Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials (1610.08983v2)

Abstract: The intense theoretical and experimental interest in topological insulators and semimetals has established band structure topology as a fundamental material property. Consequently, identifying band topologies has become an important, but often challenging problem, with no exhaustive solution at the present time. In this work we compile a series of techniques, some previously known, that allow for a solution to this problem for a large set of the possible band topologies. The method is based on tracking hybrid Wannier charge centers computed for relevant Bloch states, and it works at all levels of materials modeling: continuous k.p models, tight-binding models and ab initio calculations. We apply the method to compute and identify Chern, Z2 and crystalline topological insulators, as well as topological semimetal phases, using real material examples. Moreover, we provide a numerical implementation of this technique (the Z2Pack software package) that is ideally suited for high-throughput screening of materials databases for compounds with non-trivial topologies. We expect that our work will allow researchers to: (a) identify topological materials optimal for experimental probes, (b) classify existing compounds and (c) reveal materials that host novel, not yet described, topological states.

• The paper introduces Z2Pack, a computational toolkit that uses hybrid Wannier charge centers to calculate topological invariants like Z2 indices and Chern numbers.
• It demonstrates the method’s versatility by applying it to tight-binding, continuous k·p, and ab initio models for both insulators and semimetals.
• Z2Pack enables the identification of non-trivial topological phases in materials such as Bi₂Se₃, SnTe, and Weyl semimetals, aiding high-throughput discovery.

Overview of "Z2Pack: Numerical implementation of hybrid Wannier centers for identifying topological materials"

The paper by Gresch et al. presents a methodological and computational toolkit named Z2Pack, designed to address the complex task of identifying and classifying topological materials using hybrid Wannier charge centers (HWCCs). This toolkit is particularly relevant given the increasing importance of band topology in material characterization, driving advancements in the field of topological insulators and semimetals.

Z2Pack leverages hybrid Wannier functions—a mathematical construct that localizes in one dimension while remaining delocalized in others—to systematically calculate topological invariants across a variety of material models. The authors showcase its capability to handle continuous $k \cdot p$ models, tight-binding models, and even ab initio calculations, providing a versatile approach applicable across different computational settings.

Numerical Methodology

The core technique involves tracking HWCCs along closed paths in reciprocal space, clarifying the topological nature of the Bloch bands. Particularly, the method adeptly calculates characteristic invariants such as Chern numbers, $\mathbb{Z}_2$ indices, and mirror Chern numbers. This framework extends to the analysis of both insulators and semimetals, offering a unified approach to multiple topological classifications.

The theoretical backbone underpinning Z2Pack's calculations is robust. Importantly, the paper outlines that the computability of these invariants hinges on the smoothness and periodicity of the Bloch states over the Brillouin zone, using transformations and gauge choices to ensure HWCCs accurately reflect the system's topology.

Exemplary Applications

Gresch et al. furnish the reader with diverse examples to demonstrate Z2Pack's practical utility. They compute the $\mathbb{Z}_2$ invariant for the well-known strong topological insulator Bi$_2$Se$_3$, ascertain mirror Chern numbers in SnTe—a crystalline topological insulator—and identify topological phenomena in metallic systems such as Weyl and Dirac semimetals. They methodically explain how these calculations can reveal the presence of topological surface states, Fermi arcs, and other haLLMark signatures of non-trivial topology.

Implications and Future Developments

The implications of this work are multi-fold. Practically, Z2Pack facilitates the targeted discovery of novel topological materials by aiding high-throughput screening of materials databases, specifically seeking compounds with exotic electronic properties suitable for experimental and technological applications. Theoretically, it bridges various models, offering insights that may spur new theoretical developments in band topology.

Beyond immediate applications, the paper hints toward future developments in topological matter, whereby symmetry protection and other advanced classification criteria can be studied, potentially uncovering unconventional or elusive topological phases.

Overall, this work not only deepens our computational toolkit for studying topological phases but also exemplifies the ongoing synthesis of theory, computation, and material science in the pursuit of understanding the intricate properties of matter.