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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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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

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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

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Ordinary fermion statistics Occupation is integer Pauli exclusion principle

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

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

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An ordinary fermion out of two separated Majorana fermions

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Two vortices: Degenerate ground-state manifold and unconventional statistics |G> Ψ + |G> T 12

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Four vortices: Emergence of non- Abelian statistics

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

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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

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Zero-mode soliton Domain wall configuration

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

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

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2D generalization of Peierl instability

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Half-vortex in p+ip superconductors

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

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

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Hermitian matrix representation of Clifford algebra real imaginary

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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.

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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

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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

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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

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Broken CT, unbroken P C T P

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

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Arxiv:

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

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

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

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

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Uniform state spectrum for mixed- parity symmetry gapped

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

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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

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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

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

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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

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

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

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

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

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

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

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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

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

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

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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

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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

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

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Algebraic approach to find remaining square roots of H 2

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The desired operators do the job. Super-symmetry algebra

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Connection between spectrum and degeneracy can be shown vanishing

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Chemical potential and Zeeman field

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Perturbed spectrum

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

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Two-velocity Weyl fermions on optical lattice

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

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Chiral-block Hamiltonian Ψ Π

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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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