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for topological quantum computing. New platforms. Netanel Lindner (Caltech -> Technion ). Jerusalem, July 2013. Lessons from Yosi. Useful. Elegant. Simple. QuantumHall Effect. Topological Quantum Computing. Non- abelian fractional quantum states. - PowerPoint PPT Presentation
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Netanel Lindner
(Caltech -> Technion)
Jerusalem, July 2013
New platforms for topological quantum
computing
ElegantSimple
Useful
Lessons from Yosi
QuantumHallEffect
Topological Quantum Computing
dim NH d
Non-abelianfractional quantum states
Miller et. al, Nature Physics 3, 561 - 565 (2007) R. L. Willett et. al., arXiv:1301.2639
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)
Superconductor
Semiconductor wire
Topological 1D superconductor
Superconductor
gap SCE 0gapE
Topological 1D superconductor “Majorana fermion edge modes”
Topological SC in 1D
Superconductor
, ,
†
0 0
,
L R
i j ij
H H
Majorana Fermions:
Possible solid-state realizations
/ 2 0 / 2 𝑘𝑥
Quantum Spin Hall Effect Spin orbit coupled semiconductor wires
Superconductor
𝐵
Majorana based TQC
Advantages
• Energy gap induced by external SC and not by interactions.
• Control
Problems
• Not universal:
• Gapless electrons in the environment
1 24 1 0
0e
i
Fractionalized zero modes
Consider counter propagating edge states of a
FQH state, coupled to superconductivity
FQH=1/m
FQH=1/m
SC
Backscattering
Backscattering
Zero modes at SC/FM interfaces: Read Green (2000), Fu and Kane (2009)
FM
FM
Effectively, the ferromagnet “stitches” the two annuli into a torus
Ground state degeneracy
FTIFM
FM
(1/ , )m ,( 1/ , )m
1/ ,m
1/ ,m
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
G.S. Degeneracy = m
2 /i mx y y xW W e W W
S1
Q1
Q2
Q3
S2
S3
Ground state degeneracy
FM: Spins, / , 0,1,..2 1jS q m q m
/ , 0,1,..2 1jQ q m q m SC: Charges
S1
Q1
Q2
Q3
S2
S3
Ground state degeneracy
2N domains, fixed = Qtot, Stot
(2m)N-1 ground states
Spins, Charges
,n n
,n n
2( 1)2
Nm
j iji i S i Si mi Q i Qee e e e } { }+ i = j + 1 - i = j - 1
/ , 0,1,..2
/ , 0,1,.. 1
1
2j
j
Q
S q m q m
q m q m
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.
12ˆ( ) ( )i iU r r
Braiding
( ) ( )ij ijij
H t t H†
, , . .ij i jH h c
• Result is independent of the details of the path (topological)
• Obeys braiding relations.
( ) ( 0)H t T H t
Braiding
Some Properties of ( ) ( )ij ijij
H t t H
• Coupling two zero modes:
• Same ground state degeneracy when two or three zero modes are coupled.
• Degeneracy is lifted when four are coupled.
) 2( 1 ( )2 2
N Nm m
Braiding
Braiding interfaces :S2
Q1 Q
2
Q3
S3
S1
2
1
3
4
56• Coupling two zero modes:
• Same ground state degeneracy when two or three zero modes are coupled.
• Degeneracy is lifted when four are coupled.
) 2( 1 ( )2 2
N Nm m
Braiding
Properties of the path
• Fixed g.s. degeneracy for all
• Charge doesn’t change
• Therefore acquired phase be a function of
• Overall phase is non universal
( )H t
2Q0 t T
2Q
Braiding
Braiding interfaces 3 and 4:
22ˆ
234
ˆm
i Q k mU e
22234 1 1
ˆ ,..., ; ,..., ;i q k
mN NU q q s e q q s
22ˆ
223
ˆm
i S k mU e
Braiding 2 and 3: etc…
Braiding Relations
(Yang-Baxter equation)
Both equations hold (up to a global phase)
12U23U
The group generated by
1,ˆ
iiU
2 2m m
2 22 2
2 2 Xm
i q i ni nm me e e
0,1...,2 1
2 X
q m
q m n n
Decomposition of braid matrices
Ising anyons new non-abelian “anyon”
Two types of particles:
1 2 1 2
0 1 ... 1
mod
X X m
q q q q m
X q X
X X X
q
-q
22i m qq iXXR e e
• 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
Point particles vs. line objects
a
F(a)
a
Twist Defects in SET’s• SET: Top. Phase with
onsite finite symmetry group G
• Local Hamiltonian:
1g gU HU H
ii
H H
• 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
ag defect
Braiding defects with anyons
b
b g a
g defect
Different SETs with symmetry G, characterized by
: ( )G permutations Anyons
Permutations have to be consistent with the top. order: fusion, braiding, and with the group structure.
point particles vs. defects
c a
c d
bd
=
g
gha
a
gh
gha
a
h
𝑎
(a)
𝑎
𝑎 𝑎𝑎
(c)
𝑎
𝑎
𝑎
𝑎
(b)
𝑎
𝑎
Local G action
Suppose that G has trivial permutation of the anyons:
Projective local G action
1 1,g h gh h gV U U U
( , ; ),
i g h ag hV a e a
( , ; )( , ),
i g h ag h ae S
( , ) : .g h G G Ab Anyons
Projective local G action
1 1 1fgh h g fW U U U U
,fg h fgV V1
,f gh f gh fV U V U
( , ) ( , ) ( , ) ( , ) 0g h fg h f gh f g
Constraints from associativity:
2( , . )H G ab AMathematical terminology:
Algebraic theory of defect braiding
1. Group action on anyons
: ( )G perm A
Algebraic theory of defect braiding
1. Group action on anyons
2. Projective G- charges carried by anyons
: ( )G perm A
2( , . )H G ab A
Algebraic theory of defect braiding
1. Group action on anyons
2. Projective G- charges carried by anyons
3. Fractional charges carried by defects
2( , . )H G ab A
3( , (1))H G U
: ( )G perm A
P. Etingof, et. al. (2010)
Example 1:
1. Group action on anyons:
2. Projective G- charges carried by anyons
3. Fractional charges carried by defects
22 3( , ) {0}H Z Z
32 2( , (1))H Z U Z
2 1
1 2K
2G Z
( , ) ( , )q q q q
2ZStack a non trivial SPT
1 2 1 2
0 1 ... 1
mod
X X m
q q q q m
X q X
11 22i n K ne
Example 2:
1. Group action on anyons
2. Projective G- charges carried by anyons
3. Fractional charges carried by defects
22 2 2 2( , )H Z Z Z Z
32 2( , (1))H Z U Z
0 2
2 0K
2G Z
e m , ,e m
1 1 1X X 1e eX X
1 eX X e m
( , )g g
“Toric Code”
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!!!