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.
We have performed Monte Carlo calculations of the thermodynamic properties of the bond-dilute Ising model on the square lattice. The results are obtained for the magnetization, internal energy, specific heat, and susceptibility for several concentrations of missing bonds. The dependence of the Curie temperature on dilution is in good agreement with the results obtained by other methods. The phase transition is found to remain sharp with the same value of the susceptibility exponent as for the pure Ising model.
The techniques developed by Edwards and Anderson (1975) for the spin-glass problem are here applied to the bond-diluted quenched Ising model (S= 1 / 2 ). The n to 0 trick is used to express exactly the free energy in terms of a generalized Hamiltonian representing n different but coupled replicas. The only approximation of the theory consists of decoupling these replicas in a way that reduces the free energy of the diluted system to that of a pure system with renormalized, concentration-dependent parameters. To first order in the concentration of missing bonds the author obtains all the results previously derived by Harris (1974) using a cumulant expansion and which are exact in this limit. More generally, the theory allows a discussion of the concentration dependence of the critical temperature; in particular, the values deduced for the critical concentration agree rather well with those obtained exactly or through series expansions. The behaviour of the system near the transition is also studied and the critical exponents are found to be Fisher-renormalized.
By a recursive method numerically exact free energies are calculated for square $L\ifmmode\times\else\texttimes\fi{}L$ Ising lattices, with
6< L< 18, for several kinds of frozen-in bond disorder: (i) bonds $\ifmmode\pm\else\textpm\fi{}J$ with various concentrations of negative bonds; (ii) bonds distributed according to a Gaussian distribution. Ground states of these systems are identified, the response to "ordering fields" is studied, and the correlation function
〈S0SR〉T2 is calculated as a function of temperature for various distances
R in the lattice. This correlation is found to decay strongly (presumably exponentially) with increasing
R even at temperatures distinctly below the apparent freezing temperature
Tf of previous Monte Carlo simulations; this "freezing transition" is hence unambiguously identified as a nonequilibrium effect. However, the correlation length is found to become long ranged at low temperatures, and it is suggested that a phase transition still occurs at
T=0; while in the Gaussian model the spin-glass order parameter
q(T=0)=1, it is found that $q\ensuremath{\equiv}0$ in the $\ifmmode\pm\else\textpm\fi{}J$ model where rather a power-law decay of correlations
〈S0SR〉T=02 occurs. Performing Monte Carlo simulations for precisely the same systems, the cooling times necessary to reach the true ground states of the system are identified, as well as the simulation times necessary to reach thermal equilibrium for the correlation functions. These times are found to increase so strongly with
L that for systems of macroscopic size the correct thermal equilibrium is probably irrelevant for experimental purposes. Rather a statistical mechanics based on the many long-lived metastable states would be required.
We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J 1 on nearest neighbour and J 2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J 2 = J 1 /2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the speciflc heat and the energy distribution for various ratios of J 2 / 1 . We find a first order transition for small J 2 > J 1 /2 and the transition temperature suppressed to T C = 0 at the critical point.