[Paper Review] $(d-2)$-dimensional edge states of rotation symmetry protected topological states

Zhida Song, Zhong Fang+1 more|arXiv (Cornell University)|Aug 9, 2017

Topological Materials and Phenomena57 citations

TL;DR

The paper demonstrates that in 2D and 3D rotation-symmetric, time-reversal-invariant gapped systems, nontrivial topology manifests as $(d-2)$-dimensional edge states, and provides both noninteracting and interacting constructions for these edge modes.

ABSTRACT

We study fourfold rotation invariant gapped topological systems with time-reversal symmetry in two and three dimensions ($d=2,3$). We show that in both cases nontrivial topology is manifested by the presence of the $(d-2)$-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands, and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through explicit construction of microscopic models having robust $(d-2)$-dimensional edge states.

Motivation & Objective

• Motivate and identify a new class of SPTs protected by fourfold rotation symmetry and time-reversal symmetry.
• Characterize how $(d-2)$-dimensional edge states arise from Wannier center (WC) configurations in 2D and 3D.
• Extend the framework to strongly interacting systems via coupled wires constructions.
• Provide practical indicators, including a Fu-Kane-like formula, to diagnose the nontrivial WC flow in presence of inversion symmetry.

Proposed method

• Use the Wannier center perspective to explain 0D corner states in 2D and 1D helical edge modes in 3D.
• Introduce a $ ext{WC}$-flow classification by comparing k_z slices and identifying a $\mathbb{Z}_2$ invariant.
• Derive a Fu–Kane-like formula that relates the invariant to rotation and inversion eigenvalues at high-symmetry momenta.
• Construct explicit 2D and 3D lattice models showcasing $(d-2)$-dimensional edge states with C4 and T symmetries.
• Employ a coupled wires approach to generalize the edge-state construction to strongly interacting SPTs.
• Discuss edge-state robustness under symmetry-breaking perturbations and outline experimental signatures.

Experimental results

Research questions

• RQ1 Can a $ ext{WC}$-flow from $k_z=0$ to $k_z=\pi$ define a robust $\mathbb{Z}_2$ invariant signaling $(d-2)$-dimensional edge states?
• RQ2 Under inversion symmetry, can a Fu–Kane-like formula be established to diagnose the $ ext{WC}$ flow from symmetry eigenvalues alone?
• RQ3 Do $(d-2)$-dimensional edge modes persist in strongly interacting SPTs via a coupled-wires construction?
• RQ4 How do the edge modes change when mirror symmetries are broken but $C_4$ and time-reversal symmetries remain?
• RQ5 What are the experimental signatures of the fourfold-rotation protected $(d-2)$-dimensional edge states?

Key findings

• A 2D system with $C_4$ and T symmetry hosts 0D corner states arising from a mismatch between Wannier centers and atom positions.
• A 3D system exhibits a $ ext{Z}_2$-type WC flow between $k_z=0$ and $k_z=\pi$ slices, yielding four helical edge modes on side surfaces.
• A Fu–Kane-like formula is derived that connects the $ ext{Z}_2$ WC flow invariant to rotation and inversion eigenvalues at high-symmetry momenta for space group $P4/m$.
• Explicit 2D and 3D lattice models demonstrate the presence of robust $(d-2)$-dimensional edge states protected by $C_4$ and time-reversal symmetry.
• A coupled-wires construction extends the framework to strongly interacting bosonic and fermionic SPT states with $(d-2)$-dimensional edge modes.
• Edge modes are not strictly pinned to corners or hinges and can exist on smooth side surfaces as long as $C_4$ symmetry is preserved.