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

  • 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

  • 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

Occupation is integer

Pauli exclusion principle


Majorana fermion statistics

Majorana fermion statistics

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

Restore an ordinary fermion

from two Majorana fermions

Distinction from Majorana fermion


An ordinary fermion out of two separated majorana fermions

An ordinary fermion out of two separated Majorana fermions


Two vortices degenerate ground state manifold and unconventional statistics

Two vortices: Degenerate ground-state manifold and unconventional statistics

|G>

Ψ+|G>

T

1

2


Four vortices emergence of non abelian statistics

Four vortices: Emergence of non-Abelian statistics


N vortices braiding group in the hilbert space of dimension 2 n 2

N vortices: Braiding group in the Hilbert space of dimension 2^{N/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

Zero-mode soliton


Ssh s continuum limit

SSH’s continuum limit

component on A sublattice

component on B sublattice


Nontrivial topology and zero mode

3

1

Nontrivial topology and zero-mode

~tanh(x)


Half vortex in p ip superconductors

Half-vortex in p+ip superconductors


Cpt symmetry supersymmetry and zero mode in generalized fu kane system

2x2 second order diff. eq

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


Topological interpretation of bdg hamiltonian of p ip sc

Topological interpretation of BdG Hamiltonian of p+ip SC

full S2

μ>0

μ<0

ky

kx


Cpt symmetry supersymmetry and zero mode in generalized fu kane system

2D generalization of

Peierl instability


Discrete symmetry from hamiltonian s algebraic structure

Discrete symmetry from Hamiltonian’s algebraic structure

The beauty of Clifford and su(2) algebras


Algebraic representation of dirac hamiltonian clifford algebra

Algebraic representation of Dirac Hamiltonian: Clifford algebra

real

imaginary


Massive dirac hamiltonian and the trick of squaring

Massive Dirac Hamiltonian and the trick of squaring

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

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

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

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

Parity broken

α≠0

Metallic surface of TI


Mixed parity sc state of momentum spin helical state

Δ+

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

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

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

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

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

Degenerate Dirac vortex bound states


Hidden su 2 and super symmetry out of jackiw rossi dirac hamiltonian

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

Δ(r)

r


A simple but non trivial hamiltonian appears

A simple but non-trivial Hamiltonian appears

Fermion representation of matrix

representation of Clifford algebra

Boson representation of (x,k)


Susy form of vortex hamiltonian and its simplicity in obtaining eigenvalues

SUSY form of vortex Hamiltonian and its simplicity in obtaining eigenvalues


Degeneracy calculation fermion boson mixed harmonic oscillators

b

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

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

l=0,1/2,1,3/2,….

s=0,1/2


Degeneracy pattern

Degeneracy pattern

Lenz vector operator

J+,J-,J3


Wavefunction of vortex bound states

b

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

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

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


The desired operators do the job

The desired operators do the job.

Super-symmetry algebra


Connection between spectrum and degeneracy

Connection between spectrum and degeneracy

can be shown vanishing


Chemical potential and zeeman field

Chemical potential and Zeeman field


Perturbed spectrum

Perturbed spectrum


So 3 xso 3 algebraic structure of 4x4 hermitian matrices

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

Two-velocity Weyl fermions in optical lattice


Two velocity weyl fermions on optical lattice

Two-velocity Weyl fermions on optical lattice


Low energy effective hamiltonian

Low-energy effective Hamiltonian


Hidden so 3 xso 3 algebra from two velocity weyl fermion model

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

|u|

|v|


Chiral block hamiltonian

Chiral-block Hamiltonian

Π

Ψ


Conclusions

Conclusions

  • 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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