Efficient evaluation of the $k$-space second Chern number in four dimensions

TL;DR

This paper introduces an adaptive mesh refinement numerical method for efficiently computing the 4D topological second Chern number, improving speed and accuracy over existing techniques.

Contribution

The authors develop a novel adaptive mesh approach that enhances efficiency and accuracy in calculating the 4D second Chern number compared to traditional methods.

Findings

Achieves same accuracy with fewer diagonalizations.

Requires minimal memory, enabling larger system calculations.

Remains accurate near topological phase transitions.

Abstract

We propose an efficient numerical method to compute the $k$-space second Chern number in four-dimensional (4D) topological systems. Our approach employs an adaptive mesh refinement scheme to evaluate the Brillouin-zone integral, which automatically increases the grid density in regions where the Berry curvature is sharply peaked. We compare our method with the 4D lattice-gauge extension of the Fukui-Hatsugai-Suzuki method and a direct uniform grid integration scheme. Compared with these approaches, our method (i) achieves the same accuracy with substantially fewer diagonalizations, and thus runs faster; (ii) requires minimal memory to execute, enabling calculations for larger systems; and (iii) remains accurate even near topological phase transitions where conventional methods often face challenges. These results demonstrate that the adaptive subdivision strategy is a practical and powerful tool for calculating the $k$-space second Chern number.