Symmetry indicators vs. bulk winding numbers of topologically non-trivial bands*

The symmetry-indicators provide valuable information about the topological properties of band structures in real materials. For inversion-symmetric, non-magnetic materials, the pattern of parity eigenvalues of various Kramers-degenerate bands at the time-reversal-invariant momentum points are generally analyzed with the combination of strong Z₄, and weak Z₂ indices. Can the symmetry indicators identify the tunneling configurations of SU(2) Berry’s connections or the three-dimensional, winding numbers of topologically non-trivial bands? We perform detailed analytical and numerical calculations on various effective tight-binding models to answer this question. If the parity eigenvalues are regarded as fictitious Ising spins, located at the vertices of Miller hypercube, the strong Z₄ index describes the net ferro-magnetic moment, which is shown to be inadequate for identifying non-trivial bands, supporting even integer winding numbers. We demonstrate that an ``anti-ferromagnetic” index, measuring the staggered magnetization can distinguish between bands possessing zero, odd, and even integer winding numbers. The coarse-grained analysis of symmetry-indicators is substantiated by computing the change in rotational-symmetry-protected, quantized Berry’s flux and Wilson loops along various high-symmetry axes. By simultaneously computing ferromagnetic and anti-ferromagnetic indices, we categorize various bands of bismuth, antimony, rhombohedral phosphorus, and Bi₂Se₃.