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

Lecture 2: Majorana Fermions as an

example of non-Abelian Anyons

Lecture 1: Appearance of Anyons

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

Exchange and statistics

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

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

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)

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.)

Why dimension 2?

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

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

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

Braid group. Abelian and non-Abelian anyons

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)

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

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/

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

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/

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

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

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

Fusion rules and Hilbert space

Formal introduction of Anyons:

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

?

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

: 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

'

'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

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)

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

Pentagon relation

e

eabdb

e

cedcb

a

da

c FFFFF 12341

5

23434

5

12

5

Hexagon relation

12213

4

13123

4

1

4

231

4 acac

bba

b

cdRFRFRF

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.

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)

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

Thank you for your attention!

End of the Lecture 1