The critical exponents for T → 0 of the two-dimensional Ising spin glass model with Gaussian couplings are determined with the help of exact ground states for system sizes up to L = 50 and by a Monte Carlo study of a pseudo-ferromagnetic order parameter. We obtain: for the stiffness exponent y(= θ) = −0.281 ± 0.002, for the magnetic exponent δ = 1.48 ± 0.01 and for the chaos exponent ζ = 1.05 ± 0.05. From Monte Carlo simulations we get the thermal exponent ν = 3.6 ± 0.2. The scaling prediction y = −1/ν is fulfilled within the error bars, whereas there is a disagreement with the relation y = 1 − δ.

In this paper we study 2-dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic eld. We s h o w h o w using a technique called branch and cut, the exact ground states of grids of sizes up to 100 100 can be determined in a moderate amount of computation time, and we r e p o r t on extensive computational tests. With our method we produce results based on more than 20 000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other more popular ways to compute the ground state, like M o n te Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determined a priori. Our ground state energy estimation at zero eld is ;1:317.

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750 800 4 Wavelength (nm) Figure 8. Liquid overtone spectrum of the u = 4 N-H transition of pyrrole diluted in CC14.The other absorptions shown in Figura 2 and 3 involve the N-H stretching motion which, similar to the C-H oscilator in CX3H,lZ interacts with the bend, causing new absorptions to appear. The general pattern is a strong peak accompanied by two or three weaker peaks to lower energy. For instance, the 13 305-cm-' band in pyrrole (assigned to the 4-0 N-H transition) has three weaker transitions 103, 191, and 309 cm-' to the red. The corresponding transition in pyrrolidine a t 12 551 cm-' has three weaker peaks shifted to the red by 82, 197, and 325 cm-I. The 5-0 N-H transitions lie a t 16 316 and 15 298 cm-' for pyrrole and pyrrolidine, respectively. The transitions for pyrrole are blue-shifted due to aromaticity, while those in pyrrolidine are typical of other amines. For example, the N-H(5) level in ammonia and methylamine lies a t 15 4505 and 15 325 cm-',* respectively.The spectrum of pyrrole-d4 confirms the assignments of the N-H multiple transitions. The normal and deuterated samples have the same two transitions with similar intensity ratio at the first overtone level. The three strongest peaks in the normal third overtone spectrum are also found in the same relative intensities in the spectrum of pyrrole-d4. It is clear that the peaks common to both spectra must be due to N-H transitions and not due to overtones and combinations of the C-H stretches. Changes in band intensities and transition wavenumbers are expected since the vibrational mode responsible for the coupling (possibly the N-H: C-H bend) is altered by the deuteration. This could account for the disappearance of the two peaks in the third overtone spectrum of pyrrole-d,.Further evidence that these multiple absorption belong to intramolecular vibrational coupling in the N-H comes from the liquid spectra. The 4-0 N-H transition in the neat pyrrole spectrum shown in Figure 7 is shifted 500 cm-I to the red from the gaseous value. However, upon dilution in CC14 this broad red-shifted peak shows structure reminiscent of the gaseous spectrum ( Figure 8). Specifically, the splittings of 85, 186, and 303 cm-l appear in the dilution spectra while the splittings in the gaseous spectrum are 103, 191, and 309 cm-'. These numbers are in excellent agreement since the liquid spectra were recorded on a conventional absorption spectrometer at 1 nm (above 20 cm-I) resolution. The intensity at the red side of the broad peak in neat pyrrole decreases at higher dilution and is probably due to inhomogeneous absorption of hydrogen-bonded N-H oscillators. The values in the last column in Table I1 indicate that the entire N-H absorption at the v = 4 level shifts about 200 cm-l in the liquid 10% dilution. In contrast, the C-H v = 4 and 5 levels only shift about 100 cm-I. For pyrrolidine neat liquid the 4-0 N-H transition is split. The comparison of dilution splittings to those of the gas for pyrrolidine is not as good as for the case of pyrrole but does show th...

The critical exponents for T → 0 of the two-dimensional Ising spin glass model with Gaussian couplings are determined with the help of exact ground states for system sizes up to L = 50 and by a Monte Carlo study of a pseudo-ferromagnetic order parameter. We obtain: for the stiffness exponent y(= θ) = −0.281 ± 0.002, for the magnetic exponent δ = 1.48 ± 0.01 and for the chaos exponent ζ = 1.05 ± 0.05. From Monte Carlo simulations we get the thermal exponent ν = 3.6 ± 0.2. The scaling prediction y = −1/ν is fulfilled within the error bars, whereas there is a disagreement with the relation y = 1 − δ.