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(L⩽11) 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^θ_LV^θ_v with θ_L∼−4.7 and θ_v∼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.