ABSTRACT
We show that the infinite-dimensional representation of the recently introduced logistic algebra can be interpreted as a nontrivial generalization of the Heisenberg or oscillator algebra. This allows us to construct a quantum Hamiltonian having the energy spectrum given by the logistic map. We analyze the Hamiltonian of a solid whose collective modes of vibration are described by this generalized oscillator and compute the thermodynamic properties of the model in the two-cycle and r=3.6785 chaotic region of the logistic map.
ABSTRACT
The extensive ground-state entropy of frustrated systems on fractal lattices is investigated. Two methods of calculation are proposed, namely, recursive and factorization approaches. In the recursive approach the calculation is based on exact recursion relations for the total number of ground states. The latter procedure, which is in principle an approximation, is proposed as an alternative for dealing with complicated systems (for cases where the recursive approach may become impracticable), such as randomly frustrated models; it consists of factorizing the total number of ground states in terms of the number of ground states at each hierarchy level. Some examples of antiferromagnetic Ising models on different fractal lattices are considered, for which both procedures are applied. It is shown that the factorization approach may lead, in some cases, to the exact ground-state entropy, whereas in other cases it yields very accurate (although slightly lower) estimates.
ABSTRACT
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.