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

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

Chi-Ken Lu

Physics Department, Simon Fraser University, Canada


Acknowledgement
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
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


Ordinary fermion statistics
Ordinary fermion statistics Fu-Kane system

Occupation is integer

Pauli exclusion principle


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

Restore an ordinary fermion

from two Majorana fermions

Distinction from Majorana fermion



Two vortices degenerate ground state manifold and unconventional statistics
Two vortices: Degenerate ground-state manifold and unconventional statistics

|G>

Ψ+|G>

T

1

2




Zero mode in condensed matter system rise of study of topology
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


Cpt symmetry supersymmetry and zero mode in generalized fu kane system

Domain wall configuration topology

Zero-mode soliton


Ssh s continuum limit
SSH’s continuum limit topology

component on A sublattice

component on B sublattice


Nontrivial topology and zero mode

3 topology

1

Nontrivial topology and zero-mode

~tanh(x)



Cpt symmetry supersymmetry and zero mode in generalized fu kane system

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



Cpt symmetry supersymmetry and zero mode in generalized fu kane system

2D generalization of topology

Peierl instability


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

The beauty of Clifford and su(2) algebras



Massive dirac hamiltonian and the trick of squaring
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
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
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
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


Broken ct unbroken p
Broken CT, unbroken P of superconducting surface of TI

C

T

P


Jackiw rossi dirac hamiltonian of unconventional sc vortex on ti surface
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
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
Spin-orbital coupling in normal state: helical states pairing symmetry

Parity broken

α≠0

Metallic surface of TI


Mixed parity sc state of momentum spin helical state

Δ+ pairing symmetry

Δ-

Mixed-parity SC state of momentum-spin helical state

P-wave

S-wave


Topology associated with s wave singlet and p wave triplet order parameters

k pairing symmetry

k

-k

-k

Topology associated with s-wave singlet and p-wave triplet order parameters

s-wave limit

p-wave limit

Yip JLTP 2009

LuYip PRB 2008


Solving ode for zero mode
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
Triplet p-wave gap and zero-mode pairing symmetry

p-wave case

Zero-mode becomes un-normalizable

when chemical potential μ is zero.


Zero mode wave function and spectrum parity
Zero-mode wave function and spectrum parity pairing symmetry

s-wave case

p-wave case


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

ODE for the zero-mode

Two-gap SC


Spectrum reflection parity of zero mode in different pairing symmetry
Spectrum-reflection parity of zero-mode in different pairing symmetry

Δ+>0

p-wave like

s-wave like


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

Degenerate Dirac vortex bound states



A simple but non trivial hamiltonian appears
A simple but non-trivial Hamiltonian appears Hamiltonian

Fermion representation of matrix

representation of Clifford algebra

Boson representation of (x,k)



Degeneracy calculation fermion boson mixed harmonic oscillators

b obtaining eigenvalues

b

b

f

b

2

1

Degeneracy calculation: Fermion-boson mixed harmonic oscillators

Degeneracy =


Accidental su 2 symmetry label by angular momentum
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


Resultant degeneracy from two values of j
Resultant degeneracy from two values of j obtaining eigenvalues

l=0,1/2,1,3/2,….

s=0,1/2


Degeneracy pattern
Degeneracy pattern obtaining eigenvalues

Lenz vector operator

J+,J-,J3


Wavefunction of vortex bound states

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

±

±


Fermion representation and chiral symmetry

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 symmetry

chiral-even

,

b

b

b

,

b

f

chiral-odd

2

1


Accidental super symmetry generators
Accidental super-symmetry generators obtaining eigenvalues

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


The desired operators do the job
The desired operators do the job. obtaining eigenvalues

Super-symmetry algebra


Connection between spectrum and degeneracy
Connection between spectrum and degeneracy obtaining eigenvalues

can be shown vanishing



Perturbed spectrum
Perturbed spectrum obtaining eigenvalues


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

Two-velocity Weyl fermions in optical lattice



Low energy effective hamiltonian
Low-energy effective Hamiltonian obtaining eigenvalues




Conclusions
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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