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!!!