[論文レビュー] The Axion Insulator as a Pump of Fragile Topology

Benjamin J. Wieder, B. Andrei Bernevig|arXiv (Cornell University)|Oct 4, 2018

Dark Matter and Cosmic Phenomena被引用数 74

ひとこと要約

本論文はアクシオン絶縁体（AXI）を、2Dの壊れやすいトポロジー相の周期ポンプとして再解釈し、ネストされた Wilson loop を用いてバルク磁気電気応答とコーナー状態/ヒンジ状態を結びつける。

ABSTRACT

The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $θ=π$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of bulk bands displays no Wilson loop winding, exhibits chiral hinge states, and does not pump any previously identified quantity. Crucially, as the AXI exhibits the topological angle $θ=π$, its occupied bands cannot be formed into maximally localized symmetric Wannier functions, despite its absence of Wilson loop winding. In this letter, we revisit the AXI from the perspective of the recently introduced notion of "fragile" topology, and discover that it in fact can be generically expressed as the cyclic pumping of a "trivialized" fragile phase: a 2D inversion-symmetric insulator with no Wilson loop winding which nevertheless carries a nontrivial topological index, the nested Berry phase $γ_{2}$. We numerically show that the nontrivial value $γ_{2}=π$ indicates the presence of anomalous 0D corner charges in a 2D insulator, and therefore, that the chiral pumping of $γ_{2}$ in a 3D AXI corresponds to the presence of chiral hinge states. We also briefly generalize our results to time-reversal-symmetric higher-order TIs, and discuss the related appearance of nontrivial $γ_{2}$ protected by $C_{2} imes\mathcal{T}$ symmetry in twisted bilayer graphene, and its implications for the presence of 0D corner states.

研究の動機と目的

• Motivate and characterize AXIs as 3D magnetic topological insulators with theta = pi.
• Investigate whether AXIs exhibit Wilson loop winding and Wannier localization with many occupied bands.
• Demonstrate that AXIs realiz a cyclic pumping of a fragile 2D phase with anomalous corner charges.
• Relate nested Wilson loop flow to chiral hinge modes and magnetoelectric response.
• Generalize findings to time-reversal symmetric higher-order TIs and related C2z x T symmetric cases.

提案手法

• Construct 3D tight-binding models of inversion- and time-reversal-symmetric TIs as parent phases.
• Introduce bulk z-directed ferromagnetic potential to gap surface states and obtain AXI phase.
• Compute x-directed Wilson loops and nested y-directed Wilson loops to obtain gamma1 and gamma2.
• Show that fragile topology can be trivialized by adding inert bands, enabling nested Wilson loop analysis.
• Relate the nested Wilson loop determinant gamma2 to the magnetoelectric angle theta via theta/pi = gamma2 mod 2.
• Analyze hinge and corner states in rod and slab geometries to link bulk invariants to 1D/0D boundary modes.

実験結果

リサーチクエスチョン

• RQ1 Does an AXI with many occupied bands display Wilson loop winding or is it Wannierizable?
• RQ2 Can nested Wilson loops reveal a fragile topological pumping in AXIs despite absence of surface states?
• RQ3 How is theta = pi manifested in AXIs through bulk invariants like gamma2 and hinge/corner modes?
• RQ4 Is AXI topology robust under addition of trivial bands and symmetry relaxations (I, T, or C2z x T)?

主な発見

• AXIs can be expressed as cyclic pumping of a 2D fragile phase with anomalous corner charges.
• The nested Wilson loop winding, visible after trivial bands are added, signals the fragile topology in AXIs.
• Theta equals pi arises from the winding of the nested Berry phase gamma2 (mod 2).
• AXIs exhibit chiral hinge modes consistent with a nontrivial pumped fragile phase.
• An AXI slab behaves as an isolated Chern insulator with slab C = +1 winding in Wilson loop analysis.
• The framework extends to C2z x T symmetric AXIs and to related fragile phases in twisted bilayer graphene.