Calculation of ground-state entropies of highly frustrated systems on fractal lattices

Ground-state entropies of the Potts antiferromagnet on diamond hierarchical lattices

The ground-state degeneracies of the q-state Potts antiferromagnet on general diamond hierarchical lattices are computed, for q> or =3, by means of two distinct methods. The first method, denominated the recursive approach, is based on exact recursion relations for the total number of ground states, leading to the exact ground-state entropy in the thermodynamic limit. The second method, called the factorization approach, consists in a simple approximation, where the total number of ground states is factorized as a product of the number of ground states at each hierarchy level. The factorization approach appears to be a poor approximation for small values of q, but its accuracy improves substantially as q increases, and it becomes exact in the limit q--> infinity. In spite of the fact that such a model presents no frustration, a residual entropy at zero temperature is found for all q> or =3. Similarly to what happens on Bravais lattices, the residual entropy approaches its maximum allowed value, ln q, as q increases.

Lower bounds for the ground-state degeneracies of frustrated systems on fractal lattices

The total number of ground states for nearest-neighbor-interaction Ising systems with frustrations, defined on hierarchical lattices, is investigated. A simple method is presented, which allows one to factorize the ground-state degeneracy, at a given hierarchy level n, in terms of contributions due to all hierarchy levels. Such a method may yield the exact ground-state degeneracy of uniformly frustrated systems, whereas it works as an approximation for randomly frustrated models. In the latter cases, it is demonstrated that such an approximation yields lower-bound estimates for the ground-state degeneracies.