[Paper Review] Wilson-Loop Characterization of Inversion-Symmetric Topological Insulators

A. Alexandradinata, Xi Dai+1 more|arXiv (Cornell University)|Aug 21, 2012

Topological Materials and Phenomena3 citations

TL;DR

This paper establishes a $η^{–}$ classification for inversion-symmetric topological insulators in 1D and 2D, introducing a relative winding number $W$ protected solely by inversion symmetry. Using Wilson loops and Berry phases, it identifies a topological invariant analogous to the $ν_{2}$ invariant in time-reversal-symmetric systems, with $W$ quantized as a non-negative integer and determined by inversion eigenvalues at high-symmetry momenta.

ABSTRACT

The ground state of translationally-invariant insulators comprise bands which can assume topologically distinct structures. There are few known examples where this distinction is enforced by a point-group symmetry alone. In this paper we show that 1D and 2D insulators with the simplest point-group symmetry - inversion - have a $Z^{\geq}$ classification. In 2D, we identify a relative winding number that is solely protected by inversion symmetry. By analysis of Berry phases, we show that this invariant has similarities with the first Chern class (of time-reversal breaking insulators), but is more closely analogous to the $Z_2$ invariant (of time-reversal invariant insulators). Implications of our work are discussed in holonomy, the geometric-phase theory of polarization, the theory of maximally-localized Wannier functions, and in the entanglement spectrum.

Motivation & Objective

• To identify topological invariants in inversion-symmetric insulators that are protected solely by point-group symmetry.
• To establish a classification scheme for 1D and 2D topological insulators with only inversion symmetry.
• To define and characterize a new topological invariant, the relative winding number $W$, using Wilson loops and Berry phases.
• To connect the topological structure to geometric phases, polarization, and Wannier function localization.

Proposed method

• The study employs Wilson loops as gauge-invariant holonomies to probe the topological structure of occupied bands in momentum space.
• It analyzes the eigenspectrum of Wilson loops along closed paths in the Brillouin zone, focusing on eigenvalues fixed at $-1$ due to inversion symmetry.
• The relative winding number $W$ is defined as the net change in the phase winding of Wilson loop eigenvalues along a path, quantized as a non-negative integer.
• The method uses symmetry constraints at inversion-invariant momenta ($k=0,\pi$) to compute $W$ from the number of odd-parity bands at these points.
• It establishes a duality between $W$ and the Chern number $C_1$, showing that $W$ is robust under continuous deformation preserving inversion symmetry.
• The analysis is extended to Wannier center trajectories, proving that $W$ is isotropic in both $k_x$ and $k_y$ directions via symmetry and gauge invariance.

Figure 1: Non-Abelian Berry phases for four distinct insulators. Each figure contains three unit cells in an effective 1D-lattice along $\hat{x}$ ; we track the real-space trajectories of the phases (blue circles) in $\hat{x}$ as momentum $k_{y}$ is varied through two adiabatic cycles. (a) An insula

Experimental results

Research questions

• RQ1 Can a topological invariant be defined for inversion-symmetric insulators that is protected solely by point-group symmetry?
• RQ2 What is the nature of the topological invariant in 1D and 2D inversion-symmetric insulators, and how does it differ from invariants in time-reversal-breaking or time-reversal-invariant systems?
• RQ3 How is the relative winding number $W$ related to the Wilson loop eigenvalue spectrum and inversion eigenvalues at high-symmetry points?
• RQ4 Can the topological classification be expressed in terms of geometric phases and Wannier function localization?
• RQ5 Is the winding number $W$ invariant under adiabatic evolution and robust to perturbations preserving inversion symmetry?

Key findings

• The paper establishes a $\mathbb{Z}^{\geq}$ classification for 1D and 2D inversion-symmetric topological insulators, with the invariant $N_{(-1)}$ counting the number of Wilson loop eigenvalues fixed at $-1$.
• The relative winding number $W$ is identified as a topological invariant protected solely by inversion symmetry, quantized as a non-negative integer.
• The invariant $W$ is determined by the difference in the number of odd-parity bands at $k=0$ and $k=\pi$, with $W = \left| n_{(-)}(\pi) - n_{(-)}(0) \right|$ in the absence of complex-conjugate pairs.
• The relative winding number $W$ is shown to be isotropic: it is the same when computed along $k_x$ or $k_y$ paths, due to the combined constraints of inversion and time-reversal symmetries.
• The Wilson loop eigenvalue spectrum at $k=0$ and $k=\pi$ fully determines the topological invariant, linking it to the symmetry representation of occupied bands.
• The invariant $W$ is found to be analogous to the $\mathbb{Z}_2$ invariant in time-reversal-invariant systems, rather than the Chern number, due to its robustness under inversion symmetry alone.

Figure 2: The Wannier center flows for a 2D insulator with ${\cal I}$ symmetry. Each figure contains three unit cells in an effective 1D-lattice along $\hat{x}$ ; the positions of the primary sites are indicated by black dots. The ground state contains two occupied bands, hence there are two Wannier