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

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.

Figures (5)

• Figure 1: Schematic illustration of the adaptive mesh refinement strategy for evaluating the second Chern number. The background color gradient represents the distribution of $\epsilon^{\mu\nu\rho\sigma} \text{Tr} [\mathcal{F}_{\mu\nu} \mathcal{F}_{\rho\sigma}]$, where "High" regions (red) indicate sharp singularities. (a) Coarse estimate: The integration contribution of a hypercube is initially estimated by evaluating the Chern density at its geometric center. (b) Fine estimate: The hypercube is subdivided into 16 sub-cells to obtain a refined integral value. The discrepancy between the coarse and fine estimates serves as a local error indicator. By comparing (a) and (b), a local error indicator is constructed. (c) Adaptive refinement: Regions exhibiting a high error are subdivided (labeled by red). This comparison process is then applied iteratively between the new sub-cells and their subsequent refinements, as shown by the transition from (b) to (c), until the global error converges to the desired tolerance. This dynamic allocation of grid points ensures both numerical stability and computational efficiency.
• Figure 2: (a)-(c) The second Chern number $C_2$ calculated as a function of the parameter $m/c$ using different grid sizes $N$, demonstrating the overall stability of the three approaches. Insets provide a magnified view near the plateau edges. (d)-(g) Convergence behavior of the numerical error $\lg(\Delta C_2)$ relative to the number of diagonalizations $\lg(N_k)$ at various distances from the phase transition point: (d) far from the transition ($m/c = -3.5$), (e) approaching the transition ($m/c = -3.9$), (f) near the critical point ($m/c = -3.99$), and (g) in the immediate vicinity of the transition ($m/c = -3.999$). Here, $\Delta C_2 = |C_2 - 1|$ with $1$ being the exact theoretical value of the topological invariant and $N_k = N^4$ for Method I and Method II. The horizontal black dashed lines in (d-g) indicate a threshold of $\Delta C_2 = 10^{-3}$, which we consider to be sufficiently quantized to define a topological phase. Here, $\lg x \equiv \log_{10} x$.
• Figure 3: The evolution of the second Chern number $C_2$ and the energy spectrum as a function of the Fermi energy $E_F$. The left vertical axis (red) represents the calculated $C_2$ values, where well-defined integer plateaus indicate the quantization of topological invariants within the spectral gaps. The right vertical axis (cyan), shows the DOS distribution. The correspondence between the $C_2$ plateaus and the vanishing DOS regions highlights the bulk-gap topological protection in this 4D quantum Hall system.
• Figure 4: Convergence analysis of the second Chern number calculation for the 4D quantum Hall system with rational magnetic fluxes $\phi_z = \phi_w = 1/13$. The vertical axis displays the logarithmic deviation $\lg(\Delta C_2)$, defined as $|\Delta C_2| = |C_{2}^{\text{calc}} - C_{2}^{\text{ideal}}|$, calculated at $E_F = -4.86$ (where $C_{2}^{\text{ideal}} = -7$). The horizontal axis $\lg(N_k)$ represents the computational cost, quantified by the total number of Hamiltonian diagonalizations. The horizontal black dashed lines indicate a threshold of $\Delta C_2 = 10^{-3}$, which we consider to be sufficiently quantized to define a topological phase.
• Figure 5: Adaptive Mesh Refinement for Second Chern Number