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**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

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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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**Majorana fermion statistics**

Definition of Majorana fermion Occupation of Half? 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**

1 2

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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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**Domain wall configuration**

Zero-mode soliton

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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**

~tanh(x) 1 3

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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**

full S2 μ>0 μ<0 ky kx

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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

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

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**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

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Broken CT, unbroken P T C P

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**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

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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- P-wave S-wave

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**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

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**Pairing symmetry and spectrum in uniform state on TI surface**

gapless gapped gapped s-wave: p-wave 2 p-wave 1:

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**Uniform state spectrum for mixed-parity symmetry**

gapped

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**Localized bound state inside a single vortex**

ξ

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**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

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**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

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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 p-wave like Δ+ s-wave like 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**

Seradjeh NPB 2008 Teo Kane PRL 2010 r

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**A simple but non-trivial Hamiltonian appears**

Fermion representation of matrix representation of Clifford algebra Boson representation of (x,k)

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**SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues**

Herbut Lu PRB 2011 f1 f2 b1 b2

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**Degeneracy calculation: Fermion-boson mixed harmonic oscillators**

1 2 f b Degeneracy =

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**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

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**Resultant degeneracy from two values of j**

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

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**Wavefunction of vortex bound states**

1 2 b 1 2 f b 1 2 f b 1 2 f b

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**Fermion representation and chiral symmetry**

1 2 b 1 2 f b chiral-even , 1 2 f b b , b b b f chiral-odd 1 2

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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 H2**

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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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