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

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

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

Ordinary fermion statistics Occupation is integer Pauli exclusion principle

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

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

An ordinary fermion out of two separated Majorana fermions

Two vortices: Degenerate ground-state manifold and unconventional statistics 1 2

Four vortices: Emergence of non-Abelian statistics

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

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

Domain wall configuration Zero-mode soliton

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

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

2D generalization of Peierl instability

Half-vortex in p+ip superconductors

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

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

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

Hermitian matrix representation of Clifford algebra real imaginary

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.

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

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

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

Broken CT, unbroken P T C P

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

Arxiv:1105.0229

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

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

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 2008 2009 2010 Sato Fujimoto 2008 Yip JLTP 2009

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

Uniform state spectrum for mixed-parity symmetry gapped

Localized bound state inside a single vortex ξ

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

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

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

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 + + + -

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

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

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

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

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

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

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

Resultant degeneracy from two values of j

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

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

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

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

Algebraic approach to find remaining square roots of H2

The desired operators do the job. Super-symmetry algebra

Connection between spectrum and degeneracy can be shown vanishing

Chemical potential and Zeeman field

Perturbed spectrum

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

Two-velocity Weyl fermions on optical lattice

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

Chiral-block Hamiltonian Ψ Π

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