Exchange statistics Basic concepts Jon Magne Leinaas Department of - - PowerPoint PPT Presentation

Exchange statistics

Jon Magne Leinaas Department of Physics University of Oslo University of Oxford 12-15 April, 2016

Basic concepts

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Outline

* configuration space with identifications * from permutations to braids * effects of topology * fiber bundles and parallel transport *spin and statistics * anyons in a strong magnetic field

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The symmetrization postulate

Landau and Lifshitz on the indistinguishability of identical particles: not the full story ->

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Configuration space of identical particles:

from permutations to topology

Relative coordinates r and -r representing the same configuration Elimination of double counting creates a cone

r

• r

identify O r

• r

O

2 particles in 2D

Myrheim and Leinaas 1977

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Quantum mechanics:

from permutation to exchange symmetry In 2D:

anyons: interchange of particle positions: viewed as a periodicity condition: bosons: fermions:

• r
• r

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The fundamental group:

from permutations to braids

right-handed interchange braid representation

generators σi !, !i=1,2,..., !(N-1) ! Braid group BN(ℝ2) replaces symmetric group SN

=

k k+1 k k+1 k k l l

=

• ne-dimensional representations: σ i = eiθ

general group element:

n = winding number

g = einθ , n = 0,±1,±2,...

σ kσ l = σ lσ k, | k − l | >1 σ kσ k+1σ k = σ k+1σ kσ k+1

σ

1 1 2 2

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eiθ = e−iθ ⇒ eiθ = ±1

• nly bosons and fermions possible

In three dimensions:

back to the permutations

equivalence between right and left exchange braid group BN(ℝ3) = symmetric group SN

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Anyons

interpolating between bosons and fermions

Energy spectrum: Two anyons with harmonic

• scillator interaction

ψ l r,φ

( ) ≈ rl+θ/π, r → 0

two-body wave function

l = 0→

statistics repulsion for

1.0 0.75 0.5 0.25 0.1 0.0 bosons

fermions r (r)

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Many anyons

multivalued wave functions

Non-interacting anyons:

• no simple rule for occupation of single-particle states
• no simple generalization of BE/FD statist. distributions
• field theory: no simple def. of creation/annihilation operators

Many-anyon problem: Partial understanding based on numerical studies and approximation methods (perturbation, mean field) Anyons treated as fermions with statistics interaction, but V is long range and singular!

V = θ π   r

i ij

∑

×  r

i − 

rj  r

i − 

rj

2

see Jan Myrheim’s talk

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φ1 φ2

Effects of topology

Two particles

• n a circle

identification of points gives mixing of topological effects: torus is changed to Möbius strip

φ2

φ1

coincidence points coincidence points

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Braid group on the torus

σ 2 = τ −1ρτρ−1 σ

ρ

τ

• ne dimensional representations:

σ 2 = 1, only bosons and fermions matrix representations: ρ,τ noncommuting,

⇒ new degrees of freedom

explicit construction: quantum Hall effect on a torus

• T. Einarsson 1990

commuting

Haldane and Rezayi 1985

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Anyons on a sphere

2D sphere: orientation of loop reversed by deformation over the sphere Number N of anyons related to statistics angle θ:

n = integer

N −1

( )θ = nπ

Braid group BN(S2), constraint equation:

σ 1σ 2...σ N−1

2 ...σ 2σ 1 = 1

effects of curvature: relation modified if the anyons carry spin Explicit construction: Quantum Hall effect on a sphere

Haldane 1983

• D. Li 1993

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Non-abelian anyons

matrix representations of the braid group BN(ℝ2)

Meaningful generalization? consistent with the indistinguishable of the particles? Examples: excitations of certain plateau states of a quantum Hall fluid vortex excitations in topological superfluids New degrees of freedom: anyons as excitations in a topological fluid, the quantum state of the fluid changes Topologically protected degrees of freedom: attractive for quantum computation

X-G. Wen 1991

• G. Moore and N. Read 1991

see Pacho’s talk

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Braids and geometry

parallel transport of a quantum states

Basic formulation: fiber bundles abstract vectors section of fiber bundle parallel transport depend on path C from x to y local transport path independent derivative wave function expansion on local basis covariant derivative connection gauge field

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Braids and geometry

flat fiber bundles

simply connected space: path independent parallel basis multiply connected space: depends on homotopy classes of paths closed loops: indistinguishable particles in ℝ2: n= winding number multivalued wave function connection parallel basis (2 anyons) singlevalued basis unitary representation of braid group Aharonov-Bohm type of gauge field globally defined

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Spin and statistics

A geometric proof of the spin-statistics theorem?

