1. Soliton Defects and Quantum Numbers in One-dimensional Topological Three-band Hamiltonian Based on arXiv:1306.0998 Gyungchoon Go, Kyeong Tae Kang, Jung Hoon Han (SKKU)

2. Contents 1. Introduction Jackiw-Rebbi theory and electron fractionalization in two-band model 2. 1D Topological Three-band model Lattice construction Boundary states Continuum theory and quantum numbers (QM) QFT calculation 3. Summary

3. Introduction

4. Su, Schrieffer and Heeger PRL 42, 1698 (1979) Polyacetylene Polyacetylene story http://www.mhhe.com/physsci/chemistry/carey5e/Ch02/ch2-3-2.html

5. Polyacetylene story Peierls instability : Energy of one-dimensional lattice can be lowered by imposing the periodic lattice distortion Two degenerate vacua and two sub-lattices : displacement field

6. Polyacetylene story Soliton (anti-soliton) : localized solution with finite energy R. Jackiw arXiv:math-ph/0503039 (2005)

7. Continuum theory (Jackiw-Rebbi theory) Jackiw and Rebbi PRD 13, 3398 (1976) Energy dispersion : Question : If m(x) behaves like the soliton background, What happens in the band structure?

8. Continuum theory (Jackiw-Rebbi theory) Answer : Localized zero mode (E=0) is induced at the soliton defect From the presence of the particle-hole symmetry we can expect that the only possible mid-gap state is the zero mode.

9. Continuum theory (Jackiw-Rebbi theory) To compute the localized charge Schrodinger equation Charge density, density of state Localized charge : difference of total charge with and without soliton profile

10. Charge of the localized state Electron fractionalization One-half vacancy of the valence band Continuum theory (Jackiw-Rebbi theory)

11. In the absence of the Particle-hole symmetry Jackiw and Semenoff PRL 50, 439 (1983), M. Rice and E. Mele PRL 49, 1455 (1982) Adding one more mass term If there is a soliton non-zero mode Fractional charge On-site energy difference

12. Quauntum field theory calculation Goldstone and Wilczek PRL 47, 986 (1981) Goldstone-Wilczek method Chiral rotation (RM model )

13. Quauntum field theory calculation Induced current

14. For and Quauntum field theory calculation Induced charge QM result

15. Motivation All of these results are based on two-level system. In condensed matter system, we can consider any number of bands. Can we calculate the same quantities for the three- band model?

16. 1D Topological Three-band Model

17. Model construction General three-band model

18. Model construction Corresponding the one-dimensional lattice model? We always have the zero-energy flat band Diamond-Chain lattice Gulacsi, Kampf,Vollhardt PRL 99, 026404 (2007) ‘

19. Model construction Considering the flux model

20. Model construction Generalizing the Rice-Mele or Jackiw-Semenoff model, we can construct

21. Topological index Winding number In a model with non-trivial topological index there ‘may’ exist localized states near boundaries. Ning Wu, PLA 376, 3530 (2012)

22. Boundary state Localization condition

23. Continuum soliton Soliton configuration

24. Continuum soliton Localized solution

25. Quantum number of the soliton Comparing the number of state Two-band result Number of fractional lost from the valence band varies two times faster than two band case

26. Quantum number of the soliton

27. Quantum field theory calculation Two-band

28. Quantum field theory calculation Expectation of current operator Two-band

29. Quantum field theory calculation Since we set the Fermi energy slightly below the flat band, E=0 pole is not included in the contour integral

30. Quantum field theory calculation

31. For Quantum field theory calculation QM result

32. Summary and Discussion

33. Thank you

34. Boundary state Real-space Hamiltonian Schrodinger equation Boundary condition and localization condition : localized at one boundary