Researchers have explored the relationship between higher-order Berry curvature and second Chern numbers in four-dimensional Chern insulators. They rewrote a lattice model of these insulators as a family of translationally-invariant infinite chains over the three-dimensional Brillouin zone. Using infinite matrix product states (iMPS), they calculated the higher three-form Berry curvature, a topological concept describing how a system's quantum phases change across its parameter space.

The study focused on the topological phase diagram of the Dixmier-Douady-Kapustin-Spodyneiko (DDKS) number as a function of the model's mass term. They demonstrated that this phase diagram is exactly congruent to the one obtained from the second Chern number, whose analytical expression is known for this specific model. This agreement is crucial, as it validates the use of higher-order Berry curvature as a quantized method to compute second Chern numbers, providing a new tool for the topological characterization of materials.

Motivated by the connection between the second Chern form and the Chern-Simons axion coupling, the researchers also examined magnetoelectric coupling in three dimensions. This coupling, which relates electric and magnetic fields, is of great interest in condensed matter physics. The work suggests that higher-order Berry phases may play a fundamental role in understanding magnetoelectric phenomena, opening new avenues for the design of materials with controlled topological properties.