Topological marker in three dimensions based on kernel polynomial method

The atomic-scale influence of disorder on the topological order can be quantified by a universal topological marker, although the practical calculation of the marker becomes numerically very costly in higher dimensions. We propose that for any symmetry class in higher dimensions, the topological marker can be calculated in a very efficient way by adopting the kernel polynomial method. Using class AII in three dimensions as an example, which is relevant to realistic topological insulators like Bi2Se3 and Bi2Te3, this method reveals the criteria for the invariance of topological order in the presence of disorder, as well as the possibility of a smooth cross over between two topological phases caused by disorder. In addition, the significantly enlarged system size in the numerical calculation implies that this method is capable of capturing the quantum criticality much closer to topological phase transitions, as demonstrated by a nonlocal topological marker.

💡 Research Summary

The paper “Topological marker in three dimensions based on kernel polynomial method” addresses a central computational bottleneck in the study of topological phases of matter: the evaluation of the universal real‑space topological marker in higher dimensions. While the marker provides a local diagnostic of topological order and its response to disorder, its direct calculation requires diagonalizing the full lattice Hamiltonian, an operation whose memory and time costs scale exponentially with system size and dimension. The authors propose to overcome this limitation by employing the Kernel Polynomial Method (KPM), a Chebyshev‑polynomial expansion technique that approximates spectral projectors without explicit diagonalization.

The authors first review the formalism of the universal topological marker for Dirac‑type Hamiltonians in arbitrary dimension D and symmetry class. The marker operator (\hat C) is built from alternating products of the occupied‑state projector (P) and its complement (Q=1-P) sandwiched between position operators (\hat r_i). For odd D the operator takes the form
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