CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane system

1. CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane system Chi-Ken Lu Physics Department, Simon Fraser University, Canada

2. Acknowledgement • Collaboration with Prof. Igor Herbut, Simon Fraser University • Supported by National Science of Council, Taiwan and NSERC, Canada • Special thanks to Prof. Sungkit Yip, Academia Sinica

3. Contents of CPT talk • Motivation: Majorana zero-mode --- A half fermion • Zero-modes in condensed matter physics • Generalized Fu-Kane system,CPT symmetry, and its zero-mode • Hidden SU(2) symmetry and supersymmetry in the hedgehog-gap configuration • Two-velocity Weyl fermion in optical lattice • Conclusion

4. Ordinary fermion statistics Occupation is integer Pauli exclusion principle

5. Majorana fermion statistics Definition of Majorana fermion Occupation of Half? Exchange statistics still intact

6. Re-construction of ordinary fermion from Majorana fermion Restore an ordinary fermion from two Majorana fermions Distinction from Majorana fermion

7. An ordinary fermion out of two separated Majorana fermions

8. Two vortices: Degenerate ground-state manifold and unconventional statistics |G> Ψ+|G> T 1 2

9. Four vortices: Emergence of non-Abelian statistics

10. N vortices: Braiding group in the Hilbert space of dimension 2^{N/2}

11. Zero-mode in condensed matter system: Rise of study of topology • One-dimensional Su-Schrieffer-Heeger model of polyacetylene • Vortex pattern of bond distortion in graphene • topological superconductor vortex bound state/surface states • Superconductor-topological insulator interface • FerroM-RashbaSemiC-SC hetero-system

12. Domain wall configuration Zero-mode soliton

13. SSH’s continuum limit component on A sublattice component on B sublattice

14. 3 1 Nontrivial topology and zero-mode ~tanh(x)

15. Half-vortex in p+ip superconductors

16. 2x2 second order diff. eq Supposedly, there are 4 indep. sol.’s e component h component can be rotated into 3th component u-iv=0 from 2 of the 4 sol’s are identically zero 2 of the 4 sol’s are decaying ones

17. Topological interpretation of BdG Hamiltonian of p+ip SC full S2 μ>0 μ<0 ky kx

18. 2D generalization of Peierl instability

20. Discrete symmetry from Hamiltonian’s algebraic structure The beauty of Clifford and su(2) algebras

21. Algebraic representation of Dirac Hamiltonian: Clifford algebra real imaginary

22. Massive Dirac Hamiltonian and the trick of squaring Homogeneous massive Dirac Hamiltonian. m=0 can correspond to graphene case. 4 components from valley and sublattice degrees of freedom.

23. The Dirac Hamiltonian with a vortex configuration of mass Anti-unitary Time-reversal operator Chiral symmetry operator Particle-hole symmetry operator

24. Imposing physical meaning to these Dirac matrices: context of superconducting surface of TI Breaking of spin-rotation symmetry in the normal state represents the generator of spin rotation in xy plane Real and imaginary part of SC order parameter Represents the U(1) phase generator

25. Generalized Fu-Kane system: Jackiw-Rossi-Dirac Hamiltonian azimuthal angle around vortex center Real/imaginary s-wave SC order parameters Zeeman field along z chemical potential spin-momentum fixed kinetic energy

26. Broken CT, unbroken P C T P

27. Jackiw-Rossi-Dirac Hamiltonian of unconventional SC vortex on TI surface spin-triplet p-wave pairing i is necessary for being Hermitian {H, β3K}=0

28. Zero-mode in generalized Fu-Kane system with unconventional pairing symmetry Spectrum parity and topology of order parameter

32. Spin-orbital coupling in normal state: helical states Parity broken α≠0 Metallic surface of TI

33. Δ+ Δ- Mixed-parity SC state of momentum-spin helical state P-wave S-wave

34. k k -k -k Topology associated with s-wave singlet and p-wave triplet order parameters s-wave limit p-wave limit Yip JLTP 2009 LuYip PRB 2008

35. Solving ODE for zero-mode s-wave case purely decaying zero-mode no zero-mode oscillatory and decaying zero-mode

36. Triplet p-wave gap and zero-mode p-wave case Zero-mode becomes un-normalizable when chemical potential μ is zero.

37. Zero-mode wave function and spectrum parity s-wave case p-wave case

38. Mixed-parity gap and zero-mode: it exists, but the spectrum parity varies as… ODE for the zero-mode Two-gap SC

39. Spectrum-reflection parity of zero-mode in different pairing symmetry Δ+>0 p-wave like s-wave like

40. Accidental (super)-symmetry inside a infinitely-large vortex Degenerate Dirac vortex bound states

41. Hidden SU(2) and super-symmetry out of Jackiw-Rossi-Dirac Hamiltonian Δ(r) r

42. A simple but non-trivial Hamiltonian appears Fermion representation of matrix representation of Clifford algebra Boson representation of (x,k)

43. SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues

44. b b b f b 2 1 Degeneracy calculation: Fermion-boson mixed harmonic oscillators Degeneracy =

45. Accidental su(2) symmetry: Label by angular momentum co-rotation y α2 β2 x β1 α1 An obvious constant of motion [H,J3]=[H,J2]=[H,J1]=0 Accidental generators

46. Resultant degeneracy from two values of j l=0,1/2,1,3/2,…. s=0,1/2

47. Degeneracy pattern Lenz vector operator J+,J-,J3

48. b b b b b b b 2 1 b b b b b b b f f b b f f 2 2 1 1 2 1 Wavefunction of vortex bound states ± ±

49. b b b b b b 2 1 b b b b b f b f f 2 1 2 1 Fermion representation and chiral symmetry chiral-even , b b b , b f chiral-odd 2 1

50. Accidental super-symmetry generators Is there any other operator whose square satisfy identical commuation relation ?