CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane systemChi-Ken LuPhysics Department, Simon Fraser University, Canada

AcknowledgementCollaboration with Prof. Igor Herbut, Simon Fraser UniversitySupported by National Science of Council, Taiwan and NSERC, CanadaSpecial thanks to Prof. Sungkit Yip, Academia Sinica

Contents of CPT talkMotivation: Majorana zero-mode --- A half fermionZero-modes in condensed matter physicsGeneralized Fu-Kane system,CPT symmetry, and its zero-modeHidden SU(2) symmetry and supersymmetry in the hedgehog-gap configurationTwo-velocity Weyl fermion in optical latticeConclusion

Ordinary fermion statisticsOccupation is integerPauli exclusion principle

Majorana fermion statisticsDefinition of Majorana fermionOccupation of Half?Exchange statistics still intact

Re-construction of ordinary fermion from Majorana fermionRestore an ordinary fermionfrom two Majorana fermionsDistinction from Majorana fermion

An ordinary fermion out of two separated Majorana fermions

Two vortices: Degenerate ground-state manifold and unconventional statistics|G>+|G>T12

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 study of topology One-dimensional Su-Schrieffer-Heeger model of polyacetyleneVortex pattern of bond distortion in graphenetopological superconductor vortex bound state/surface statesSuperconductor-topological insulator interface FerroM-RashbaSemiC-SC hetero-system

Zero-mode solitonDomain wall configuration

SSHs continuum limitcomponent on A sublatticecomponent on B sublattice

Nontrivial topology and zero-mode~tanh(x)

Half-vortex in p+ip superconductors

e componenth component2x2 second order diff. eqSupposedly, there are 4indep. sol.su-iv=0from 2 of the 4sols are identicallyzero2 of the 4 sols are decaying onescan be rotated into 3th component

Topological interpretation of BdG Hamiltonian of p+ip SC>0

2D generalization ofPeierl instability

Discrete symmetry from Hamiltonians algebraic structureThe beauty of Clifford and su(2) algebras

Algebraic representation of Dirac Hamiltonian: Clifford algebrarealimaginary

Massive Dirac Hamiltonian and the trick of squaringHomogeneous massiveDirac Hamiltonian.m=0 can correspond to graphene case.4 components from valley and sublatticedegrees of freedom.

The Dirac Hamiltonian with a vortex configuration of mass Chiral symmetry operatorAnti-unitary Time-reversal operatorParticle-hole symmetry operator

Imposing physical meaning to these Dirac matrices: context of superconducting surface of TIBreaking of spin-rotation symmetryin the normal staterepresents the generator of spinrotation in xy planeReal and imaginary part of SCorder parameterRepresents the U(1) phasegenerator

Generalized Fu-Kane system: Jackiw-Rossi-Dirac Hamiltonianspin-momentum fixed kinetic energyZeeman field along zReal/imaginary s-wave SC order parameterschemical potentialazimuthal angle aroundvortex center

Broken CT, unbroken P CTP

Jackiw-Rossi-Dirac Hamiltonian of unconventional SC vortex on TI surfacespin-triplet p-wave pairingi is necessary for being Hermitian{H, 3K}=0

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

Spin-orbital coupling in normal state: helical statesParity broken0Metallic surface of TI

Mixed-parity SC state of momentum-spin helical stateS-waveP-wave

Topology associated with s-wave singlet and p-wave triplet order parameterss-wave limitp-wave limitLuYip PRB 2008Yip JLTP 2009

Solving ODE for zero-modepurely decaying zero-modeoscillatory and decayingzero-modeno zero-modes-wave case

Triplet p-wave gap and zero-modep-wave caseZero-mode becomes un-normalizablewhen chemical potential is zero.

Zero-mode wave function and spectrum paritys-wave casep-wave case

Mixed-parity gap and zero-mode: it exists, but the spectrum parity varies asODE for the zero-modeTwo-gap SC

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

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

Hidden SU(2) and super-symmetry out of Jackiw-Rossi-Dirac Hamiltonian(r)r

A simple but non-trivial Hamiltonian appearsBoson representation of (x,k)Fermion representation of matrixrepresentation of Clifford algebra

SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues

Degeneracy calculation: Fermion-boson mixed harmonic oscillatorsDegeneracy =

Accidental su(2) symmetry: Label by angular momentum 1212xy[H,J3]=[H,J2]=[H,J1]=0An obvious constant ofmotionAccidental generatorsco-rotation

Resultant degeneracy from two values of js=0,1/2l=0,1/2,1,3/2,.

Degeneracy patternJ+,J-,J3Lenz vector operator

Wavefunction of vortex bound states

Fermion representation and chiral symmetry,12fbbbb,chiral-evenchiral-odd

Accidental super-symmetry generatorsIs there any other operator whose square satisfy identical commuation relation ?

The desired operators do the job.Super-symmetry algebra

Connection between spectrum and degeneracycan be shown vanishing

Chemical potential and Zeeman field

Perturbed spectrum

so(3)xso(3) algebraic structure of 4x4 Hermitian matricesTwo-velocity Weyl fermions in optical lattice

Two-velocity Weyl fermions on optical lattice

Low-energy effective Hamiltonian

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

Chiral-block Hamiltonian

ConclusionsLinear dispersion and lessons from high-energy physics: Zoo of mass in condensed matter physicsDirac bosons: One-way propagation EM mode at the edge of photonic crystal