Cpt symmetry supersymmetry and zero mode in generalized fu kane system
Sponsored Links
This presentation is the property of its rightful owner.
1 / 60

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


  • 54 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

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

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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

  • Special thanks to Prof. Sungkit Yip, Academia Sinica


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

Occupation is integer

Pauli exclusion principle


Majorana fermion statistics

Definition of Majorana fermion

Occupation of Half?

Exchange statistics still intact


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


Two vortices: Degenerate ground-state manifold and unconventional statistics

|G>

Ψ+|G>

T

1

2


Four vortices: Emergence of non-Abelian statistics


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


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

Zero-mode soliton


SSH’s continuum limit

component on A sublattice

component on B sublattice


3

1

Nontrivial topology and zero-mode

~tanh(x)


Half-vortex in p+ip superconductors


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

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


Algebraic representation of Dirac Hamiltonian: Clifford algebra

real

imaginary


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

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

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

C

T

P


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

Spectrum parity and topology of order parameter


Spin-orbital coupling in normal state: helical states

Parity broken

α≠0

Metallic surface of TI


Δ+

Δ-

Mixed-parity SC state of momentum-spin helical state

P-wave

S-wave


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

s-wave case

purely decaying zero-mode

no zero-mode

oscillatory and decaying

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

s-wave case

p-wave case


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

Δ+>0

p-wave like

s-wave like


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

Δ(r)

r


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


b

b

b

f

b

2

1

Degeneracy calculation: Fermion-boson mixed harmonic oscillators

Degeneracy =


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

l=0,1/2,1,3/2,….

s=0,1/2


Degeneracy pattern

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

Wavefunction of vortex bound states

±

±


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

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


The desired operators do the job.

Super-symmetry algebra


Connection between spectrum and degeneracy

can be shown vanishing


Chemical potential and Zeeman field


Perturbed spectrum


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

Two-velocity Weyl fermions in optical lattice


Two-velocity Weyl fermions on optical lattice


Low-energy effective Hamiltonian


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

|u|

|v|


Chiral-block Hamiltonian

Π

Ψ


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


  • Login