[Paper Review] Rotation Anomaly and Topological Crystalline Insulators

Chen Fang, Liang Fu|arXiv (Cornell University)|Sep 6, 2017

Topological Materials and PhenomenaPhysics and Astronomy47 references52 citations

TL;DR

The paper proves a rotation-multiplication anomaly for 2D systems with time-reversal and n-fold rotation, explains its bulk-boundary implications, and introduces new classes of topological crystalline insulators protected by C2, C4, and C6 rotations.

ABSTRACT

We show that in the presence of $n$-fold rotation symmetries and time-reversal symmetry, the number of fermion flavors must be a multiple of $2n$ ($n=2,3,4,6$) on two-dimensional lattices, a stronger version of the well-known fermion doubling theorem in the presence of only time-reversal symmetry. The violation of the multiplication theorems indicates anomalies, and may only occur on the surface of new classes of topological crystalline insulators. Put on a cylinder, these states have $n$ Dirac cones on the top and on the bottom surfaces, connected by $n$ helical edge modes on the side surface.

Motivation & Objective

• Establish a fermion multiplication (anomaly) theorem for 2D systems with time-reversal and n-fold rotation
• Show that certain Dirac cone configurations are anomalous and surface-bound to 3D TCIs
• Construct bulk Z2 invariants and real-space interpretations for rotation-protected TCIs
• Propose material platforms (e.g., SnTe, Sr3PbO) where anomalous surface states can be realized
• Connect dimensional reduction to understand interacting generalizations of rotation-protected SPT phases

Proposed method

• Analyze Berry phases along symmetry-determined contours in the Brillouin zone to derive Dirac-cone counting constraints
• Use symmetry eigenvalues at high-symmetry points to compute Θ2, Θ4, Θ6 and show they equal 0 mod 2π
• Construct surface Hamiltonians by superposing two TI surface states with opposite helicities under C2, C4, C6 to realize anomalous TCIs
• Develop real-space mass-domain-wall arguments to explain side-edge modes and surface domain walls
• Provide bulk Z2 invariants via Wannier-center flows between kz=0 and kz=π slices
• Outline a dimensional-reduction framework to relate 3D rotation-TCI to stacks of 2D TIs
• Discuss candidate materials and perturbations that preserve Cn and T while realizing anomalous surface states

Experimental results

Research questions

• RQ1 What constraints do C2, C4, and C6 rotation symmetries plus time-reversal impose on the number and arrangement of Dirac cones in 2D?
• RQ2 Can configurations with a single Dirac cone (per symmetry-related multiplet) exist in a symmetric 2D lattice, or are they anomalous?
• RQ3 How are the anomalous 2D surface states of rotation-protected TCIs manifested on the 3D bulk boundary (top/bottom surfaces and side edges)?
• RQ4 What are the bulk Z2 invariants and Wannier-center flows that classify these rotation-protected TCIs?
• RQ5 Which real materials can realize the predicted anomalous surface states protected by rotation and time-reversal, and how can perturbations affect them?

Key findings

• In 2D with T^2 = -1 and Cn rotation (n=2,4,6), the number of stable massless Dirac fermions must be a multiple of 4m (m=1,2,3 for n=2,4,6), enforcing an anomaly for a single Dirac cone per symmetry family
• Anomalous Dirac configurations (one Dirac cone enclosed by symmetry-determined loops) can only exist as surface states of 3D TCIs protected by rotation and time-reversal
• Surface states for C2, C4, C6 TCIs consist of n Dirac cones on top and bottom surfaces connected by n helical edge modes on the side surface; these edge modes are symmetry-related and cannot be gapped without breaking the protecting symmetries
• Two constructive pictures are provided: (1) superposition of two TIs with opposite helicity to realize surface states, predicting a Z2×Z2 classification for C2 and analogous tables for C4/C6; (2) real-space domain-wall mass pictures yield protected helical modes along domain walls
• Bulk invariants are captured by Z2 Wannier-center flows between kz=0 and kz=π planes, with specific modulo conditions in Wyckoff positions
• Dimensional-reduction framework shows the 3D rotation-protected TCIs as stacks of 2D TIs arranged around the rotation axis; this connects to both non-interacting and interacting SPT perspectives
• Proposes material realizations in SnTe surfaces (110 and 001) and Sr3PbO-based systems, with perturbations like strain retaining C2/T while breaking mirror symmetry