cpt-symmetry, supersymmetry, and zero-mode in generalized fu-kane system
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CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane system. 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 - PowerPoint PPT PresentationTRANSCRIPT
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CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane systemChi-Ken LuPhysics Department, Simon Fraser University, Canada
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AcknowledgementCollaboration with Prof. Igor Herbut, Simon Fraser UniversitySupported by National Science of Council, Taiwan and NSERC, CanadaSpecial thanks to Prof. Sungkit Yip, Academia Sinica
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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
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Ordinary fermion statisticsOccupation is integerPauli exclusion principle
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Majorana fermion statisticsDefinition of Majorana fermionOccupation of Half?Exchange statistics still intact
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Re-construction of ordinary fermion from Majorana fermionRestore an ordinary fermionfrom two Majorana fermionsDistinction 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>+|G>T12
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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 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
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Zero-mode solitonDomain wall configuration
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SSHs continuum limitcomponent on A sublatticecomponent on B sublattice
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Nontrivial topology and zero-mode~tanh(x)
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Half-vortex in p+ip superconductors
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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
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2D generalization ofPeierl instability
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Discrete symmetry from Hamiltonians algebraic structureThe beauty of Clifford and su(2) algebras
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Algebraic representation of Dirac Hamiltonian: Clifford algebrarealimaginary
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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.
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The Dirac Hamiltonian with a vortex configuration of mass Chiral symmetry operatorAnti-unitary Time-reversal operatorParticle-hole symmetry operator
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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
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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
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Broken CT, unbroken P CTP
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Jackiw-Rossi-Dirac Hamiltonian of unconventional SC vortex on TI surfacespin-triplet p-wave pairingi is necessary for being Hermitian{H, 3K}=0
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Zero-mode in generalized Fu-Kane system with unconventional pairing symmetrySpectrum parity and topology of order parameter
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Spin-orbital coupling in normal state: helical statesParity broken0Metallic surface of TI
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Mixed-parity SC state of momentum-spin helical stateS-waveP-wave
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Topology associated with s-wave singlet and p-wave triplet order parameterss-wave limitp-wave limitLuYip PRB 2008Yip JLTP 2009
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Solving ODE for zero-modepurely decaying zero-modeoscillatory and decayingzero-modeno zero-modes-wave case
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Triplet p-wave gap and zero-modep-wave caseZero-mode becomes un-normalizablewhen chemical potential is zero.
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Zero-mode wave function and spectrum paritys-wave casep-wave case
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Mixed-parity gap and zero-mode: it exists, but the spectrum parity varies asODE for the zero-modeTwo-gap SC
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Spectrum-reflection parity of zero-mode in different pairing symmetry+>0s-wave likep-wave like
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Accidental (super)-symmetry inside a infinitely-large vortexDegenerate Dirac vortex bound states
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Hidden SU(2) and super-symmetry out of Jackiw-Rossi-Dirac Hamiltonian(r)r
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A simple but non-trivial Hamiltonian appearsBoson representation of (x,k)Fermion representation of matrixrepresentation of Clifford algebra
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SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues
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Degeneracy calculation: Fermion-boson mixed harmonic oscillatorsDegeneracy =
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Accidental su(2) symmetry: Label by angular momentum 1212xy[H,J3]=[H,J2]=[H,J1]=0An obvious constant ofmotionAccidental generatorsco-rotation
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Resultant degeneracy from two values of js=0,1/2l=0,1/2,1,3/2,.
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Degeneracy patternJ+,J-,J3Lenz vector operator
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Wavefunction of vortex bound states
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Fermion representation and chiral symmetry,12fbbbb,chiral-evenchiral-odd
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Accidental super-symmetry generatorsIs there any other operator whose square satisfy identical commuation relation ?
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The desired operators do the job.Super-symmetry algebra
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Connection between spectrum and degeneracycan 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 of 4x4 Hermitian matricesTwo-velocity Weyl fermions in optical lattice
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Two-velocity Weyl fermions on optical lattice
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Low-energy effective Hamiltonian
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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
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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