Magnetization-induced phase transitions on the surface of three-dimensional topological insulators

From the low-energy model, the topological field theory indicates that the surface magnetization can open a surface gap in three-dimensional (3D) topological insulators (TIs), resulting in a half-quantized Hall conductance. Here, by employing the realistic lattice model, we show the occurrence of the surface phase transitions, accompanied by the sharp changes of the surface Chern number from $\frac{1}{2}$ to $-\frac{3}{2}$, and finally to $-\frac{1}{2}$, in 3D TIs induced by surface magnetization. These surface phase transitions lead to the sudden jumps in the magneto-electric coefficient and the quantum Hall conductance, which are experimentally observable. Furthermore, we present the phase diagram that elucidates the behavior of the 3D TI surface Chern numbers under surface magnetization for different $Z_4$ topological numbers. Our study highlights the presence of the new phases with broken bulk-boundary correspondence and enriches the understanding of the properties of TIs.