Higher Berry curvature, second Chern numbers and magnetoelectric coupling in crystalline insulators

Abstract: We rewrite a lattice model of the four-dimensional Chern insulator as a family of translationally-invariant infinite chains over the three-dimensional Brillouin zone and compute its higher three-form Berry curvature using infinite matrix product states (iMPS). We calculate the topological phase diagram of the associated Dixmier--Douady--Kapustin--Spodyneiko (DDKS) number as a function of the model's mass term, and show that it is exactly congruent to the phase diagram in terms of the second Chern number, the analytic expression of which is known for this particular model. This agreement demonstrates that higher Berry curvature can be used to compute second Chern numbers in a manifestly quantized manner. Motivated by the connection between the second Chern form and the Chern--Simons axion coupling, we study magnetoelectric coupling in three dimensions and its relation to higher Berry phases.