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Home , Anyon, Identical particles

University of Oxford 12-15 April, 2016

Exchange statistics Basic concepts

Jon Magne Leinaas Department of University of Oslo

søndag 10. april 16 Outline

* con!guration space with identi!cations * from permutations to braids * effects of * !ber bundles and parallel transport * and statistics * in a strong magnetic !eld

søndag 10. april 16 The symmetrization postulate

Landau and Lifshitz on the indistinguishability of identical :

not the full story ->

søndag 10. april 16 Con!guration space of : r from permutations to topology -r

2 particles in 2D

O r r O -r -r identify

Relative coordinates r and -r Elimination of O representing the same double counting r con!guration creates a cone

O -r identify Myrheim and Leinaas 1977

søndag 10. april 16 : from permutation to exchange

In 2D:

r r

interchange of positions:

viewed as a periodicity condition:

: : anyons:

søndag 10. april 16 The : from permutations to braids

1 2

2

right-handed braid representation interchange 1 σ 2 BN(ℝ ) k k+1 k k+1 k l k l replaces SN = = generators

σ i , i=1,2,..., (N-1)

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

iθ one-dimensional representations: σ i = e

in general group element: g = e θ , n = 0,±1,±2,... n = winding number

søndag 10. april 16 In three : back to the permutations

equivalence between right and left exchange

eiθ = e−iθ ⇒ eiθ = ±1

only bosons and fermions possible

braid group BN(ℝ3) = symmetric group SN

søndag 10. april 16 Anyons interpolating between bosons and fermions

0.0 bosons

0.1 two-body 0.25

0.5 (r) r, rl+θ /π , r 0 0.75 ψ l ( φ) ≈ → 1.0 fermions statistics repulsion for l = 0→

r

Energy spectrum: Two anyons with harmonic oscillator interaction

søndag 10. april 16 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 - !eld theory: no simple def. of creation/annihilation operators

Many- problem: Partial understanding based on numerical studies and approximation see Jan Myrheim’s talk methods (perturbation, mean !eld)

Anyons treated as fermions   θ  ri − rj V = r × with statistics interaction, π ∑ i   2 ij ri − rj but V is long range and singular!

søndag 10. april 16 Effects of topology

Two particles φ 2 φ1 on a circle

coincidence points

φ1

φ2 coincidence points

identi!cation of points gives mixing of topological effects: torus is changed to Möbius strip

søndag 10. april 16 Braid group on the torus

ρ σ τ σ 2 = τ −1ρτρ −1 one dimensional representations:

2 T. Einarsson 1990 σ = 1 , only bosons and fermions

representations: ρ,τ noncommuting, commuting

⇒ new degrees of freedom

explicit construction: quantum Hall effect on a torus Haldane and Rezayi 1985

søndag 10. april 16 Anyons on a sphere

2D sphere: orientation of loop reversed by deformation over the sphere

Braid group BN(S2), constraint equation:

2 σ 1σ 2 ...σ N−1...σ 2σ 1 = 1

Number N of anyons related to statistics angle θ:

(N −1)θ = nπ n = effects of curvature: relation modi!ed if the anyons carry spin

Explicit construction: Haldane 1983 Quantum Hall effect on a sphere D. Li 1993

søndag 10. april 16 Non-abelian anyons

matrix representations of the braid group BN(ℝ2)

Meaningful generalization? consistent with the indistinguishable of the particles?

New degrees of freedom: anyons as excitations in a topological $uid, the of the !uid changes

Topologically protected degrees of freedom: attractive for quantum computation see Pacho’s talk

Examples: excitations of certain plateau states of a quantum Hall $uid vortex excitations in topological super$uids X-G. Wen 1991 G. Moore and N. Read 1991

søndag 10. april 16 Braids and geometry parallel transport of a quantum states

Basic formulation: !ber bundles

abstract vectors section of !ber 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 !eld

søndag 10. april 16 Braids and geometry $at !ber bundles

simply connected space: path independent

parallel basis globally de!ned

multiply connected space: depends on classes of paths

closed loops: unitary representation of braid group

indistinguishable particles in ℝ2: n= winding number

parallel basis multivalued (2 anyons) wave function

singlevalued basis

connection Aharonov-Bohm type of gauge !eld

søndag 10. april 16 Spin and statistics A geometric proof of the spin-statistics theorem?

