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CPT-symmetry, supersymmetry, and zero-mode in generalized Fu-Kane systemPowerPoint Presentation

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

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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 Fu-Kane system

- 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 CPT talk Fu-Kane system

- Motivation: Majorana zero-mode --- A half fermion
- Zero-modes in condensed matter physics
- Generalized Fu-Kane system,CPT symmetry, and its zero-mode
- Hidden SU(2) symmetry and supersymmetry in the hedgehog-gap configuration
- Two-velocity Weyl fermion in optical lattice
- Conclusion

Majorana fermion statistics Fu-Kane system

Definition of Majorana fermion

Occupation of Half?

Exchange statistics still intact

Re-construction of ordinary fermion from Majorana fermion Fu-Kane system

Restore an ordinary fermion

from two Majorana fermions

Distinction from Majorana fermion

An ordinary fermion out of two separated Majorana fermions Fu-Kane system

Four vortices: Emergence of non-Abelian statistics unconventional statistics

Zero-mode in condensed matter system: Rise of study of topology

- One-dimensional Su-Schrieffer-Heeger model of polyacetylene
- Vortex pattern of bond distortion in graphene
- topological superconductor vortex bound state/surface states
- Superconductor-topological insulator interface
- FerroM-RashbaSemiC-SC hetero-system

Domain wall configuration topology

Zero-mode soliton

Half-vortex in p+ip superconductors topology

2x2 second order diff. eq topology

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

2D generalization of topology

Peierl instability

Discrete symmetry from Hamiltonian’s algebraic structure topology

The beauty of Clifford and su(2) algebras

Massive Dirac Hamiltonian and the trick of squaring algebra

Homogeneous massive

Dirac Hamiltonian.

m=0 can correspond to

graphene case.

4 components from

valley and sublattice

degrees of freedom.

The Dirac Hamiltonian with a vortex configuration of mass algebra

Anti-unitary Time-reversal operator

Chiral symmetry operator

Particle-hole symmetry operator

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

Generalized Fu-Kane system: Jackiw-Rossi-Dirac Hamiltonian of superconducting surface of TI

azimuthal angle around

vortex center

Real/imaginary s-wave SC order parameters

Zeeman field along z

chemical potential

spin-momentum fixed kinetic energy

Jackiw-Rossi-Dirac Hamiltonian of unconventional SC vortex on TI surface

spin-triplet p-wave pairing

i is necessary for being Hermitian

{H, β3K}=0

Zero-mode in generalized Fu-Kane system with unconventional pairing symmetry

Spectrum parity and topology of order parameter

Spin-orbital coupling in normal state: helical states pairing symmetry

Parity broken

α≠0

Metallic surface of TI

k pairing symmetry

k

-k

-k

Topology associated with s-wave singlet and p-wave triplet order parameterss-wave limit

p-wave limit

Yip JLTP 2009

LuYip PRB 2008

Solving ODE for zero-mode pairing symmetry

s-wave case

purely decaying zero-mode

no zero-mode

oscillatory and decaying

zero-mode

Triplet p-wave gap and zero-mode pairing symmetry

p-wave case

Zero-mode becomes un-normalizable

when chemical potential μ is zero.

Mixed-parity gap and zero-mode: it exists, but the spectrum parity varies as…

ODE for the zero-mode

Two-gap SC

Accidental (super)-symmetry inside a infinitely-large vortex symmetry

Degenerate Dirac vortex bound states

A simple but non-trivial Hamiltonian appears Hamiltonian

Fermion representation of matrix

representation of Clifford algebra

Boson representation of (x,k)

SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues

b obtaining eigenvalues

b

b

f

b

2

1

Degeneracy calculation: Fermion-boson mixed harmonic oscillatorsDegeneracy =

Accidental su(2) symmetry: Label by angular momentum obtaining eigenvalues

co-rotation

y

α2

β2

x

β1

α1

An obvious constant of

motion

[H,J3]=[H,J2]=[H,J1]=0

Accidental generators

b obtaining eigenvalues

b

b

b

b

b

b

2

1

b

b

b

b

b

b

b

f

f

b

b

f

f

2

2

1

1

2

1

Wavefunction of vortex bound states±

±

b obtaining eigenvalues

b

b

b

b

b

2

1

b

b

b

b

b

f

b

f

f

2

1

2

1

Fermion representation and chiral symmetrychiral-even

,

b

b

b

,

b

f

chiral-odd

2

1

Accidental super-symmetry generators obtaining eigenvalues

Is there any other operator whose square satisfy identical commuation relation ?

The desired operators do the job. obtaining eigenvalues

Super-symmetry algebra

Connection between spectrum and degeneracy obtaining eigenvalues

can be shown vanishing

Chemical potential and Zeeman field obtaining eigenvalues

Perturbed spectrum obtaining eigenvalues

so(3)xso(3) algebraic structure of 4x4 Hermitian matrices obtaining eigenvalues

Two-velocity Weyl fermions in optical lattice

Two-velocity Weyl fermions on optical lattice obtaining eigenvalues

Low-energy effective Hamiltonian obtaining eigenvalues

Conclusions model

- 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