Part II: Witten effect and $\mathbb{Z}$-classification of axion angle $θ=n π$

Abstract: The non-trivial third homotopy class of three-dimensional topological insulators leads to quantized, magneto-electric coefficient or axion angle $\theta= n \pi$, with $n \in \mathbb{Z}$. In Part I, we developed tools for computing $n$ from a staggered symmetry-indicator $\kappa_{AF,j}$ and Wilson loops of non-Abelian, Berry connection in momentum-space, which clearly distinguished between magneto-electrically trivial ($n=0$), and non-trivial ($n=2s$) topological crystalline insulators. In this work, we perform $\mathbb{Z}$-classification of real-space, topological response or $\theta$ by carrying out thought experiments with magnetic, Dirac monopoles. We demonstrate this for non-magnetic and magnetic topological insulators by computing induced electric charge on monopoles or Witten effect. We show that both first- and higher- order topological insulators can exhibit quantized, magneto-electric response, irrespective of the presence of gapless surface-states, and corner-states. Special attention is paid to the response of octupolar higher-order topological insulator, which was originally predicted to be magneto-electrically trivial. The important roles of fermion zero-modes, $\mathcal{CP}$, and flavor symmetries are critically addressed. Our work outlines a unified theoretical framework for addressing dc topological response and topological quantum phase transitions, which cannot be reliably predicted by symmetry-based classification scheme.