Wannier representation of Z_2 topological insulators (1009.1415v2)

Abstract: We consider the problem of constructing Wannier functions for Z_2 topological insulators in two dimensions. It is well known that there is a topological obstruction to the construction of Wannier functions for Chern insulators, but it has been unclear whether this is also true for the Z_2 case. We consider the Kane-Mele tight-binding model, which exhibits both normal (Z_2-even) and topological (Z_2-odd) phases as a function of the model parameters. In the Z_2-even phase, the usual projection-based scheme can be used to build the Wannier representation. In the Z_2-odd phase, we do find a topological obstruction, but only if one insists on choosing a gauge that respects the time-reversal symmetry, corresponding to Wannier functions that come in time-reversal pairs. If instead we are willing to violate this gauge condition, a Wannier representation becomes possible. We present an explicit construction of Wannier functions for the Z_2-odd phase of the Kane-Mele model via a modified projection scheme followed by maximal localization, and confirm that these Wannier functions correctly represent the electric polarization and other electronic properties of the insulator.

• The paper demonstrates that a topological obstruction prevents constructing Wannier functions for Z₂-odd phases under a time-reversal symmetric gauge.
• It introduces a modified projection scheme with maximal localization to overcome this barrier and accurately capture electronic properties.
• Numerical analysis using the Kane–Mele model validates the approach by reproducing expected polarization and topological characteristics.

Wannier Representation of $\mathbb{Z}_2$ Topological Insulators

The paper by Soluyanov and Vanderbilt explores the construction of Wannier functions in two-dimensional $\mathbb{Z}_2$ topological insulators, specifically within the context of the Kane-Mele tight-binding model. This model is of particular interest because it includes both conventional insulating ($\mathbb{Z}_2$-even) and topological insulating ($\mathbb{Z}_2$-odd) phases, depending on the chosen parameters.

The primary question addressed in this work is whether there exists a similar topological obstruction to constructing Wannier functions for $\mathbb{Z}_2$ topological insulators as with Chern insulators, where such an obstruction is well established. The authors confirm that a topological obstruction does indeed exist for the $\mathbb{Z}_2$-odd phase if one insists on maintaining a gauge that respects time-reversal symmetry. However, they present an alternative approach: by allowing a gauge that violates this condition, one can construct a Wannier representation even in the $\mathbb{Z}_2$-odd regime.

The paper involves a detailed analytical and numerical exploration of the Kane-Mele model. It is shown that in the $\mathbb{Z}_2$-even phase, Wannier functions can be constructed using conventional methods. However, in the $\mathbb{Z}_2$-odd phase, implementing a time-reversal-symmetric gauge leads to singularities, rendering the construction impossible. The authors resolve this by introducing a modified projection scheme, followed by maximal localization, which ultimately yields Wannier functions that accurately describe the electronic properties of the $\mathbb{Z}_2$-odd insulator.

Key numerical results illustrate these findings, such as the behavior of the determinant of the overlap matrix used in the Wannier construction, which confirms the obstruction under the symmetric gauge for the topologically non-trivial phase. The authors further confirm the validity of the constructed Wannier functions by demonstrating that they reproduce the expected electric polarization and electronic properties consistent with the topology of the insulator.

From a theoretical perspective, this work highlights the nuanced differences in the topological characterization between $\mathbb{Z}_2$ and Chern insulators. While both categories of materials are bulk insulators with protected edge states, the underlying topological invariants dictate the feasible representations and computational methodologies one can employ.

Practically, the implications of this research extend to better understanding the properties of topological insulators, which have received significant interest due to their robust edge states protected against disorder. This understanding is crucial for potential applications in spintronics and quantum computing, where the manipulation of topological states could lead to breakthroughs in device performance and functionality.

Looking to the future, developments in AI and machine learning may provide new computational tools to extend the methods utilized in this paper, possibly allowing for faster or more accurate determinations of Wannier functions across a range of complex materials. Additionally, exploring the generalization of these findings to three-dimensional topological insulators opens new avenues in both theoretical studies and practical applications. This paper lays a foundation for these explorations by tackling the critical problem of Wannier representation for $\mathbb{Z}_2$ topological insulators.