New Technologies and Platforms Jennifer Tieman, Flinders University
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Netanel Lindner
(Caltech -> Technion)
Jerusalem, July 2013
New platforms for topological quantum
computing
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ElegantSimple
Useful
Lessons from Yosi
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QuantumHallEffect
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Topological Quantum Computing
dim NH d
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Non-abelianfractional quantum states
Miller et. al, Nature Physics 3, 561 - 565 (2007) R. L. Willett et. al., arXiv:1301.2639
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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
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Superconductor
gap SCE 0gapE
Topological 1D superconductor “Majorana fermion edge modes”
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Topological SC in 1D
Superconductor
, ,
†
0 0
,
L R
i j ij
H H
Majorana Fermions:
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Possible solid-state realizations
/ 2 0 / 2 𝑘𝑥
Quantum Spin Hall Effect Spin orbit coupled semiconductor wires
Superconductor
𝐵
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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
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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
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Effectively, the ferromagnet “stitches” the two annuli into a torus
Ground state degeneracy
FTIFM
FM
(1/ , )m ,( 1/ , )m
1/ ,m
1/ ,m
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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
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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
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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
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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
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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
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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
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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
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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
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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…
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Braiding Relations
(Yang-Baxter equation)
Both equations hold (up to a global phase)
12U23U
The group generated by
1,ˆ
iiU
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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”
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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
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Point particles vs. line objects
a
F(a)
a
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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)
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Braiding defects with anyons
ag defect
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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.
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point particles vs. defects
c a
c d
bd
=
g
gha
a
gh
gha
a
h
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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:
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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
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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:
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Algebraic theory of defect braiding
1. Group action on anyons
: ( )G perm A
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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
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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)
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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
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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”
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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)
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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.
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Happy Birthday!!!