Exploring metamagnetism in triangular Ising networks: Insights from further-neighbor interactions with a case study on ${\mathrm{ErGa}}_2$

Recent attention to frustrated magnetism has revived interest in most fundamental models of the latter, such as the classical triangular Ising lattice (I-3). Despite the long history, our understanding of the physics of I-3 is still patchy, and even the mean-field magnetic phase diagram has been addressed only in specific cases. Here we present a study considerably more exhaustive than those available in the literature, including arbitrary Heisenberg interaction up to 4th nearest neighbors (NN) identifying all inequivalent magnetic patterns up to the 8th NN (1901 of those), and calculating the phase diagrams in the external magnetic field. We then connect these findings with a real-life material, ${\mathrm{ErGa}}_2$, which our calculations show to be a very good approximation to the I-3, despite its crystallographically 3D character. We calculate, from first principles, the exchange parameters, and show that they reproduce well the observed magnetization steps. We also find a possibility of additional narrow steps, which may be observed in future experiments.