We developed a hybrid genetic algorithm and studied the ground state and low-lying excitations of the $\ifmmode\pm\else\textpm\fi{}J$ Heisenberg model on a three-dimensional $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L(L\ensuremath{\leqslant}11)$ lattice. Our results suggest occurrence of a Parisi state. That is, there exists a metastable state with a finite energy of $\ensuremath{\Delta}E\ensuremath{\sim}O(J)$ in the thermodynamic limit and the energy barrier $\ensuremath{\Delta}W$ between the metastable state and the ground state is $\ensuremath{\Delta}W\ensuremath{\propto}J{L}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\sim}0.53$. Our results also suggest that the energy $\ensuremath{\Delta}E$ 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., $\ensuremath{\Delta}E\ensuremath{\propto}J{L}^{{\ensuremath{\theta}}_{L}}{V}^{{\ensuremath{\theta}}_{v}}$ with ${\ensuremath{\theta}}_{L}\ensuremath{\sim}\ensuremath{-}4.7$ and ${\ensuremath{\theta}}_{v}\ensuremath{\sim}2.0$. This fact suggests that similarity exists in the ground state structure between the $\ifmmode\pm\else\textpm\fi{}J$ and the ferromagnetic Heisenberg models, though the spin directions of those two models differ greatly from one another.
Thermodynamic properties of antiferromagnetic spin chain with various AFM nearest-neighbor and FM next-nearest-neighbor exchange interactions are studied using functional integral method in J 1 – J 2 Heisenberg model. Besides, the excitation spectrum of the system is also given. The results show that the system has two distinct excitations with no energy gap from the ground state for any permissible values of J 1 and J 2 without single-ion anisotropy. Changing the exchange parameters influences on the thermodynamic properties and the excitation spectra of the system.