Weak ferromagnetism in the square-lattice Heisenberg $J_1$-$J_2$-$J_2^\prime$ model with easy-axis single-ion anisotropy
Motivated by recent experimental realizations of the Heisenberg model and two-dimensional altermagnetism using ultracold atoms in optical lattices, we systematically investigate the magnetic properties of the square-lattice Heisenberg $J_1$-$J_2$-$J_2^\prime$ model in the presence of easy-axis single-ion anisotropy $D$. Here, $J_1 (>0)$ is the nearest-neighbor antiferromagnetic exchange parameter, while $J_2$ and $J_2^\prime$ are two distinct next-nearest-neighbor superexchange parameters that alternate on the lattice. We perform numerical calculations of the sublattice magnetization, net magnetization, and critical temperature. Interestingly, the net magnetization is not identically zero at finite temperatures; its absolute value first increases and then decreases with increasing temperature, revealing a temperature-dependent weak ferromagnetism. We find that $J_2\not=J_2^\prime$ is the primary factor responsible for weak ferromagnetism at finite temperatures, while $D$ provides the necessary background for two-dimensional long-range magnetic ordering. Moreover, for a fixed $J_2+J_2^\prime$, we observe that a larger $|J_2-J_2^\prime|$ leads to a greater maximum net magnetization and critical temperature, as well as a broader temperature range for weak ferromagnetism. The difference between $J_2$ and $J_2^\prime$ originates from the distinct superexchange pathways between magnetic atoms, with these pathways mediated by two different types of nonmagnetic atoms. This mechanism, which differs from conventional ones such as the Dzyaloshinskii-Moriya interaction, may be detectable in real materials.