No abstract

We use low-density series expansions to calculate critical exponents for the behavior of random resistor networks near the percolation threshold as a function of the spatial dimension d. By using scaling relations, we obtain values of the conductivity exponent μ. For d=2 we find μ=1.43±0.02, and for d=3, μ=1.95±0.03, in excellent agreement with the experimental result of Abeles et al. Our results for high dimensionality agree well with the results of ε-expansion calculations.

Heat bath Monte Carlo simulations have been used to study a four-state clock model with a type of random field on simple cubic lattices. The model has the standard nonrandom two-spin exchange term with coupling energy J and a random field which consists of adding an energy D to one of the four spin states, chosen randomly at each site. This Ashkin-Teller-like model does not separate; the two random-field Ising model components are coupled. When D/J = 3, the ground states of the model remain fully aligned. When D/J ≥ 4, a different type of ground state is found, in which the occupation of two of the four spin states is close to 50%, and the other two are nearly absent. This means that one of the Ising components is almost completely ordered, while the other one has only short-range correlations. A large peak in the structure factor S(k) appears at small k for temperatures well above the transition to long-range order, and the appearance of this peak is associated with slow, "glassy" dynamics. The phase transition into the state where one Ising component is long-range ordered appears to be first order, but the latent heat is very small.

Monte Carlo simulations have been used to study a discretized Heisenberg ferromagnet (FM) with random uniaxial single-site anisotropy on L × L × L simple cubic lattices, for L up to 64. The spin variable on each site is chosen from the twelve [110] directions. The random anisotropy has infinite strength and a random direction on a fraction x of the sites of the lattice, and is zero on the remaining sites. In many respects the behavior of this model is qualitatively similar to that of the corresponding random-field model. Due to the discretization, for small x at low temperature there is a [110] FM phase. For x > 0 there is an intermediate quasi-long-range ordered (QLRO) phase between the paramagnet and the ferromagnet, which is characterized by a |k| −3 divergence of the magnetic structure factor S(k) for small k, but no true FM order. At the transition between the paramagnetic and QLRO phases S(k) diverges like |k| −2 . The limit of stability of the QLRO phase is somewhat greater than x = 0.5. For x close to 1 the low temperature form of S(k) can be fit by a Lorentzian, with a correlation length estimated to be 11 ± 1 at x = 1.0 and 25 ± 5 at x = 0.75.

No abstract