Computing topological invariants without inversion symmetry

Abstract: We consider the problem of calculating the weak and strong topological indices in noncentrosymmetric time-reversal (T) invariant insulators. In 2D we use a gauge corresponding to hybrid Wannier functions that are maximally localized in one dimension. Although this gauge is not smoothly defined on the two-torus, it respects the T symmetry of the system and allows for a definition of the Z_2 invariant in terms of time-reversal polarization. In 3D we apply the 2D approach to T-invariant planes. We illustrate the method with first-principles calculations on GeTe and on HgTe under [001] and [111] strain. Our approach differs from ones used previously for noncentrosymmetric materials and should be easier to implement in ab initio code packages.

• The paper introduces an algorithm using hybrid Wannier functions to compute ℤ₂ invariants in noncentrosymmetric systems.
• It reformulates the invariant computation for both 2D and 3D insulators, validating the method with first-principles calculations.
• The method streamlines ab initio computations, accelerating the discovery of topological materials for quantum computing and spintronics.

Insights into Computing Topological Invariants without Inversion Symmetry

The paper by Alexey A. Soluyanov and David Vanderbilt addresses the computation of topological indices in noncentrosymmetric time-reversal invariant insulators. The focus is on efficiently determining the weak and strong $\mathbb{Z}_2$ topological invariants, which distinguish trivial insulators from topological ones. The approach is applicable to both two-dimensional and three-dimensional systems and is designed to be easily implementable in {\it ab initio} calculations.

Context and Motivation

The classification of time-reversal invariant insulators into topological and trivial categories has become an essential aspect of condensed matter physics. Traditional methods for computing topological invariants often rely on inversion symmetry or are computationally intensive, requiring visual inspections or extensive surface state calculations. The paper introduces an algorithm centered on hybrid Wannier functions, tailoring it for systems that lack inversion symmetry, thus expanding the computational toolkit available for identifying topological phases.

Methodological Advances

The authors harness the concept of time-reversal polarization (TRP) and hybrid Wannier charge centers to define the $\mathbb{Z}_2$ invariant. Their method focuses on the evolution of Wannier charge centers as an indicator of topological phase transitions.

1. Two-dimensional Case: The approach reformulates the problem in terms of Wannier charge centers, calculated by diagonalizing the position operator in the band subspace. The procedure tracks these centers as parameters are varied, determining the topological phase based on their evolution.
2. Three-dimensional Extension: The methodology is extended to three-dimensional systems by considering $\mathbb{Z}_2$ invariants of planes within the BZ, thus identifying the weak and strong topological indices. This requires evaluating slices of the BZ and aggregating results to classify the overall topology.

Numerical Implementation

A significant contribution is the development of a practical numerical algorithm that automates the computation of topological invariants by following the largest gap between Wannier charge centers across the Brillouin zone. The solution circumvents the need for visual inspection, making it scalable for systems with many bands.

The algorithm is validated through first-principles calculations on various materials, including centrosymmetric examples like Bi and Bi$_2$Se$_3$, and more challenging noncentrosymmetric examples such as GeTe and strained HgTe. The results concur with known topological classifications, reinforcing the method's reliability and efficiency.

Implications and Future Directions

The paper's methodology provides a robust, computationally efficient tool for identifying nontrivial topological phases in materials lacking inversion symmetry. By streamlining the process to fit within ab initio frameworks, this approach could significantly facilitate the discovery and analysis of topological materials in both existing databases and novel compounds.

Practically, this could invigorate exploration within materials science, particularly concerning quantum computing and spintronics, where topological materials hold promise. Theoretically, the automatable computation of topological indices opens new pathways for understanding complex band structures and may inspire further refinements and applications in various symmetry classes of insulators.

Future research could focus on applying these techniques to a broader range of materials and exploring the implications of structural, electronic, or external perturbations (such as strain or magnetic fields) on the topological properties of these systems. Additionally, expanding the framework to account for interactions or disorder could provide richer insights into real-world materials.

In conclusion, the approach delineated by Soluyanov and Vanderbilt broadens the scope and accessibility of topological classifications in condensed matter physics, particularly for noncentrosymmetric systems, paving the way for new investigations in both theoretical and applied domains.