Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Rui Chen, Bin Zhou

TL;DR

This work addresses real-space computation of higher-dimensional Chern numbers in disordered, non-translationally invariant systems. It adopts the kernel polynomial method to approximate the projector and compute the real-space second Chern number $C_2$ for a 4D Wilson-Dirac model, and it presents an exploratory real-space calculation of the third Chern number $C_3$ in 6D. The key results show that $C_2$ converges to the quantized value with exponential finite-size scaling up to system sizes of $30^4$ and that disorder phase diagrams align with the self-consistent Born approximation, while $C_3$ exhibits qualitative agreement but remains non-quantized due to finite-size effects; tensor networks are suggested to enable larger 6D calculations. Collectively, the paper establishes a scalable real-space framework for higher-dimensional topological invariants and points toward future methods to achieve quantization in 6D.

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.

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

TL;DR

This work addresses real-space computation of higher-dimensional Chern numbers in disordered, non-translationally invariant systems. It adopts the kernel polynomial method to approximate the projector and compute the real-space second Chern number for a 4D Wilson-Dirac model, and it presents an exploratory real-space calculation of the third Chern number in 6D. The key results show that converges to the quantized value with exponential finite-size scaling up to system sizes of and that disorder phase diagrams align with the self-consistent Born approximation, while exhibits qualitative agreement but remains non-quantized due to finite-size effects; tensor networks are suggested to enable larger 6D calculations. Collectively, the paper establishes a scalable real-space framework for higher-dimensional topological invariants and points toward future methods to achieve quantization in 6D.

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 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.

Paper Structure

This paper contains 11 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and Method
  3. Wilson-Dirac model
  4. Real-space Chern number
  5. Kernel polynomial method
  6. Second Chern number in 4D
  7. Clean Limit
  8. Disorder effect
  9. Third Chern number in 6D
  10. Conclusion
  11. Born approximation

Figures (3)

  • Figure 1: (a) Numerically computed real-space second Chern number $C_2$ as a function of mass parameter $m$ with with different side lengths $L=4$ (gray), $L=12$ (blue), and $L=20$ (red), respectively. The green curve represents the theoretical prediction obtained from momentum-space calculations. (b) Logarithmic plot of the absolute deviation from the quantized value, $\ln\left(\left|C_2+3\right|\right)$, as a function of system size for $m=-1$. The red dashed line indicates a linear fit, demonstrating exponential convergence of the real-space Chern number toward the quantized value $C_2=-3$.
  • Figure 2: (a)-(b) Numerically calculated real-space second Chern number as a function of disorder strength $W$ for different side length $L$ with (a) $m=-1$ and (b) $m=-3$. Here, the error bar indicates the standard deviation for 200 samples. (c) The disorder averaged $C_2$ as functions of $W$ and $m$ with $L=12$. Here, the black dashed lines are obtained by Born approximation in Appendix \ref{['Sec:Born']}.
  • Figure 3: (a) Numerically computed real-space third Chern number $C_3$ as a function of mass parameter $m$ with with different side lengths $L=3$ (gray), $L=4$ (blue), and $L=5$ (red), respectively. The green curve represents the theoretical prediction obtained from momentum-space calculations.