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The Puzzling Boundaries of Topological Quantum Matter

The Puzzling Boundaries of Topological Quantum Matter. Michael Levin. Collaborators : Chien-Hung Lin (University of Chicago) Chenjie Wang (University of Chicago) Zheng-Cheng Gu (Perimeter Institute). Protected boundary modes. Gapless excitations. Gapped excitations.

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The Puzzling Boundaries of Topological Quantum Matter

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  1. The Puzzling Boundaries of Topological Quantum Matter Michael Levin Collaborators: Chien-Hung Lin (University of Chicago) Chenjie Wang (University of Chicago) Zheng-Cheng Gu (Perimeter Institute)

  2. Protected boundary modes Gapless excitations Gapped excitations

  3. Outline I. Example: 2D topological insulators II. General non-interacting fermion systems III. The puzzle of interactions

  4. Model a Non-interacting electrons in a periodic potential

  5. Solving the model Discrete translational symmetry  kx, ky are good quantum numbers (mod 2/a) /a k -/a -/a /a

  6. Energy spectrum E -/a kx /a

  7. Energy spectrum E Conduction band  Valence band -/a kx /a

  8. Energy spectrum E Conduction band  Valence band -/a kx /a Band insulator

  9. 1928: Theory of band insulators (Bloch + others)

  10. 1928: Theory of band insulators (Bloch + others) 2005: Two fundamentally distinct classes of time reversal invariant band insulators: 1. Conventional insulators 2. “Topological insulators” (Kane, Mele + others)

  11. Insulators with an edge Vacuum

  12. Energy spectrum with an edge E Conduction  Valence -/a kx /a

  13. Energy spectrum with an edge E Conduction Edge modes  Valence -/a kx /a

  14. Two types of edge spectra E 0 kx /a

  15. Two types of edge spectra E 0 kx /a

  16. Two types of edge spectra E E 0 kx /a 0 kx /a

  17. Two types of edge spectra E E 0 kx /a 0 kx /a

  18. Two types of edge spectra E E 0 kx /a 0 kx /a

  19. Two types of edge spectra E E EF EF 0 kx /a 0 kx /a

  20. Two types of edge spectra E E EF EF 0 kx /a 0 kx /a Conventional insulator “Topological insulator”

  21. Experimental Realization (HgCdTe) (Koenig et al, Science, 2007) I. d < dc II. d < dc III. d > dc IV. d > dc

  22. Experimental Realization (HgCdTe) (Koenig et al, Science, 2007) I. d < dc II. d < dc III. d > dc IV. d > dc

  23. Other examples • 3D topological insulators • “Topological superconductors” (1D/2D/3D) • Quantum Hall states (2D) • Many others…

  24. The main question What is the general theory of protected boundary modes? In general, which systems have protected boundary modes and which do not?

  25. Formalism for non-interacting case Step 1: Specify symmetry and dimensionality of system e.g. “2D with charge conservation symmetry” Step 2: Look up corresponding “topological band invariants” e.g. “Chern number”

  26. Formalism for non-interacting case Bulk band structure Boundary is protected Topological band invariant Boundary is not protected

  27. Formalism for non-interacting case Bulk band structure Boundary is protected Topological band invariant Boundary is not protected

  28. Topological insulator boundaries are also protected against interactions! xxxxxxxxx Arbitrary local interactions

  29. Theory of interacting boundaries

  30. Theory of interacting boundaries Boundary is protected Boundary is not protected

  31. Theory of interacting boundaries Boundary is protected Bulk Hamiltonian Boundary is not protected

  32. Theory of interacting boundaries Boundary is protected Bulk Hamiltonian ??? Boundary is not protected

  33. Phase space of interacting systems with energy gap Anti-unitary symmetry Unitary symmetry Symmetry No symmetry No fractional statistics Fractional statistics Statistics of excitations

  34. Phase space of interacting systems with energy gap Anti-unitary symmetry Unitary symmetry Symmetry No symmetry Case 1 No fractional statistics Fractional statistics Statistics of excitations

  35. Phase space of interacting systems with energy gap Anti-unitary symmetry Unitary symmetry Symmetry Case 2 No symmetry No fractional statistics Fractional statistics Statistics of excitations

  36. Case 1: No symmetry

  37. Case 1: No symmetry Example: Integer quantum Hall states B

  38. Energy spectrum E c kx

  39. Energy spectrum E c kx

  40. Integer quantum Hall edge

  41. A more general result nR = 2 nL = 1

  42. A more general result (Kane, Fisher, 1997) nR = 2 nL = 1

  43. Yes! (ML, PRX 2013)

  44. Examples Superconductor

  45. Examples Superconductor Superconductor

  46. Fractional statistics in 2D

  47. Fractional statistics in 2D

  48. Fractional statistics in 2D

  49. (ML, PRX 2013) General criterion for protected edge l m

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