Fiber bundle approach ( 2 particles in 3D)

local spin space

product basis order arbitrarily chosen

Exchange of particle positions: S S2 1

S1 S2 transposition

Pe undetermined sign individual spins unchanged under parallel transport!

søndag 10. april 16 Spin-statistics theorem

speci!es the sign

change basis to total spin

symmetry of CG coeff.:

gives symmetries M=m1+m2

⇒ independent of individual spins s

Example: spin 1 = vector is that of importance? same as for relative position vectors

søndag 10. april 16 Charge-monopole composite example of coupling between spin and statistics

eg n S Dirac condition µ ≡ = n integer e c 2 s 1/ 2 charge and monopole minimal value n=1 -> spin = as spinless bosons g is it a ?

Calculation of the exchange effect by deforming the path

λ = 0 : exchange of of tightly bound eg pairs g d λ → ∞ : monopoles g removed far from the charges e

g phase angle determined by surface integral r e ∞ 1    ⎛ ∂r ∂r ⎞ r θ = dλ dτ ⎜ × ⎟ ⋅ 3 = −π it is a fermion! e ∫ ∫ ⎝ ∂τ ∂λ ⎠ r d 0 0     r = 2ρ cos(πτ )e + sin(πτ )e + λe Leinaas 1978 ( 1 2 ) 3

søndag 10. april 16 Spin and statistics in 2D Rotation group SO(2) continuous spin s Spin-statistics relation: (mod 1) ? Example: charge - $uxtube composite, s-s relation satis!ed Wilczek 1982

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

a a statistics angles:

spin: }

Einarsson, Sondhi, Girvin and Arovas 1995

søndag 10. april 16 Anyons in a strong magnetic !eld

Quasiholes in a quantum Hall $uid: vortex like excitations - physical space like a

Use a semiclassical decription multi-vortex state

quantum Lagrangian  = 1 restricted to manifold of vortex states Berry connection potential

Two-vortex system Arovas, Schrieffer and Wilczek 1984 Integral determines: 1) Berry phase associated with the the loop Hansson, Isakov, Leinaas 2) phase space area within the loop and Lindström 2001

søndag 10. april 16 Anyons in the lowest landau level Two anyons with relative coordinate z (complex): anyon

fermion symplectic form ω = − fzz d z ∧ dz 2 R metric ds = −2i fzz d z dz

with fzz = ∂z Az − ∂z Az

N+1 particles: available phase space area smoothened of last particle added within relative space radius R

Reduction in number of quantum states θ exclusion statistics with statistics parameter g = π Haldane 1991

søndag 10. april 16 An alternative description of identical particles Heisenberg quantization (in 1d)

Observables are symmetric in particle indices

Basic for two particles relative coordinates

1 2 2 1 2 2 1 A = (p + x ), B = (p − x ), C = (px + xp) A 4 4 4 The fundamental algebra

[A, B] = −iC, [B,C] = −iA, [C, A] = iB su(1,1) B B Quantization as irreducible representations - +

A a = a a , a = a0 + n n = integer

B± a = b± a ±1 , B± = B ± iC 1/4 a0 statistics parameter a > 0 1/4 1/2 3/4 0 Bose Fermi

bosons: a0 = 1/ 4 fermions: a0 = 3 / 4 Myrheim and Leinaas 1988

søndag 10. april 16 Heisenberg quantization deriving the Berry connection

s(1,1) coherent states

∞ n 2a 1 β β 0 − B− β = β β ⇒ β = Nβ n,a0 N = ∑ β I (2 β ) n=0 n!Γ(n + 2a0 ) 2a0 −1

Berry connection I (2 ) 2a0 β Aφ = 2 β (β ∂β − β ∂β ) β = 2 β I2a −1(2 β ) 1 1 0 β ≈ ( p2 + x2 ) = R2 1 4 4 for β >> 1 A = 2 β − 2(a − )) φ 0 4

1 θ statistics parameters 2(a − ) = = g 0 4 π

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

søndag 10. april 16 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 $uids

➭ 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: ➟ ....

søndag 10. april 16