Topological marker in three dimensions based on kernel polynomial method

TL;DR

The paper develops a scalable framework for quantifying how disorder affects topological order in higher dimensions by combining the universal topological marker with the kernel polynomial method (KPM). It defines a topological operator $\hat{\cal C}$ constructed from projectors and position operators and extracts both local and nonlocal markers to capture bulk invariants and correlations, enabling efficient calculations in large systems. In a 3D class AII TI model, it shows robustness of the spatially averaged marker to impurities that modify nonzero matrix elements, and demonstrates a disorder-induced smooth crossover between topological phases via mass-term impurities, with the nonlocal marker revealing criticality through a diverging decay length as the transition is approached. Overall, the method enables high-precision exploration of topological quantum criticality in higher dimensions and can be extended to other symmetry classes and dimensionalities, enhancing the practical study of disorder effects in topological materials.

Abstract

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.

Figures (4)

• Figure 1: Scaling of $\Delta \mathcal{C}$ with system size with the red vertical bars denoting the error in the extrapolated value. Figures (a), (b), (c) and (d) (left and central panel) correspond to impurities varying non-zero elements of the Hamiltonian matrix whereas figures (e) and (f) correspond to impurities varying zero elements of the Hamiltonian matrix (a) Onsite impurity (b)$A_0- \Gamma^1$ impurity (c) $A_0-\Gamma^2$ impurity, (d) $B_0$ impurity, (e) $S_x$ impurity, (f) Fourth nearest neighbour hopping
• Figure 2: Spatial pattern of the marker near the impurity site. In (a), impurity is introduced by modifying the hopping strength ($A_0$) for the hopping from the central site to the nearest neighbour in $+y$ direction whereas in (b), a single additional hopping to the fourth nearest neighbour (2a units) in $+y$ direction from the central site is introduced.
• Figure 3: Phase diagram as a function of concentration of impurities for different mass parameters. Top panel : $M_0=-0.5$. Bottom panel : $M_0=-2.0$
• Figure 4: Non-local topological marker for different values of mass parameter $M$