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

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

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 talk
Contents of talk
  • Motivation: Majorana fermion --- A half fermion
  • Realization of Majorana fermion in superconducting system: Studies of zero-modes.
  • Pairing between Dirac fermions on TI surface: Zero-mode inside a vortex of unconventional symmetry
  • Full vortex bound spectrum in Fu-Kane vortex Hamiltonian: Hidden SU(2) symmetry and supersymmetry
  • Realization of two-Fermi-velocity graphene in optical lattice: Hidden SO(3)XSO(3) symmetry of 4-site hopping Hamiltonian.
  • 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

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 topology
Zero-mode in condensed matter system: Rise of topology
  • 1D case: Peierl instability in polyacetylene.
  • 2D version of Peierls instability: Vortex pattern of bond distortion in graphene.
  • 2D/3D topological superconductors: Edge Andreev states and vortex zero-modes.
  • 2D gapped Dirac fermion systems: Proximity-indeuced superconducting TI surface
slide12

Domain wall configuration

Zero-mode soliton

ssh s continuum limit
SSH’s continuum limit

component on A sublattice

component on B sublattice

slide15

2D generalization of

Peierl instability

slide18

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

discrete symmetry from hamiltonian s algebraic structure

Discrete symmetry from Hamiltonian’s algebraic structure

The beauty of Clifford and su(2) algebras

from dirac equation to klein gordon equation square
From Dirac equation to Klein-Gordon equation: Square!

Homogeneous massive

Dirac Hamiltonian.

m=0 can correspond to

graphene case.

4 components from

valley and sublattice

degrees of freedom.

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

orderparameter

Represents the U(1) phase

generator

cpt from dirac hamiltonian with a mass vortex
CPT from Dirac Hamiltonian with a mass-vortex

Anti-unitary Time-reversal operator

Chiral symmetry operator

Jackiw Rossi NPB 1981

n zero-modes for vortex

of winding number n

Particle-hole symmetry operator

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

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

pairing symmetry on helicity based band
Pairing symmetry on helicity-based band

Parity broken

α≠0

Metallic surface of TI

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

Trivial superconductor

Nontrivial Z2 superconductor

p-wave limit

s-wave limit

LuYip PRB 2008 2009 2010

Sato Fujimoto 2008

Yip JLTP 2009

pairing symmetry and spectrum in uniform state on ti surface
Pairing symmetry and spectrum in uniform state on TI surface

gapless

gapped

gapped

s-wave:

p-wave 2

p-wave 1:

solving ode for zero mode
Solving ODE for zero-mode

Orbital coupling

To magnetic field

s-wave case

Lu Herbut PRB 2010

μ≠0 and gapped

Winding number odd:

1 zero-mode

Winding number even:

0 zero-mode

See also Fukui PRB 2010

Zeeman coupling

triplet p wave gap and zero mode
Triplet p-wave gap and zero-mode

p-wave case

h2>μ2

Zero-mode becomes un-normalizable

when chemical potential μ is zero.

p-wave sc op

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

smoothly

connected

at Fermi surface

+

+

-

+

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

Δ+

s-wave like

Δ-

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)

Seradjeh NPB 2008

Teo Kane PRL 2010

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

Herbut Lu PRB 2011

f1

f2

b1

b2

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

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 super symmetric representation of quaternion algebra
Accidental super-symmetry generators:Super-symmetric representation of quaternion algebra

Lu Herbut JPhysA 2011

the desired operators do the job
The desired operators do the job.

Super-symmetry algebra

so 3 xso 3 algebraic structure within 4x4 hermitian matrices

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

Two-velocity Weyl fermions in optical lattice

conclusions and prospects
Conclusions and prospects
  • Clifford algebra and su(2) algebra help gain insight into hidden symmetry
  • Zero-modes of Fu-Kane Hamiltonian survive when gap in uniform state is not closed
  • Ordinary fermion representation of Gamma matrices and super-symmetric form of Fu-Kane Hamiltonian
  • 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