J. Vannimenus

Theory of the frustration effect. II. Ising spins on a square lattice

For pt.I see Commun. Phys., vol.2, p.115 (1977). Presents numerical results for the ground-state properties of the simplest system showing the frustration effect, Ising spins on a square lattice with a variable concentration x of antiferromagnetic interactions. Ferromagnetism disappears at a concentration xc approximately 0.09. The ground-state energy and degeneracy have markedly different behaviours below and above xc. The authors compare the results with other calculations and stress the advantages of the frustration approach.

Simple frustrated systems: chains, strips and squares

The ground-state properties of several systems showing the Toulouse frustration effect are investigated analytically. First, the energy and entropy of the random-bond Ising chain in a uniform field are obtained for all concentrations of antiferromagnetic bonds using the transfer-matrix method. The susceptibility has discontinuities for an infinite number of critical values of the field, where the entropy shows spikes in addition to discontinuities. These effects are related to the physics of frustration. The second system studied consists of frustrated strips, which it is argued are the proper one-dimensional limit of spin glasses. Two kinds of strips are considered and the results are compared with recent numerical works on two-dimensional spin glasses and with the exact results for random (3*3) squares.

Ground-state correlations in the two-dimensional Ising frustration model

The ground states of the Ising spin glass with random +or-J bonds on a square lattice have been studied numerically. The distribution of effective fields is obtained as a function of the concentration of negative bonds, together with the proportion of living bonds (i.e. bonds which are frustrated in some ground states only). The comparison of different ground states shows the existence of large packets of solitary spins which keep the same relative orientation in all cases. For the (18*18) samples, the largest packet contains on average more than half the spins, even for maximum bond disorder. The results reveal a striking rigidity of the system at zero temperature, which is very different from the behavior of simple paramagnets and fully frustrated models.

Anisotropic conductivity in fractal models of percolation networks

Anisotropy effects are studied on various fractal resistor lattices. The macroscopic conductivity becomes isotropic on large scales and the anisotropy decays with a scaling exponent, as L- lambda . For the d-dimensional Sierpinski gaskets this exponent is given by lambda =ln((d+3)/(d+2))/ln 2. The decay of the anisotropy is faster in two dimensions than in three, in agreement with recent numerical results on percolation networks.