introduction to the physics of anyons with majorana...
TRANSCRIPT
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M. Baranov
Introduction to the Physics of Anyons
with
Majorana Fermions as an Example
Kaiserslautern 10 – 15 December 2018
Anyon Physics of Ultracold Atomic Gases
Institute for Quantum Optics and Quantum Information,
Center for Quantum Physics, University of Innsbruck
IQOQI
AUSTRIAN ACADEMY OF SCIENCESUNIVERSITY OF INNSBRUCK
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Lecture 2: Majorana Fermions as an
example of non-Abelian Anyons
Lecture 1: Appearance of Anyons
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Lecture 1: Appearance of Anyons
Formal introduction of anyons:
Fusion of anyons and anyon Hilbert space
Braid group. Abelian and no-Abelian anyons
Why dimension 2?
Exchange and statistics
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Exchange and statistics
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Particle exchange and statistics
),,(?),,(),,( 211221
rrrrrr
Exchange of two (quasi)particles
StatisticsBehavior of the state (wave function)
under the exchange of two
identical (quasi)particles
1 2
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has to be single-valued with respect to particle coordinates
Properties of many-body wave functions:
),,;,,( 2121
rrRR
jriR
positions of quasiparticles
positions of particles
but not necessarily with respect to quasiparticle coordinates iRjr
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Exchange as adiabatic dynamical evolution
General statements:
1. Adiabatic theorem: States in a (possibly degenerate) energy subspace
separated from others by a gap remain in the subspace when the system
is changed adiabatically without closing the gap.
2. Change under adiabatic transport = combination of Berry’s phase/matrix
and transformation of instantaneous energy eigenstate (explicit monodromy)
3. Change under adiabatic transport is invariant, but Berry’s phase/matrix and
eigenstate transformation depend on choice of gauge (and can be shifted from
one to the other)
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Exchange as adiabatic dynamical evolution
1
dynamical phase
)(1
tEdt
For a unique ground state (single-valued)
tim
e t
2
1 2
separated by a gap from excited states
Exchange statistics
2 1
, ie
Berry phase
g (path) geometrical phase
(topology of the path) statistical angle - of interest!
ie
)path(| gdt
ddti
(single-valued w.f.)
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Why dimension 2?
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1
tim
e t
2
2
relative position
1
12R
fR ,12
0
exchange2
1 2
1 2
0
Constraint on the exchange
double exchange
2Is an identity?
fR ,12
iR ,12
if RR ,12,12
iR ,12
iR ,12
iR ,12
if RR ,12,12
fR ,12
fR ,12
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Conclusion: in 3D only bosons ( ) or fermions ( )
3D case:
The contour C can be deformed
0
0
1ˆ 2 (identity!)
C
without crossing the origin
)(const ,1212 iRR
012 R
to a point
0
(i.e., to the case when nothing happens) fR ,12
iR ,12
iR ,12
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2D case:
The contour C cannot be deformed
0
C
without crossing the origin
to a point
0
(i.e., to the case when nothing happens)
Conclusion: in 2D more possibilities, not only bosons or fermions
1ˆ 2
iR ,12
iR ,12
fR ,12
)(const ,1212 iRR
012 R
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Braid group. Abelian and non-Abelian anyons
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Particle exchange in 2D: Braid group (for N particles)
i
Trajectories that wind around starting from initial positions
Note:
NRR
,,1to final positions (the same set – identical particles)
NRR
,,1
generated by defining relations
tim
e t
1iR
1iR
iR
2iR
1iR
1iR
iR
2iR
ijji ˆˆˆˆ for
111ˆˆˆˆˆˆ iiiiii
2 ji
1. Braid group is infinite dimensional ( !) in contrast
to finite-dimensional permutation group ( )
1ˆ 2 1ˆ 2 p
2. Braid group is non-Abelian iiii ˆˆˆˆ11
(braiding of particles i and i+1)
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Representations of the braid group: statistics of particles
1. One-dimensional representations: unique (ground) state
Elements of the braid group
(trajectories of particles)
2. Higher-dimensional representations: degenerate (ground) state
Changes of the states under the evolution
(particle statistics)
representation
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1. One-dimensional (Abelian) representations
1. bosons ( ) and fermions ( )
Transformation under braiding operation
Unique (ground) state
ieˆ
0 Examples:
2. quasiholes in the at FQHE Laughlin state
with arbitrary - Abelian anyons
M/1
M/
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N
lk
zM
lk
n N
i
i
n ZM
Mi
M
N
j
j
n
ezzzZeZZzZ 1
2
1
2
4
1
1 1
4
11
1 )()()(),(
Exchanging two quasiholes give a phase of
from eigenstate transformation (explicit monodromy)
Berry’s phase = Aharonov-Bohm phase (geometric)
of a charge encircling flux of
Trial wave function for N fermions at positions
with n quasiholes at positions :
Example 2. quasiholes in the FQHE Laughlin state M/1
M/ D. Arovas, J.R. Schrieffer,F.Wilczek,1984
R. Laughlin, 1987
B. Blok, X.G. Wen, 1992
cM
e Bg
)path(
Meq / BAB
R. Laughlin, 1983
ir
R
)(1|),(| 2
1
1
2
ZZ
i
M
N
i
i eOzZzd
Bjjj liyxz /)( BliYXZ /)(
normalization quasihole Laughlin w.f.
