Witten effect and integer classification of three-dimensional topological insulators

The non-trivial third homotopy class of three-dimensional topological insulators leads to quantized, magneto-electric coefficient or axion angle θ=n π, with n being an integer valued three-dimensional, winding number. We probe integer classification of θ for non-magnetic and magnetic topological insulators by computing induced electric charge on test, magnetic monopoles or Witten effect. We show that both first-order and higher-order topological insulators can exhibit quantized, magneto-electric response, irrespective of the presence of gapless surface-states, and corner-localized-states. The important roles of fermion zero-modes, CP, and flavor symmetries are critically addressed. Our work outlines a unified theoretical framework for addressing topological response and topological quantum phase transitions of three-dimensional materials.