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

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Presentation on theme: "CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane systems Chi-Ken Lu Physics Department, Simon Fraser University, Canada."— Presentation transcript:

1 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 Definition of Majorana fermion Occupation of Half? Majorana fermion statistics 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 12

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 Zero-mode soliton Domain wall configuration

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

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

15 2D generalization of Peierl instability

16 Half-vortex in p+ip superconductors

17 Topological interpretation of BdG Hamiltonian of p+ip SC μ>0 μ<0 full S 2 kx ky

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

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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 Particle-hole symmetry operator Jackiw Rossi NPB 1981 n zero-modes for vortex of winding number n

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

26 Broken CT, unbroken P C T P

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

28 Arxiv:1105.0229

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

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

33 -k k k s-wave limit p-wave limit Topology associated with s-wave singlet and p-wave triplet order parameters LuYip PRB 2008 2009 2010 Yip JLTP 2009 Trivial superconductorNontrivial Z2 superconductor Sato Fujimoto 2008

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

35 Uniform state spectrum for mixed- parity symmetry gapped

36 Localized bound state inside a single vortex Δ(r) r ξ

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

38 p-wave case Triplet p-wave gap and zero-mode Zero-mode becomes un-normalizable when chemical potential μ is zero. p-wave sc op h 2 >μ 2

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 s-wave like p-wave like Δ+ 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 Δ(r) r Seradjeh NPB 2008 Teo Kane PRL 2010

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

45 SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues Herbut Lu PRB 2011 f1f2b1b2

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

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

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

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

50 Wavefunction of vortex bound states 1 2 f b b b b 1 2 b b b b b ± 1 2 f b b b f ± 1 2 f b b b b

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

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

53 Algebraic approach to find remaining square roots of H 2

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 Hamiltonian Ψ 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


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