Transcript Non-Abelian Anyons on Fractional Topological Insulator Edges

New platforms
for topological quantum computing
Netanel Lindner
(Caltech -> Technion)
Jerusalem, July 2013
Lessons from Yosi
Useful
Elegant
Simple
Quantum
Hall
Effect
Topological Quantum Computing
dim H
d
N
Non-abelian
fractional quantum states
Miller et. al, Nature Physics 3, 561 - 565 (2007)
R. L. Willett et. al.,
arXiv:1301.2639
Topological 1D superconductor
Semiconductor
wire
Superconductor
Two degenerate ground states:
● The two states correspond to a total even or odd
number of electrons in the system.
● Ground state degeneracy is “topological”: no local
measurement can distinguish between the two states!
Read and Green (2000), Kitaev (2002), Sau et al. (2010), Oreg et al. (2010)
Topological 1D superconductor
“Majorana fermion edge modes”
Egap  0
Egap   SC
Superconductor
Topological SC in 1D
Majorana Fermions:  L, H   0  R , H   0
 †    i ,  j    ij
Superconductor
Possible solid-state realizations
Quantum Spin Hall Effect

 / 2
0
 /2

Spin orbit coupled
semiconductor wires
𝑘𝑥
𝐵
Superconductor
Majorana based TQC
Advantages
• Energy gap induced by
external SC and not by
interactions.
Problems
• Not universal:
• Control
• Gapless electrons in the
environment
e
 4  1 2
1 0


0 i 
Fractionalized zero modes
Consider counter propagating edge states of a
FQH state, coupled to superconductivity
SC
Backscattering
FM
FQH
=1/m
FQH
=1/m
FM
Backscattering
Zero modes at SC/FM interfaces: Read Green (2000), Fu and Kane (2009)
Ground state degeneracy
FM
FTI
  (1/ m, ),(1/ m, )
  1/ m,

FM
  1/ m, 
Effectively, the ferromagnet
“stitches” the two annuli into a torus
Ground state degeneracy
Spin on outer edge
(el. spin=1)
Sout = 2n/m, n = 0,...,m-1
Assuming no q.p. in the bulk:
Sin = - Sout
WxWy  e
i 2 / m
WyWx
G.S. Degeneracy = m
Ground state degeneracy
FM: Spins,
S1
Q1
Q2
S j  q / m, q  0,1,..2m  1
SC: Charges
S3
S2
Q3
Q j  q / m, q  0,1,..2m  1
Ground state degeneracy
Spins, Charges
S1
Q1
S j  q / m, q  0,1,..2m  1
Q2
Q j  q / m, q  0,1,..2m  1
n , n
S3
e
S2
i Qi i S j
+
Q3
e
e
i = j+1}
 i m i S j i Qi
e
{- i = j - 1}
2N domains, fixed n , n = Qtot, Stot
 
N-1
(2m)
ground states


2m

2( N 1)
e
Non-abelian statistics:
1) Degenerate number of ground states, depending on
the number of particles.
2) Exchanging two particles, yields a topologically
protected unitary transformation in the ground state
manifold.
 (ri )  Uˆ12  (ri )
Braiding
H (t )   ij (t ) H ij
ij
H ij  i , 
†
j ,
 h.c.
H (t  T )  H (t  0)
• Result is independent of the details of
the path (topological)
• Obeys braiding relations.
Braiding
Some Properties of
H (t )   ij (t ) H ij
ij
• Coupling two zero modes:
 2m 
( N 1)
  2m 
( N 2)
• Same ground state degeneracy
when two or three zero modes are
coupled.
• Degeneracy is lifted when four are
coupled.
Braiding
2
S2
3
Braiding interfaces
Q1
Q2
1
4
S1
S3
6
Q3
5
:
• Coupling two zero modes:
 2m 
( N 1)
  2m 
( N 2)
• Same ground state degeneracy when
two or three zero modes are coupled.
• Degeneracy is lifted when four are
coupled.
Braiding
Properties of the path
H (t )
• Fixed g.s. degeneracy for all
0t T
• Charge Q2 doesn’t change
• Therefore acquired phase be
a function of Q2
• Overall phase is non universal
Braiding
Braiding interfaces 3 and 4:
Uˆ 34  e
i

m ˆ
i
Q2  k m
Uˆ 34 q1,..., qN ; s  e 2m
Braiding 2 and 3:
Uˆ 23  e
2

 q2 k 
2
q1,..., qN ; s
m ˆ
i
S2  k m

2

2

2
etc…
Braiding Relations
The group generated by
Uˆ i ,i 1
(Yang-Baxter
equation)
Uˆ 12
Uˆ 23
Both equations hold (up to a global phase)
Decomposition of braid matrices
2m  2 m
q  0,1...,2m  1
q  m n  2nX
e
i