Fiber bundle approach ( 2 particles in 3D) Exchange of particle positions:

S1 S2 S1 S2 Pe

local spin space

• rder arbitrarily chosen

product basis

transposition

individual spins unchanged under parallel transport!

undetermined sign

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Spin-statistics theorem

change basis to total spin symmetry of CG coeff.: gives symmetries

⇒

Example: spin 1 = vector independent of individual spins s same as for relative position vectors M=m1+m2

specifies the sign

is that of importance?

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Charge-monopole composite

example of coupling between spin and statistics

Dirac condition

µ ≡ eg c = n 2

minimal value n=1 -> spin s = 1/ 2 Calculation of the exchange effect by deforming the path

charge and monopole as spinless bosons

is it a fermion?

 r = 2ρ cos(πτ ) e1 + sin(πτ ) e2

( )+ λ

e3

λ = 0 :

exchange of of tightly bound eg pairs

λ → ∞ :

monopoles g removed far from the charges e

phase angle determined by surface integral

e g S

e e g g d d

• r

n integer

θ = dλ dτ

1

∫

∞

∫

∂ r ∂τ × ∂ r ∂λ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⋅  r r3 = −π

it is a fermion!

Leinaas 1978

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Spin and statistics in 2D

Spin-statistics relation: (mod 1) ? Rotation group SO(2) continuous spin s Example: charge - fluxtube composite, s-s relation satisfied

a a a

generalized spin-statistics relation a = anyon, a = anti-anyon, no long range effects from an aa pair statistics angles: spin:

}

Einarsson, Sondhi, Girvin and Arovas 1995 Wilczek 1982

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Anyons in a strong magnetic field

Quasiholes in a quantum Hall fluid: vortex like excitations - physical space like a phase space Use a semiclassical decription quantum Lagrangian restricted to manifold of vortex states Berry connection potential Two-vortex system Integral determines: 1) Berry phase associated with the the loop 2) phase space area within the loop multi-vortex state

Arovas, Schrieffer and Wilczek 1984 Hansson, Isakov, Leinaas and Lindström 2001

 = 1

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Anyons in the lowest landau level

Two anyons with relative coordinate z (complex):

ω = − fzz d z ∧ dz ds2 = −2i fzz d z dz

symplectic form metric

R

smoothened relative space

boson anyon fermion

exclusion statistics with statistics parameter

Haldane 1991

fzz = ∂z Az − ∂z Az

with Reduction in number of quantum states

g = θ π

N+1 particles: available phase space area

• f last particle added

within radius R

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An alternative description of identical particles

Heisenberg quantization (in 1d)

Basic observables for two particles

A = 1 4 (p2 + x2), B = 1 4 (p2 − x2), C = 1 4 (px + xp)

The fundamental algebra

A,B

[ ] = −iC,

B,C

[ ] = −iA,

C,A

[ ] = iB

su(1,1)

Quantization as irreducible representations A a = a a , a = a0 + n

B± a = b± a ±1 , B± = B ± iC

n = integer

a0 > 0 statistics parameter

a0 = 1/ 4

fermions: a0 = 3/ 4 Observables are symmetric in particle indices

relative coordinates Myrheim and Leinaas 1988

bosons:

A

Bose

a0

1/4 1/4 1/2 3/4 Fermi

B+ B-

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Heisenberg quantization

deriving the Berry connection

B− β = β β ⇒ β = Nβ β n n!Γ(n + 2a0)

n=0 ∞

∑

n,a0

Nβ = β

2a0−1

I2a0−1(2 β )

s(1,1) coherent states Berry connection

Aφ = 2 β (β ∂β − β ∂β ) β = 2 β I2a0(2 β ) I2a0−1(2 β )

for β >>1

Aφ = 2 β − 2(a0 − 1 4))

statistics parameters

β ≈ 1 4 p2 + x2

( ) = 1

4 R2

2(a0 − 1 4) = θ π = g

three different approaches, give the same result (but not in higher dimensions!)

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A brief summary

➭ Exchange statistics

➟ from permutations to braids ➟ relates quantum statistic to topology ➟ anyons in 2D

➭ The importance of topology

➟ anyons as excitations in topological fluids

➭ Spin and statistics

➟ a «natural» relation, but no proof

➭ 2D space as a phase space

➟ statistics phase -> phase space reduction

➭ Furthermore

➟ Many anyons: unsolved problems ➟ Non-abelian anyons: quantum computing ➟ ....

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