CPT-symmetry, supersymmetry, and
zero-mode in generalized Fu-Kane
systems
Chi-Ken Lu
Physics Department,
Simon Fraser University,
Canada
Acknowledgement
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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
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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

1
{c , c1}  1 {c , c2}  {c1, c2}  0

1
1
C C1  
0
C C  0 C C  C C
Occupation is integer
Pauli exclusion principle

1

1

1

1

2

2

1
Majorana fermion statistics

1
  1
Definition of Majorana fermion

1
 
1 1

1 1
{ ,  1}  1      1 1     12

1
{ ,  }  0

2
Occupation of Half?
Exchange statistics still intact
Re-construction of ordinary fermion
from Majorana fermion

 1  i 2
2

, 
{,   }  1


     0
Distinction from Majorana fermion
 1  i 2
2
Restore an ordinary fermion
from two Majorana fermions
An ordinary fermion out of two
separated Majorana fermions
Two vortices: Degenerate ground-state
manifold and unconventional statistics

 
|G>
T
1
2
Ψ+|G>
 1  i 2
2
Four vortices: Emergence of nonAbelian statistics
G

1
 G

2
 G

1

2
 G

N vortices: Braiding group in the
Hilbert space of dimension 2^{N/2}
Zero-mode in condensed matter
system: Rise of topology

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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: Proximityindeuced superconducting TI surface
Domain wall configuration
Zero-mode soliton
SSH’s continuum limit
component on A sublattice
component on B sublattice
Nontrivial topology and zero-mode
~tanh(x)
H  [k 3  ( x) 1 ]

{H ,  2 }  0
 2 E   E
 2 0  0

 Beff  s
3
1
1 (S 1 )  Z
2D generalization of
Peierl instability
Half-vortex in p+ip superconductors
 
 ( R, k )  e
i ( )


d ( )   (i 2 )
Topological interpretation of BdG
Hamiltonian of p+ip SC



 
H k  (c , ck ) ( k   ) 3  (k     k    )  B s



k
μ>0
ky
kx
μ<0
full S2
2x2 second order diff. eq
Supposedly, there are 4
indep. sol.’s
u 
 
v
e component
h component
can be rotated into 3th component
2 of the 4 sol’s are decaying ones
u-iv=0
from 2 of the 4
sol’s are identically
zero
Discrete symmetry from
Hamiltonian’s algebraic structure
The beauty of Clifford
and su(2) algebras
Hermitian matrix representation of
Clifford algebra
 3
1  
0
 0
1  
  iI 2
0 
  1 0 
0
,  2  
,  3  
3 
 0 1 
 I2
iI 2 
 2
,  2  
0
0
0 

 2 
I2 

0
real
imaginary
{ i ,  j }  { i ,  j }  2 ij I 4
{ i ,  j }  0
 3  1 2 1 2
From Dirac equation to Klein-Gordon
equation: Square!
H  k x1  k y 2  m1
 H  (k  m ) I 4
2
2
2
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
Breaking of spin-rotation symmetry
in the normal state
{1 ,  2 , i1 2 }
represents the generator of spin
rotation in xy plane
Real and imaginary part of SC
order parameter
{1 ,  2 , i1 2 }
Represents the U(1) phase
generator
CPT from Dirac Hamiltonian with a
mass-vortex
H  kx1  k y2  m(r )[1 cos  2 sin  ]
Chiral symmetry operator
Anti-unitary Time-reversal operator
{H , 3}  0, [ H , K ]  0
 {H , 3 K}  0
Particle-hole symmetry operator
Jackiw Rossi NPB 1981
n zero-modes for vortex
of winding number n
Generalized Fu-Kane system: JackiwRossi-Dirac Hamiltonian
Real/imaginary s-wave SC order parameters
azimuthal angle around
vortex center
H  kx1  k y2  h(i12 )  m(r)[1 cos  2 sin  ]  (i12 )
Zeeman field along z
chemical potential
spin-momentum fixed kinetic energy
Broken CT, unbroken P
H  (k x  A1i1 2 )1  (k y  A2i1 2 ) 2 
m(r )[1 cos   2 sin  ]  h(i1 2 )   (i1 2 )
T
C
{H 0 , 3}  0, [ H 0 , K ]  0
 {H 0 , 3 K}  0
P
i1 2
i1 2
[
,  3 ]  0,{
, K}  0
i1 2
i1 2
i1 2
{
, 3 K}  0
i1 2
Zero-mode in generalized Fu-Kane
system with unconventional pairing
symmetry
Spectrum parity and
topology of order
parameter
Arxiv:1105.0229
Pairing symmetry on helicity-based
band
  
