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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

- 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

- 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

Occupation is integer

Pauli exclusion principle

Definition of Majorana fermion

Occupation of Half?

Exchange statistics still intact

Restore an ordinary fermion

from two Majorana fermions

Distinction from Majorana fermion

|G>

Ψ+|G>

T

1

2

- 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

Zero-mode soliton

component on A sublattice

component on B sublattice

3

1

~tanh(x)

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

full S2

μ>0

μ<0

ky

kx

2D generalization of

Peierl instability

Discrete symmetry from Hamiltonian’s algebraic structure

The beauty of Clifford and su(2) algebras

real

imaginary

Homogeneous massive

Dirac Hamiltonian.

m=0 can correspond to

graphene case.

4 components from

valley and sublattice

degrees of freedom.

Anti-unitary Time-reversal operator

Chiral symmetry operator

Particle-hole symmetry operator

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

azimuthal angle around

vortex center

Real/imaginary s-wave SC order parameters

Zeeman field along z

chemical potential

spin-momentum fixed kinetic energy

C

T

P

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

Parity broken

α≠0

Metallic surface of TI

Δ+

Δ-

P-wave

S-wave

k

k

-k

-k

s-wave limit

p-wave limit

Yip JLTP 2009

LuYip PRB 2008

s-wave case

purely decaying zero-mode

no zero-mode

oscillatory and decaying

zero-mode

p-wave case

Zero-mode becomes un-normalizable

when chemical potential μ is zero.

s-wave case

p-wave case

ODE for the zero-mode

Two-gap SC

Δ+>0

p-wave like

s-wave like

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

Degenerate Dirac vortex bound states

Δ(r)

r

Fermion representation of matrix

representation of Clifford algebra

Boson representation of (x,k)

b

b

b

f

b

2

1

Degeneracy =

co-rotation

y

α2

β2

x

β1

α1

An obvious constant of

motion

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

Accidental generators

l=0,1/2,1,3/2,….

s=0,1/2

Lenz vector operator

J+,J-,J3

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

±

±

b

b

b

b

b

b

2

1

b

b

b

b

b

f

b

f

f

2

1

2

1

chiral-even

,

b

b

b

,

b

f

chiral-odd

2

1

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

Super-symmetry algebra

can be shown vanishing

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

Two-velocity Weyl fermions in optical lattice

|u|

|v|

Π

Ψ

- 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