Non-Abelian Anyon Interferometry
Kirill Shtengel, UC Riverside
Collaborators: Parsa Bonderson, Microsoft Station Q
Alexei Kitaev, Caltech
Joost Slingerland, DIAS
Exchange statistics in (2+1)D
• Quantum-mechanical amplitude for particles at x1, x2, …, xn
at time t0 to return to these coordinates at time t.
Feynman: Sum over all trajectories, weighting each one by eiS.
Exchange statistics:
What are the relative
amplitudes for these
trajectories?
Exchange statistics in (2+1)D
• In (3+1)D, T −1 = T, while in (2+1)D T −1 ≠ T !
• If T −1 = T then T 2 = 1, and the only types of particles are
bosons and fermions.
Exchange statistics in (2+1)D:
The Braid Group
Another way of saying this: in (3+1)D the particle statistics
correspond to representations of the group of permutations.
In (2+1)D, we should consider the braid group instead:
Abelian Anyons
• Different elements of the braid group correspond to disconnected
subspaces of trajectories in space-time.
• Possible choice: weight them by different overall phase factors
(Leinaas and Myrrheim, Wilczek).
• These phase factors realise an Abelian representation of the braid group.
E.g q = p/m for a Fractional Quantum Hall state at a filling factor ν = 1/m.
• Topological Order is manifested in the ground state degeneracy on higher
genus manifolds (e.g. a torus): m-fold degenerate ground states for FQHE
(Haldane, Rezayi ’88, Wen ’90).
Example (toy model) of Abelian Anyons :
charge q - flux Φ composites (Wilczek, '82)
The Aharonov - Bohm phase 2q  qΦ  qΦ  2qΦ
(in the units   c  1)
E.g. for q  e, Φ  2p/ne, the statistica l angle q  2π/n.
Non-Abelian Anyons
Exchanging particles 1 and 2:
Exchanging particles 2 and 3:
• Matrices M12 and M23 need not commute, hence Non-Abelian Statistics.
• Matrices M form a higher-dimensional representation of the braid-group.
• For fixed particle positions, we have a non-trivial multi-dimensional Hilbert
space where we can store information
The Quantum Hall Effect
e2

h
 
p
q
Eisenstein, Stormer, Science 248, 1990
“Unusual” FQHE states
Pan et al. PRL 83,1999
Gap at 5/2 is 0.11 K
Xia et al. PRL 93, 2004
Gap at 5/2 is 0.5K, at 12/5: 0.07K
FQHE Quasiparticle interferometer
Chamon, Freed, Kivelson, Sondhi, Wen 1997
Fradkin, Nayak, Tsvelik, Wilczek 1998
Measure longitudinal conductivity  xx due
to tunneling of edge current quasiholes
n
General Anyon Models
(unitary braided tensor categories)
Describes two dimensional systems with an energy gap.
Allows for multiple particle types and variable particle number.
1. A finite set of particle types or anyonic “charges.”
2. Fusion rules (specifying how charges can combine or split).
3. Braiding rules (specifying behavior under particle exchange).
There is a particle type I correspond ing to " vacuum."
Each particle type a has a " charge conjugate" a that is
the unique particle type which may fuse with a to give I .
The fusion rules are specified as :
where the integer N
c
ab
ab   N c
is the number of
c
ab
c
channels for fusing a and b into c. Diagrammatically,
this gives a set of allowed trivalent vertices :
Associativity relations for fusion:
 [F
abc
d
ef
]
f
Braiding rules:
a
b
b
R
ab
c
* These are all subject to consistency conditions.
a
Consistency conditions
A. Pentagon equation:
Consistency conditions
B. Hexagon equation:
Topological S-matrix
M ab
S ab S II


S Ia S Ib
M ab  1 corresponds to Abelian braiding and
M ab  1 iff the braiding is non - Abelian.
Back to the experiment al setup :
Need to calculate  xx

