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Soliton Defects and Quantum Numbers in One-dimensional Topological Three-band Hamiltonian

Soliton Defects and Quantum Numbers in One-dimensional Topological Three-band Hamiltonian. Gyungchoon Go , Kyeong Tae Kang, Jung Hoon Han. (SKKU). Based on arXiv:1306.0998. Contents. 1. Introduction Jackiw-Rebbi theory and electron fractionalization in two-band model

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Soliton Defects and Quantum Numbers in One-dimensional Topological Three-band Hamiltonian

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

  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. Polyacetylene story Su, Schrieffer and HeegerPRL 42, 1698 (1979) Polyacetylene • Quasi-one-dimensional lattice • 3 valence electron , 1 conduction electron 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. JackiwarXiv:math-ph/0503039(2005)

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

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

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

  13. Quauntum field theory calculation Induced current

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

  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 We always have the zero-energy flat band Corresponding the one-dimensional lattice model? Diamond-Chain lattice ‘ Gulacsi,Kampf ,VollhardtPRL 99, 026404 (2007)

  19. Model construction Considering the flux model

  20. Model construction Generalizing the Rice-Meleor Jackiw-Semenoff model, we can construct comes from the Peierls instability

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

  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 Number of fractional lost from the valence band varies two times faster than two band case Two-band result

  26. Quantum number of the soliton • No state loss from the flat-band • Sudden change from 0 to 1 at: flat-band • At , it is natural to take average : ½ reproduced in the case of PHS

  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. Quantum field theory calculation For QM result

  32. Summary and Discussion We constructed the one-dimensional three-band lattice model with non-trivial topological index. In the presence of solitonic background the fractional charge is induced. From the existence of the zero-energy flat band, we know that the fractional number is quite different from the two-band model. QFT result agrees with the QM result. Possible candidate : -form of PdCl2? For general Hamiltonian?

  33. Thank you

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

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