Topological selection rules in momentum resolved spectroscopies

We analyze the scattering amplitudes in spin-resolved angle resolved photoemission spectroscopy and inelastic neutron scattering. We reveal that (i) the Chern number of Weyl nodes, (ii) the Berry phase of nodal lines and (iii) the Z2 charge of nodal points or nodal lines can produce observable spectroscopic signatures in both electronic and bosonic bands. Our findings indicate that on a closed manifold in momentum space enclosing the band nodes having non-trivial topological charge, there will be zeros of scattering amplitudes. This phenomenon is dubbed the "topological selection rule", as these zeros are protected by the topological nodes mentioned above. Our work provides a new approach to detecting the topological charge of band nodes by measuring bulk wave functions instead of band energy dispersion or boundary modes.