1. Chi-Ken Lu Physics Department, Simon Fraser University, Canada
CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane systems
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 talk Motivation: Majorana fermion --- A half fermion
Realization of Majorana fermion in superconducting system: Studies of zero-modes.
Pairing between Dirac fermions on TI surface: Zero-mode inside a vortex of unconventional symmetry
Full vortex bound spectrum in Fu-Kane vortex Hamiltonian: Hidden SU(2) symmetry and supersymmetry
Realization of two-Fermi-velocity graphene in optical lattice: Hidden SO(3)XSO(3) symmetry of 4-site hopping Hamiltonian.
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 statistics1 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 topology
1D case: Peierl instability in polyacetylene.
2D version of Peierls instability: Vortex pattern of bond distortion in graphene.
2D/3D topological superconductors: Edge Andreev states and vortex zero-modes.
2D gapped Dirac fermion systems: Proximity-indeuced superconducting TI surface

12. Domain wall configuration
Zero-mode soliton

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

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

15. 2D generalization of
Peierl instability

16. Half-vortex in p+ip superconductors

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

18. 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

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

21. Hermitian matrix representation of Clifford algebra
real
imaginary

22. From Dirac equation to Klein-Gordon equation: Square!
Homogeneous massive
Dirac Hamiltonian.
m=0 can correspond to
graphene case.
4 components from
valley and sublattice
degrees of freedom.

23. 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

24. CPT from Dirac Hamiltonian with a mass-vortex
Chiral symmetry operator
Anti-unitary Time-reversal operator
Jackiw Rossi NPB 1981
n zero-modes for vortex
of winding number n
Particle-hole symmetry operator

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 T C P

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

28. Arxiv:

31. Pairing symmetry on helicity-based band
Parity broken
α≠0
Metallic surface of TI

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

33. Topology associated with s-wave singlet and p-wave triplet order parameters
Trivial superconductor
Nontrivial Z2 superconductor -k k
p-wave limit
s-wave limit
LuYip PRB
Sato Fujimoto 2008
Yip JLTP 2009

34. Pairing symmetry and spectrum in uniform state on TI surface
gapless
gapped
gapped
s-wave:
p-wave 2
p-wave 1:

35. Uniform state spectrum for mixed-parity symmetry
gapped

36. Localized bound state inside a single vortexξ

37. Solving ODE for zero-mode
Orbital coupling
To magnetic field
s-wave case
Lu Herbut PRB 2010
μ≠0 and gapped
Winding number odd:
1 zero-mode
Winding number even:
0 zero-mode
See also Fukui PRB 2010
Zeeman coupling

38. Triplet p-wave gap and zero-mode
p-wave case
h2>μ2
Zero-mode becomes un-normalizable
when chemical potential μ is zero.
p-wave sc op

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

40. Mixed-parity gap and zero-mode: it exists, but the spectrum parity varies as…
ODE for the zero-mode
Two-gap SC
smoothly
connected
at Fermi surface + + + -

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

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

43. Hidden SU(2) and super-symmetry out of Jackiw-Rossi-Dirac Hamiltonian
Seradjeh NPB 2008
Teo Kane PRL 2010 r

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

45. SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues
Herbut Lu PRB 2011 f1 f2 b1 b2

46. Degeneracy calculation: Fermion-boson mixed harmonic oscillators1 2 f b
Degeneracy =

47. 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

48. Resultant degeneracy from two values of j

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

50. Wavefunction of vortex bound states1 2 b 1 2 f b 1 2 f b 1 2 f b

51. Fermion representation and chiral symmetry1 2 b 1 2 f b
chiral-even , 1 2 f b b , b b b f
chiral-odd 1 2

52. Accidental super-symmetry generators: Super-symmetric representation of quaternion algebra
Lu Herbut JPhysA 2011

53. Algebraic approach to find remaining square roots of H2

54. The desired operators do the job.
Super-symmetry algebra

55. Connection between spectrum and degeneracy
can be shown vanishing

56. Chemical potential and Zeeman field

57. Perturbed spectrum

58. so(3)xso(3) algebraic structure within 4x4 Hermitian matrices
Two-velocity Weyl fermions in optical lattice

59. Two-velocity Weyl fermions on optical lattice

60. Hidden so(3)xso(3) algebra from two-velocity Weyl fermion model
|u|
|v|

61. Chiral-block HamiltonianΨ Π

62. Conclusions and prospects
Clifford algebra and su(2) algebra help gain insight into hidden symmetry
Zero-modes of Fu-Kane Hamiltonian survive when gap in uniform state is not closed
Ordinary fermion representation of Gamma matrices and super-symmetric form of Fu-Kane Hamiltonian
Linear dispersion and lessons from high-energy physics: Zoo of mass in condensed matter physics
Dirac bosons: One-way propagation EM mode at the edge of photonic crystal