimensions, since the expectation value of the magnetization is zero and spatial correlation functions decay exponentially. In three dimensions, our data strongly support the absence of spontaneous magnetization in the infinite-volume limit