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

Technical Summary: Evaluation of Real-Space Second Chern Number Using the Kernel Polynomial Method

Problem Statement
Topological phases in $2n$-dimensional systems, such as four-dimensional (4D) Chern insulators characterized by the second Chern number ($C_2$) and six-dimensional (6D) systems characterized by the third Chern number ($C_3$), are traditionally defined in momentum space. This definition relies on translational invariance, which is often absent in realistic systems containing disorder or quasiperiodicity. While real-space formulations exist to address this, standard approaches relying on direct matrix diagonalization are limited to system sizes of approximately $10^4$ degrees of freedom. This size constraint is particularly problematic for higher-dimensional systems, where the Hilbert space dimension grows exponentially ($L^{2n}$), making it difficult to achieve the large system sizes necessary for the quantization of topological invariants to emerge clearly. Previous attempts to calculate real-space $C_2$ using supercell approximations or Kitaev's formula were limited to system sizes of $4^4$ and $8^4$, respectively, and often failed to achieve full quantization due to finite-size effects.

Methodology
The authors employ the Kernel Polynomial Method (KPM) to evaluate real-space Chern numbers for significantly larger system sizes. The core of their approach involves:

1. Model: They utilize the $2n$-dimensional Wilson-Dirac lattice model, discretized on a hypercubic lattice with lattice constant $a=1$. The Hamiltonian is defined using gamma matrices satisfying the Clifford algebra.
2. Real-Space Formula: They adopt a general universal topological marker formula for the $n$-th Chern number:
   $$C_n = \frac{(2\pi i)^n}{n!} \epsilon_{j_1 j_2 \dots j_{2n}} \text{Tr} (P X_{j_1} P X_{j_2} \dots P X_{j_{2n}} P)$$
   where $P$ is the projector onto occupied bands and $X_j$ are coordinate operators.
3. KPM Implementation: To avoid expensive diagonalization, the projector $P$ is approximated by expanding the step function using Chebyshev polynomials up to order $M=256$. The trace is estimated using a stochastic trace approximation with $R=5$ random-phase vectors. The Hamiltonian is rescaled to the interval $[-1, 1]$ to ensure numerical stability.
4. Disorder: Anderson-type disorder is introduced via random on-site energies uniformly distributed in $[-W, W]$.

Key Contributions and Results

• 4D Second Chern Number ($C_2$) in Clean Systems:
  The authors performed real-space calculations on 4D systems with up to $30^4$ sites (304 total sites in the text, though the notation $30^4$ implies a side length $L=30$).

  • Quantization: As the system size ($L$) increased from 4 to 20, the numerically computed $C_2$ converged toward the theoretical momentum-space prediction ($C_2 = -3$ for specific mass parameters).
  • Convergence Rate: Analysis of the deviation $\ln(|C_2 + 3|)$ versus system size $L$ revealed an exponential decay, confirming that finite-size effects become negligible in larger systems.

• 4D Second Chern Number ($C_2$) in Disordered Systems:

  • Robustness: The method successfully captured the stability of the topological phase under weak disorder. The quantized value of $C_2$ remained stable for moderate disorder strengths ($W$).
  • Phase Diagram: The disorder-averaged $C_2$ phase diagram agreed well with predictions from the self-consistent Born approximation.
  • Critical Behavior: Near critical points (where the bulk gap is small), even weak disorder drove the topological marker away from quantized values, whereas systems with larger gaps remained robust against stronger disorder.

• 6D Third Chern Number ($C_3$) Exploratory Calculation:
  The authors extended the method to 6D systems to calculate the real-space third Chern number.

  • Qualitative Agreement: For system sizes up to $L=5$, the numerical results showed qualitative agreement with theoretical momentum-space predictions, correctly identifying phase transition positions.
  • Limitations: Full quantization was not achieved. The authors attribute this to significant finite-size effects and the extreme computational complexity involved. The calculation of $C_3$ requires summing over 720 non-zero terms of the 6D Levi-Civita tensor, compared to only 24 terms for the 4D case, drastically increasing resource demands.

Significance and Claims
The paper claims to represent a "step forward" in the real-space characterization of higher-dimensional topological phases. Its primary significance lies in demonstrating that the Kernel Polynomial Method enables the calculation of real-space second Chern numbers for system sizes ($30^4$) that are orders of magnitude larger than previous real-space approaches ($8^4$). This scale is sufficient to observe the exponential convergence toward quantized values in 4D systems, validating the method's utility for disordered topological insulators.

Regarding the 6D extension, the authors maintain a modest tone, acknowledging that while the method captures the qualitative behavior of the third Chern number, current computational limits prevent full quantization. They suggest that future integration with tensor network techniques may be necessary to access the larger system sizes required for quantized $C_3$ values in 6D. The work does not propose new experimental setups but rather provides a robust numerical tool for analyzing existing theoretical models in the presence of disorder.