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 slates 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, N-(n). In this case, for all lattices investigated, it is shown that the ground-state degeneracy behaves like exp[h(d)N-(n)], in the limit N-(n) 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, F-(n)(p), is calculated analytically (three last hierarchy levels for d=2 and the last hierarchy of the d=3 lattice, with 0 less than or equal to p less than or equal to 1), as well as numerically (all other cases, with p = 1/2). Except for d=2, in which case F-(n)(1/2) increases by decreasing the hierarchy level, all other dimensions investigated present an exponential decrease in F-(n)(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 d(l) (d(l) approximate to 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 much greater than 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. [S1063-651X(99)07910-6].