Fractional charge bound to a vortex in topological crystalline insulators

1. Fractional charge bound to a vortex in topological crystalline insulators arXiv:1903.02737 Akira Furusaki (RIKEN) Eunwoo Lee and Bohm-Jung Yang Seoul National University Euler Symposium on Theoretical and Mathematical Physics

2. 理化学研究所　RIKEN established in 1917 Center for Emergent Matter Science K computer Tomonaga worked in RIKEN in 1930s. 3000 researchers including postdocs Dirac and Heisenberg visited RIKEN and Univ. of Tokyo in 1930.

3. Introduction symmetry protected topological phases • A unique ground state and gapped excitations (periodic boundary conditions) • Gapless excitations on the boundaries • Topological index • Classification depends on symmetry and spatial dimension. Examples of free-fermion systems: • Integer quantum Hall states (2D) • Quantum spin Hall insulators (2D, time-reversal symmetry) • Topological insulators (3D, time-reversal symmetry) • ……. • p-wave superconductors • ……..

4. Energy band structure: topological numbers (e.g., winding number) Band structures are said to be topologically equivalent, if they can be continuously deformed into one another without closing the energy gap. , , , , ….

5. Table of topological insulators/superconductors for d=1,2,3 10 Symmetry Classes TRS PHS CS d=1 d=2 d=3 QSHE d+id SC p+ip SC 3He-B Z2TI IQHE A (unitary) AI (orthogonal) AII (symplectic) • - Z -- • - -- -- • - Z2Z2 polyacetylene (SSH) 0 0 1 +1 +1 1 1 1 1 0 0 0 +1 0 0 1 0 0 p SC Standard (Wigner-Dyson) AIII (chiral unitary) BDI (chiral orthogonal) CII (chiralsymplectic) Z -- Z Z -- -- Z -- Z2 Chiral (Verbaarschot) Z2 Z -- -- Z -- Z2Z2 Z -- -- Z D (p-wave SC) C (d-wave SC) DIII (p-wave TRS SC) CI (d-wave TRS SC) 0 +1 0 0 1 0 1 +1 1 +1 1 1 BdG Altland & Zirnbauer, PRB (1997) Schnyder, Ryu, AF, and Ludwig, PRB (2008)

6. Crystalline symmetries of solids (230 space group symmetries) topological crystalline insulators Example: SnTe (Chern numbers defined on mirror-invariant planes in BZ) Topological quantum chemistry (Bernevig et al., …) Symmetry-based indicators (Po, Watanabe, Vishwanath, Fang, ….) Higher-order topological insulators (Schindler et al., Langbehn et al., …, 2017) 3rd order TI 2nd order TI 2D 3D

7. Plan • A zero mode localized at a topological defect • 1D: a domain wall in the Su-Schrieffer-Heeger (SSH) model • 2D: a vortex of Kekule textured graphene the Hou-Chamon-Mudry model • Generalization

8. Su-Schrieffer-Heeger(SSH) model … … spinless fermions hopping on the 1D lattice Two insulating phases … … Su, Schrieffer, and Heeger 1979 1/2 charge at a domain wall How to describe the emergence of a fractional charge bound to a domain wall? ( )

9. A A B B A unit cell Time-reversal symmetry: : complex conjugation “Chiral symmetry” (sublattice symmetry): Inversion symmetry :

10. 1st approach: Low-energy theory (Dirac Hamiltonian) Consider the low energy Hamiltonian of SSH model changes sign at the domain wall. There is a zeromode: soliton charge quantized into ! Jackiw and Rebbi, 1976 ( )

11. 2nd approach: bulk topological property … … Under inversion symmetry, electric polarization is quantized to or (). Nontopological Wannier Center Wannier Center … … Topological Wannier Center (mod 1) fractional bound charge at the DW ( ) If we break chiral symmetry, then the bound state remains but .

12. K K’ K’ K K’ K E ky K K’ kx 2D: graphene with kekule texture Graphene: a tight-binding model of spinless fermions hopping on the honeycomb lattice A B A B

13. Graphene with Kekule texture A sublattice Bsublattice from Hou et al., PRL 2007 (constant) strong bond An energy gap is open at the Dirac points, . weak bond

14. Zero mode in a vortex ofthe Kekule texture (Hou-Chamon-Mudry model) Hou, Chamon and Mudry, 2007 A fractional charge bound to a vortex in Kekule textured graphene. When , charge is localized at the vortex core. How can we understand the emergence of fractional charge?

15. 1st approach: Low energy Hamiltonian (HCM 2007) Low-energy Hamiltonian A vortex in the Dirac (Kekule) mass: Zero mode Jackiw & Rossi (1981) normalizable zero mode states (index theorem) E. J. Weinberg, PRD (1981), J. Goldstone and F. Wilczek, PRL (1981). Electron fractionalization: ( )

16. … … Canweexplainthe fractional charge in terms of bulk topological property? 2nd approach: bulk topological property Remember the 1D case: Nontopological Under inversion symmetry, electric polarization is quantized to or (). Wannier Center … … Wannier Center Topological Wannier Center (mod 1) fractional bound charge at DW! ( )

17. … … Canweexplainthe fractional charge in terms of bulk topological property? 2nd approach: bulk topological property Compare two different uniformly textured structures = const. Trivial (), : strong : weak … Inter > intra Intra > inter Both structures are invariant under rotation . … ( rotation ) Topological (, index

18. Kekule textured graphene is a second order TI. : 2nd order topological insulator : # of occupied states with eigenvalue of at time-reversal invariant momentum : strong : weak parity of # of times nested Wilson loop spectra cross : trivial insulator Wilson loop operator

19. Kekule textured graphene is a second order TI! electron density 3.5 3 2.5 : 2nd order topological insulator E : strong : weak 28 28 0 0 electron density : trivial insulator 3.1 3 2.9 E 28 28 0 0 Example: unit cells with open boundaries

20. Spectral flow 3 0 -3 - 0

21. Topological spectral flow around the vortex The vortex can be understood as a charge pumping process of a higher order topological insulator along the adiabatic variation of . : , : The spectral flow implies the existence of a bound state at the vortex core E E 3 n=1 0 -3 -> a half charge at the vortex! - 0

22. 1 0.5 0 -0.5 -1 3 a vortex-antivortex pair in the background of the Kekule pattern 0 -3 - 0

23. Topological vortex & Axion insulator trivial topological Consider a fictitious 3D insulator: This 3D insulator is a 2nd order topological insulator, called a 3D axion insulator. Topological invariant of axioninsulators magneto-electric polarizability n=1 BJ Wieder and BA Bernevig, 1810.02373 J Ahn and BJ Yang, 1810.05363

24. Generalization trivial insulator A vortex in a 2D crystalline topological insulator carries a fractional charge, if the 2D vortex Hamiltonian can be implemented as a 3D axion insulator Hamiltonian . quadrupole moment insulator A : strong 2 : weak Axion insulator protected by symmetry n=1 C.-Y. Houet al, 2008 G van Miert, 2018 2

25. Classification Table The correspondence between the symmetry of exhibiting quantized magneto-electric polarizability and that of the Hamiltonian.

26. Summary • The fractional charge localized at a vortex is related to the change in the 2D bulk topological invariant (defined at ) around the vortex. • The associated spectral flow during the adiabatic process of changing corresponds to chiral hinge modes of a 3D axion insulator. • Generalization and classification.