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

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.

💡 Research Summary

The paper presents a comprehensive study of real‑space topological invariants in higher‑dimensional Chern insulators using the Kernel Polynomial Method (KPM). The authors focus on the second Chern number (C₂) of four‑dimensional (4D) Wilson‑Dirac lattice models and extend the approach to the third Chern number (C₃) in six dimensions (6D).

Model and Real‑Space Formulation
A generic 2n‑dimensional Wilson‑Dirac Hamiltonian is introduced, with gamma matrices satisfying the Clifford algebra. For the 4D case (n = 2) the Hamiltonian depends on a Dirac mass m, hopping t, and spin‑orbit coupling λ, which are set to unity in the simulations. The real‑space n‑th Chern number is expressed as a fully antisymmetric trace involving the projector onto occupied states, P = Θ(E_F − H), and coordinate operators X_j. Explicit formulas for C₁, C₂, and C₃ are given (Eqs. 5‑7).

Kernel Polynomial Method Implementation
Instead of diagonalizing the Hamiltonian, the projector is approximated by a Chebyshev polynomial expansion of order M = 256. The Jackson kernel suppresses Gibbs oscillations, and the expansion coefficients µ_χ are analytic functions of the Fermi energy. The Hamiltonian is rescaled to fit the Chebyshev domain