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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

### Discrete symmetry from Hamiltonian’s algebraic structure topology

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

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

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

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

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

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

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

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

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