Choice 1
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Example 2. quasiholes in the FQHE Laughlin state M/1
Berry’s phase = Aharonov-Bohm phase + statistical angle
(different “gauge”: single-valued )Choice 2
N
lk
zM
lk
n N
i
i
n ZM
Mi
M
N
j
j
n
ezzzZeZZzZ 1
2
1
2
4
1
1 1
4
11
1 )()(),(
Eigenstate transformation (analytic continuation) = trivial
M/
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𝑈 𝜎 = 𝑃𝑒𝑥𝑝(𝑖 𝑑𝑡 𝑚)ℳ
Examples: Ising anyons (Majorana fermions, qH-state),
Fibonacci anyons
for at least two and
2. Higher-dimensional representations
Degenerate ground state
g,,1, with an orthonormal basis
Transformation under braiding operation
)ˆ(ˆU
with matrix
1Particles are non-Abelian anyons if
)ˆ()ˆ()ˆ()ˆ( 1221 UUUU
2(do not commute!)
dt
dim |)ˆ(
excited
states
gap
Berry matrix explicit monodromy of the w.f.
𝜈 = 5/2
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Conditions for non-Abelian anyons:
Robust degeneracy of the ground state:
CV loc
The degeneracy cannot be lifted by local perturbations
(which are needed, i.e., for braiding)
Degenerate ground states cannot be distinguished
by local measurements
Nonlocal measurements: parity measurements (for Majorana fermions), etc.
Require topological states of matter with
topological degeneracy and long-range entanglement
Braiding of identical particles changes state within the degenerate manifold,
but should not be visible for local observer
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Comment: In real world (finite systems, etc.): GS degeneracy is lifted
Condition on the time of operations:
T
excited
states
gap
~
slow enough to be adiabatic
fast enough to NOT resolve the GS manifold
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Fusion rules and Hilbert space
Formal introduction of Anyons:
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Fusion of anyons
Set of particles (anyons) cba ,,,1 1 - vacuum
b
a
“Observer”l
r
how do they behave as a combined object seen from distances
much large than the separation between them
Fusion of two Anyons :ba,
lr
?
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Fusion rules
c
c
ab cNba c
abN - integers
b
a
“Observer”l
rc
c
c
abN 1Non-Abelian anyons if for some a and b
a b
c
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: n anyons, where fuse into
and into and into , …,
Hilbert space – fusion chains
𝑒1, ⋯ , 𝑒𝑛−3
𝑎1, ⋯ , 𝑎𝑛−1 𝑎𝑛
𝑎2
𝑎1
𝑎4𝑎3 𝑎𝑛−1
𝑎𝑛𝑒1
⋯
𝑒2 𝑒𝑛−3
𝑎𝑛−2
1c
abN( for simplicity)
𝑎1 𝑎2and fuse into ,𝑒1 𝑒1 𝑎3 𝑒2 𝑒𝑛−3 𝑎𝑛−1 𝑒𝑛uniquely specified by the intermediate fusion outcomes
Dimension dimℋ𝑛 =
𝑒1…𝑒𝑛−3
𝑁𝑎1𝑎2⋯𝑒1 𝑁𝑒𝑛−3𝑎𝑛−1
𝑒𝑛
𝑎1, ⋯ , 𝑎𝑛(⋯ (𝑎1× 𝑎2) × 𝑎3) × ⋯) × 𝑎𝑛−1 = 𝑎𝑛
Basis vectors
(no creation and annihilations operators!)
ℋ𝑛
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'
'e
ee
abc
dF
a b c
e
d
a b c
d
'e
Associativity of braiding:
a,b fuse to e;
e,c fuse to d
b,c fuse to e’;
e’,a fuse to d
F-matrix (basis change)
𝑎 × 𝑏 × 𝑐 = 𝑎 × 𝑏 × 𝑐 = 𝑑
guaranties the invariance of the Hilbert space construction:
Different fusion ordering is equivalent to the change of the basis vectors
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R-matrix (braiding in a given fusion channel)
ab
cR
a b
c
a b
c
Exchange properties of anyons (in a given fusion channel)
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Consistency conditions for F- and R-matrices
'
'e
ee
abc
dF
a b c
e
d
a b c
d
'e ab
cR
a b
c
a b
c
F- and R-matrices satisfy the pentagon and hexagon
consistency relations
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Pentagon relation
e
eabdb
e
cedcb
a
da
c FFFFF 12341
5
23434
5
12
5
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Hexagon relation
12213
4
13123
4
1
4
231
4 acac
bba
b
cdRFRFRF
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If no solution exists the hypothetical set of anyons and fusion rule
are incompatible with local quantum physics.
Alternative approach: topological field theories
Set of anyons with F- and R-matrices satisfying the pentagon
and hexagon equations completely determine an Anyon model.
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Examples:
2: Ising anyons
1: Fibonacci anyons
,,1
1
1 },,1{},,1{1
Fusion rules:
,1
1
},1{},1{1
Fusion rules:
- non-Abelian anyon - fermion
(more in the next lecture)
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1. Split in pairs
Hilbert space for Majorana fields ( )m Nm 2,,1
N
N2
𝛾2𝑗−1, 𝛾2𝑗
2. Specify fusion channels for each pair j),1( Nj ,,1( )
The fusion tree is now uniquely define. The resulting “charge” is
either or depending on the number of the fusion outputs:
1
1 },,1{},,1{1
1 𝑛𝜓
1 for even 𝑛𝜓 for odd 𝑛𝜓
Fusion rules:
…𝛾1 𝛾2 𝛾3 𝛾4 𝛾2𝑁−1 𝛾2𝑁𝛾5 𝛾6
1 or 1 or
1 or1 or
1 or
1 or
Equivalent to the Hilbert space for fermionic modesN
1 or
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Thank you for your attention!
End of the Lecture 1