Evaluation of real-space second Chern number using the kernel polynomial method

TL;DR

This paper demonstrates the use of the kernel polynomial method to evaluate the real-space second Chern number in four-dimensional Chern insulators, including disordered systems, and extends the approach to six dimensions for the third Chern number.

Contribution

It introduces a real-space computational approach for higher-dimensional Chern numbers and validates it against theoretical predictions, including disorder effects and extensions to six dimensions.

Findings

Numerical results agree with theoretical expectations for clean systems.

The method captures disorder effects consistent with the self-consistent Born approximation.

Qualitative agreement with theory observed in six-dimensional calculations despite finite-size effects.

Abstract

We evaluate the real-space second Chern number of four-dimensional Chern insulators using the kernel polynomial method. Our calculations are performed on a four-dimensional system with $30^4$ sites, and the numerical results agree well with theoretical expectations. Moreover, we show that the method is capable of capturing the disorder effects. This is evidenced by the phase diagram obtained for disordered systems, which agrees well with predictions from the self-consistent Born approximation. Furthermore, we extend the method to six dimensions and perform an exploratory real-space calculation of the third Chern number. Although finite-size effects prevent full quantization, the numerical results show qualitative agreement with theoretical expectations. The study represents a step forward in the real-space characterization of higher-dimensional topological phases.