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

Ground-state clusters of two-, three-, and four-dimensional +/-J Ising spin glasses

A huge number of independent true ground-state configurations is calculated for two-, three- and four-dimensional +/-J spin-glass models. Using the genetic cluster-exact approximation method, system sizes up to N=20(2),8(3),6(4) spins are treated. A "ballistic-search" algorithm is applied, which allows even for large system sizes to identify clusters of ground states that are connected by chains of zero-energy flips of spins. The number of clusters n(C) diverges with N going to infinity. For all dimensions considered here, an exponential increase of n(C) appears to be more likely than a growth with a power of N. The number of different ground states is found to grow clearly exponentially with N. A zero-temperature entropy per spin of s(0)=0.078(5)k(B) (2D), s(0)=0.051(3)k(B) (3D), respectively, s(0)=0.027(5)k(B) (4D) is obtained.

Ground-state degeneracies of Ising spin glasses on diamond hierarchical lattices

The total number of ground states for short-range Ising spin glasses, defined on diamond hierarchical lattices of fractal dimensions d=2, 3, 4, 5, and 2.58, is estimated by means of analytic calculations (three last hierarchy levels of the d=2 lattice) and numerical simulations (lower hierarchies for d=2 and all remaining cases). It is shown that in the case of continuous probability distributions for the couplings, the number of ground states is finite in the thermodynamic limit. However, for a bimodal probability distribution (+/-J with probabilities p and 1-p, respectively), the average number of ground states is maximum for a wide range of values of p around p=1 / 2 and depends on the total number of sites at hierarchy level n, Nn. In this case, for all lattices investigated, it is shown that the ground-state degeneracy behaves like exp[h(d)Nn], in the limit Nn large, where h(d) is a positive number which depends on the lattice fractal dimension. The probability of finding frustrated cells at a given hierarchy level n, Fn(p), is calculated analytically (three last hierarchy levels for d=2 and the last hierarchy of the d=3 lattice, with 0<or=p<or=1), as well as numerically (all other cases, with p=1 / 2). Except for d=2, in which case Fn(1 / 2) increases by decreasing the hierarchy level, all other dimensions investigated present an exponential decrease in Fn(1 / 2) for decreasing values of n. For d=2 our results refer to the paramagnetic phase, whereas for all other dimensions considered [which are greater than the lower critical dimension dl (dl approximately 2.5)], our results refer to the spin-glass phase at zero temperature; in the latter cases h(d) increases with the fractal dimension. For n>>1, only the last hierarchies contribute significantly to the ground-state degeneracy; such a dominant behavior becomes stronger for high fractal dimensions. The exponential increase of the number of ground states with the total number of sites is in agreement with the mean-field picture of spin glasses.

Large Clusters of Metal Ions: The Transition from Molecular to Bulk Magnets

Clusters of metal ions are a class of compounds actively investigated for their magnetic properties, which should gradually change from those of simple paramagnets to those of bulk magnets. However, their interest lies in a number of different disciplines: chemistry, which seeks new synthetic strategies to make larger and larger clusters in a controlled manner; physics, which can test the validity of quantum mechanical approaches at the nanometer scale; and biology, which can use them as models of biomineralization of magnetic particles.