1 2
h
k   z k  
2m
Parity broken
α≠0
Metallic surface of TI
1
0
m
Mixed-parity SC state of momentumspin helical state


h  vF n 
ΔΔ+
P-wave


M  I 2  vF n 
S-wave
Topology associated with s-wave
singlet and p-wave triplet order
parameters
Trivial superconductor
Nontrivial Z2 superconductor
k
p-wave limit
s-wave limit
k
  ( pF  )    ( pF  )
  ( pF  )    ( pF  )
pF   pF 
pF   pF 
  ( pF  )    ( pF  )  0
pF   pF 
  ( pF  )    ( pF  )  0
pF   pF 
-k
-k
LuYip PRB 2008 2009 2010
Sato Fujimoto 2008 Yip JLTP 2009
Pairing symmetry and spectrum in
uniform state on TI surface
gapped
gapless
gapped
s-wave:
M  I 2
p-wave 2
p-wave 1:


M  n 


M  n  
Uniform state spectrum for mixedparity symmetry
gapped
Localized bound state inside a single
vortex
Δ(r)
ξ
r
Solving ODE for zero-mode
s-wave case
Orbital coupling
To magnetic field
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
p-wave case
h2>μ2
Zero-mode becomes un-normalizable
when chemical potential μ is zero.
p-wave sc op
Zero-mode wave function and
spectrum parity
3 K 0  0
s-wave case
3 K 0  sgn() 0
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
+
+
smoothly
connected
at Fermi surface
+
-
Spectrum-reflection parity of zeromode in different pairing symmetry
Δ+>0
3 K 0   0
p-wave like
s-wave like
Δ+
Δ-
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)
Seradjeh NPB 2008
Teo Kane PRL 2010
r
A simple but non-trivial Hamiltonian
appears
H  kx1  k y2  x1  y2
Boson representation of (x,k)
Fermion representation of matrix
representation of Clifford algebra
SUSY form of vortex Hamiltonian and
its simplicity in obtaining eigenvalues
Herbut Lu PRB 2011
E N
N  N f  Nb
f1
f2
b1
b2
Degeneracy calculation: Fermionboson mixed harmonic oscillators
N  N f  Nb
b
b
b
 N f  0,1,2
 N b  N , N  1, N  2
Degeneracy =
f
b
1
2
( N  1)  2 N  N  1
 2N
2
Accidental su(2) symmetry: Label by
angular momentum
H  kx1  k y2  x1  y2
co-rotation
y
α2
J 3  L3  S3


β2
β1
α1
( xk y  ykx )  i(1 2  1 2 )
1
2
2
a1 a1  a2 a2  b1 b1  b2 b2

x
An obvious constant of
motion
[H,J3]=[H,J2]=[H,J1]=0

Accidental generators
Resultant degeneracy from two values
of j
 
J   a1


1
2

 a1  1  
a     2  b1
 a2  


2
s=0,1/2



 b1 
b   
 b2 

2


l=0,1/2,1,3/2,….
Degeneracy pattern
Lenz vector operator
J+,J-,J3
Wavefunction of vortex bound states
b
b
b
b
b
1
b
±
2
b
b
f
b
1
b
f
f
1
2
2
b
b
b
 E, j , m
±
b
b
f
b
1
2
 E, j , m
Fermion representation and chiral
symmetry
b
b
chiral-even
  1 2 1 2
b
b
{,  }  {,  }  0
b
b
1
{, a1, 2 }  0
,
2
b
b
f
f
b
[, b1, 2 ]  0
chiral-odd
b
b
f
b
1
2
2
1
b
,
b
b
b
f
1
2
Accidental super-symmetry generators:
Super-symmetric representation of
quaternion algebra
 1  i, j , k
Lu Herbut JPhysA 2011
 I 2  iI 2 , i x , i y , i z
H  H , Ax , Ay , Az
2
Algebraic approach to find remaining
square roots of H2

1 1

2 2

1 1

2 2
H  a a a a b b b b
2

i

i

i

i
[H , a ]  a ,[H , b ]  b
2

i
2

i
A  U ij (a b j  b a j )
[ A2 , X  ]  { A, [ A, X  ]}  [ A,{ A, X  }]
U  I 2  A  H
2
2
2
The desired operators do the job.
Super-symmetry algebra
Connection between spectrum and
degeneracy
j , j
j ,( j 1)
j , j 1
j , j
can be shown vanishing
Chemical potential and Zeeman field
Perturbed spectrum
so(3)xso(3) algebraic structure
within 4x4 Hermitian matrices
Two-velocity Weyl
fermions in optical
lattice
Two-velocity Weyl fermions on optical
lattice
Hidden so(3)xso(3) algebra from twovelocity Weyl fermion model
|u|
|v|
Chiral-block Hamiltonian
HW  k (i1 2 )  k (i1 2 )
i1 2  i1 2
i1 2  i1 2
 (1   )k
 (1   )k
2
2
Ψ
Π
  1 2 1 2
Conclusions and prospects
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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