1 Uq2ab ab
tUt tM
 xxxx  t11U1t 2t Re
t cosU
2
2
2ab
2
*
ab ab
2 1 22 1 2ab
U1 ab
1
U 2 ab
b
i
a
e R ab
2
ab
M ab  ab R ab  M ab e
2
iq ab

 xx  t1  t 2  2 t1t 2 M ab cos  q ab 
2
    argt2 / t1 
2
is a parameter that can be experimentally varied.
|Mab| < 1 is a smoking gun that indicates non-Abelian braiding.
  5 / 2 is believed to be MR  U(1)  Ising
U(1) is a familiar Abelian factor due to electric charge
Ising particle types : I ,  , 
Fusion rules :
I  I  I,
I    ,
I    ,
    I  ,     ,    I
1 1 1 


Monodromy : M  1 0  1


1  1 1 
quasiholes carry anyonic charge : (e / 4,  )
electrons carry anyonic charge : (e,  )
n quasiholes carry anyonic charge : (ne / 4,  ) for n odd
(ne / 4, I or  ) for n even
Combined anyonic state of the antidot
Adding non-Abelian “anyonic charges” for the Moore-Read state:
Even-Odd effect
(Stern & Halperin; Bonderson, Kitaev & KS, PRL 2006):
  12 / 5 is believed to be RR 3  U(1)  Pf3  U(1)  Fib
(Note : Fibonacci anyons can simulate a universal quantum computer.)
Fib particle types : I , 
Fusion rules :
I  I  I,
I   ,    I  
Monodromy :
1 
1
M 
2 
1




where  
is the golden ratio (
1 5
2
2
 .38)
quasiholes carry anyonic charge : (e / 5,  )
electrons carry anyonic charge : ( e, I )
n quasiholes carry anyonic charge : (ne / 5, I or  )
For   5 / 2 :
n odd :  xx  t1  t 2
2
2
n even :  xx  t1  t 2  2 t1t 2  1
2
2
N
cos  np / 4 
where N  0 for I and N  1 for 
For   12 / 5 (Bonderson , KS & Slingerlan d PRL 2006;
Chung & Stone PRB 2006) :
 xx  t1  t 2  2 t1t 2  
2
2

 2 N
cos  n4p / 5
where N  0 for Fibonacci charge I and N  1 for 
Note :   2  .38
Topological Quantum Computation
(Kitaev, Preskill, Freedman, Larsen, Wang)
0 
a
b
a
c0
1 
b
c1

(Bonesteel, et. al.)
 Interferom eter
Topological Qubit
(Das Sarma, Freedman, Nayak)
One (or any odd number) of quasiholes per antidot. Their combined state
can be either I or , we can measure the state by doing interferometry:
The state can be switched by sending another quasihole through the middle constriction.
Topological Qubit
(the Fibonacci kind)
  12 / 5 qubit : the combined state of two antidots may be I or 
 xx  t1  t2  2 t1t2  
2
2

 2 N
cos  n4p / 5,
N  0 (1) for I ( )
The two states can be discriminated by the amplitude of the interference term!
Wish list
Get experimentalists to figure out how to perform these
very difficult measurements and, hopefully,
• Confirm non-Ableian statistics.
• Find a computationally universal topological phase.
• Build a topological quantum computer.
Conclusions
If we had bacon, we could have bacon and eggs, if
we had eggs…
Part Two: Measurements and Decoherence:
State vectors may be expressed diagrammatically in an
isotopy invariant formalism:


1/ 4
* Normalization factors d c / d a d b
that make the diagrams
isotopy invariant will be left implicit to avoid clutter.
Anyonic operators are formed by fusing and splitting
anyons (conserving anyonic charge).
For example a two anyon density matrix is:
Traces are taken by closing loops on the endpoints:
Mach-Zehnder
Interferometer
t j
Tj   *
rj
2
rj 
 t *j 
2
t j  rj  1
Applying an (inverse) F-move to the target system, we
have contributions from four diagrams:
Remove the loops by using:
0
to get…
M ab
S ab S 00

S 0 a S 0b
… the projection:

for the measurement outcome s ,  where
Result:
Interferometry with many b probes gives binomial
distributions in p s for the measurement outcomes, and
collapses the target onto fixed states with a definite
s
p
value of
. Hence, superpositions of charges a and a'
survive only if:
M ab  M a 'b
and only in fusion channels corresponding to difference
charges e with:
M eb  1