Cubic ferromagnet and its emergent phenomenon in the vicinity of phase boundary

Among the natural materials displaying magnetism, it is well-known that the magnetic anisotropy can be widely found and it affects the fundamental physics. However, the technical difficulty have been keeping the problem out of reach for a long time. In this work, we study the simplest lattice spin model for the two-dimensional~(2D) cubic ferromagnet by means of mean-field analysis and tensor network calculation. While both methods give rise to similar results in detecting related phases, the 2D infinite projected entangled-pair state~(iPEPS) calculation provides more accurate values of transition points. Moreover, our iPEPS results indicate that it is more difficult in pinning down the direction of magnetic easy axes near the phase boundary, and we interpret it as the easy-axis softening. By constructing a low-energy effective model in the reduced Hilbert basis, an emergence of $U(1)$ symmetry is diagnosed which explains the softening of easy axes. We argue that, from the perspective of field theory and renormalization group, the emergence of continuous symmetry becomes exact right on the critical points and the criticality is governed by the 3D XY universality class. Numerical correlation functions obtained by iPEPS also support this physical scenario. Our study, to the best of our knowledge, provides the very first systematic analysis upon this lattice spin model beyond the mean-field consideration. We also address the practical importance of such emergent phenomenon and its possible application in manipulating the magnetic easy axes for spinful semiconductors.