Ground state and low-lying excitations in a Heisenberg spin-glass model in three dimensions

We developed a hybrid genetic algorithm and studied the ground state and low-lying excitations of the ±J Heisenberg model on a three-dimensional L×L×L(L11) lattice. Our results suggest occurrence of a Parisi state. That is, there exists a metastable state with a finite energy of ΔE~O(J) in the thermodynamic limit and the energy barrier ΔW between the metastable state and the ground state is ΔW∝JL^θ with θ~0.53. Our results also suggest that the energy ΔE of small scale excitations are related to a scale of the excitation V in a similar manner to those of the ferromagnetic model, i.e., ΔE∝JL^θLV^θᵥ with θL~-4.7 and θᵥ~2.0. This fact suggests that similarity exists in the ground state structure between the ±J and the ferromagnetic Heisenberg models, though the spin directions of those two models differ greatly from one another.