Quantized helicity of Berry connection and band topology of magneto-electric systems

In classical electrodynamics, the flux and the helicity are two important physical quantities, which can be used to characterize topological properties of electromagnetic fields. They can also play important roles for defining bulk invariants of three-dimensional topological materials. The non-degenerate bands of time-reversal-symmetry breaking Chern insulators are known to support quantized flux of Abelian Berry connections or Chern numbers. Can generic three-dimensional insulators support quantized helicity of Berry connections? To answer this question, we discuss the general principles for constructing tight-binding Hamiltonians of N-band systems, which can exhibit quantized helicity as bulk topological invariants. Based on such model Hamiltoians, we address various physical properties of magneto-electric topological insulators, including those of topological Hopf insulators.