2m
q
2

m 2 2 2
i
n i nX
m
2
e
e
Ising anyons
new non-abelian
“anyon”
Two types of particles:
X  X  0  1  ...m  1
X
-q
q1  q2  q1  q2 mod m
X q  X
q
RXX
•
•
•
•
•
i i 2 m q
e e
2
M. Barkeshli, C-M. Jian, X-L. Qi (2013)
D. Clarke, J. Alicea, K. Shtengel, (2013)
M. Cheng, PRB 86, 195126 (2012)
NHL, E. Berg, G. Refael, A. Stern, (2012)
A. Kapustin, N. Sauling, (2011)
X X X
q
Point particles vs. line objects
F(a)
a
a
Twist Defects in SET’s
• SET: Top. Phase with
onsite finite symmetry
group G
1
U g HU g  H
• Local Hamiltonian:
H   Hi
i
•
•
•
•
L. Bombin (2010)
A. Kitaev and L. Kong (2012)
M. Barkeshli,, X-L. Qi (2012)
Y.-Z. You and X.-G. Wen (2012)
Braiding defects with anyons
g
defect
a
Braiding defects with anyons
Different SETs with symmetry
G, characterized by
 : G  permutations( Anyons)
g
defect
b
b  g a
Permutations have to be
consistent with the top. order:
fusion, braiding, and with the
group structure.
point particles vs. defects
c
a d
b
gha
a
g
=
gha
c
d
a
gh
h
Local G action
Suppose that G has trivial permutation of the anyons:
(c)
(b)
(a)

𝑎
𝑎
𝑎
𝑎
𝑎
𝑎
𝑎 𝑎
𝑎
𝑎
𝑎
Projective local G action
1 1
Vg ,h  U ghU h U g
Vg ,h a  e
e
i ( g ,h;a )
i ( g ,h;a )
a
 S ( g ,h ),a
 ( g , h ) : G  G  Ab. Anyons
Projective local G action
Constraints from associativity:
W U
1 1 1
fghU h U g U f
 V fg ,hV fg
1
 V f , ghU f VghU f
Mathematical terminology:
2
H (G, ab. A)
 ( g , h)   ( fg , h)   ( f , gh)   ( f , g )  0
Algebraic theory of defect braiding
1. Group action on anyons
 : G  perm( A)
Algebraic theory of defect braiding
1. Group action on anyons
 : G  perm( A)
2. Projective G- charges
carried by anyons
2
H (G, ab. A)
Algebraic theory of defect braiding
P. Etingof, et. al. (2010)
1. Group action on anyons
 : G  perm( A)
2. Projective G- charges
carried by anyons
2
H (G, ab. A)
3. Fractional charges
carried by defects
3
H (G,U (1))
Example 1:
2 1
K 

1 2
G  Z2
1. Group action on anyons:
(q, q)  (q, q)
e
i 2 n1K 1n2
2. Projective G- charges
carried by anyons
H ( Z 2 , Z3 )  {0}
2
3. Fractional charges
carried by defects
H ( Z 2 ,U (1))  Z 2
3
X  X  0  1  ...m  1
q1  q2  q1  q2 mod m
X q  X
Stack a non trivial
Z2
SPT
“Toric
Code”
0 2
K 

2 0
1. Group action on anyons
e, m, 
Example 2:
e  m  
2. Projective G- charges
carried by anyons
2
H (Z2 , Z2  Z2 )  Z2
3. Fractional charges
carried by defects
3
H ( Z 2 ,U (1))  Z 2
G  Z2
X1  X1  1  
Xe  Xe  1 
X1  X e  e  m
( g, g )  
Collaborators
• Erez Berg, Gil Refael, Ady Stern
PRX 2, 041002 (2012)
• Lukasz Fidkowski, Alexei Kitaev
(to be published soon)
• Jason Alicea, David Clarke, Kirril Stengel
•
•
•
•
•
•
•
M. Barkeshli, C-M. Jian, X-L. Qi (2013)
D. Clarke, J. Alicea, K. Shtengel, (2013)
M. Cheng, PRB 86, 195126 (2012)
M. Lu, A. Vishwanath, arXiv:1205.3156v3
M. Levin and Z.-C. Gu. PRB 86, 115109 (2013)
A M. Essin and M.Hermele, PRB 87, 104406 (2013)
X. Chen, Z-C. Gu, Z-X. Liu, and X-G. Wen, PRB, 87, 155114 (2013)
Summary
• Zero modes yielding non-Abelian statistics emerge on
abelian FQH edges coupled to a superconductor.
• The braiding rules are akin to those of defects in a
symmetry enriched topological phase: a route for
engineering new non-Abelian systems.
• Projective quantum numbers carried by anyons lead
to a modified braiding theory for defects.
• Finite number of consistent braiding theories,
classified by three physically measurable invariants:
each theory corresponds to a different class of SETs.
• Advantages to TQC: Braid universality*, enhanced
robustness.
Happy Birthday!!!