The modified spin-wave theory is used to investigate the one-dimensional Heisenberg ferromagnet with the nearest-neighbor (NN) and next-nearest-neighbor (NNN) exchange anisotropies. The ground-state and low-temperature properties of the system are studied within the self-consistent method. It is found that the effect of the NN anisotropy on the thermodynamic quantities is stronger than that of the NNN anisotropy in the low-temperature region. The anisotropy dependence behaviors (such as the power, exponential and linear laws) are obtained for the position and the height of the maximum of the specific heat and its coefficient, as well as the susceptibility coefficient. The specific heat and its coefficient both display the low-temperature double maxima which are induced by the anisotropies and the NNN interaction. In the very low temperatures the specific heat and the susceptibility behave severally as T[Formula: see text] and T[Formula: see text] at the critical point J2/J1 = −0.25, where J1 and J2 are the NN and NNN interactions, respectively.

In this paper, the spin wave theory is applied to the one-dimensional Heisenberg antiferromagnet in the coexistence of two different anisotropies [Formula: see text] and [Formula: see text], which are separately the easy-axis single-ion anisotropies for sublattice [Formula: see text] and sublattice [Formula: see text] of the system. Both the ground-state and low-temperature properties of the system are strongly affected by the competition between these two anisotropies. Two kinds of the competition in terms of the deviation parameter [Formula: see text] are discussed for the uniform anisotropy taking the values of [Formula: see text] and [Formula: see text], respectively. The [Formula: see text]-dependent behaviors (such as the power, exponential and linear laws) are obtained for the total magnetization, the staggered magnetizations, the internal energy, the specific heat and the susceptibility. It is found that at zero-temperature, the interplay between these two anisotropies induces the antiferromagnetic-disorder phase transition in the small anisotropy region with [Formula: see text]. For the selected cases of [Formula: see text], our results for are in agreement with the findings obtained by the existing theories and the quantum Monte Carlo data.