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

University of Oxford 12-15 April, 2016 Exchange statistics Basic concepts Jon Magne Leinaas Department of Physics University of Oslo søndag 10. april 16

Outline * con fi guration space with identi fi cations * from permutations to braids * e ff ects of topology * fi ber bundles and parallel transport *spin and statistics * anyons in a strong magnetic fi eld søndag 10. april 16

The symmetrization postulate Landau and Lifshitz on the indistinguishability of identical particles: not the full story -> søndag 10. april 16

Con fi guration space of identical particles: from permutations to topology 2 particles in 2D O r r O -r -r identify Relative coordinates r and -r Elimination of representing the same double counting con fi guration creates a cone Myrheim and Leinaas 1977 søndag 10. april 16

Quantum mechanics: from permutation to exchange symmetry In 2D: r r � � interchange of particle positions: viewed as a periodicity condition: bosons: fermions: anyons: søndag 10. april 16

The fundamental group: from permutations to braids 1 2 2 right-handed braid representation interchange 1 σ Braid group B N ( ℝ 2 ) k l k l k k+1 k k+1 replaces symmetric group S N = = generators σ i ! , !i=1,2,..., !(N-1) ! σ k σ k + 1 σ k = σ k + 1 σ k σ k + 1 σ k σ l = σ l σ k , | k − l | > 1 one-dimensional representations: σ i = e i θ g = e in θ , n = 0, ± 1, ± 2,... general group element: n = winding number søndag 10. april 16

In three dimensions: back to the permutations equivalence between right and left exchange e i θ = e − i θ ⇒ e i θ = ± 1 only bosons and fermions possible braid group B N ( ℝ 3 ) = symmetric group S N søndag 10. april 16

Anyons interpolating between bosons and fermions ������ 0.0 bosons 0.1 two-body wave function 0.25 0.5 � (r) ( ) ≈ r l + θ / π , r → 0 ψ l r , φ 0.75 1.0 fermions l = 0 → statistics repulsion for 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 - fi eld theory: no simple def. of creation/annihilation operators Many-anyon problem: Partial understanding based on see Jan Myrheim’s talk numerical studies and approximation methods (perturbation, mean fi eld)  i −  Anyons treated as fermions V = θ  r r j ∑  × r  i −  with statistics interaction, π i 2 r r j ij but V is long range and singular! søndag 10. april 16

E ff ects of topology Two particles φ 2 φ 1 on a circle coincidence points φ 1 φ 2 coincidence points identi fi cation of points gives mixing of topological e ff ects: torus is changed to Möbius strip søndag 10. april 16

Braid group on the torus ρ σ τ σ 2 = τ − 1 ρτρ − 1 one dimensional representations: σ 2 = 1 , only bosons and fermions T. Einarsson 1990 matrix representations: ρ , τ noncommuting, commuting ⇒ new degrees of freedom explicit construction: Haldane and Rezayi 1985 quantum Hall e ff ect on a torus søndag 10. april 16

Anyons on a sphere 2D sphere: orientation of loop reversed by deformation over the sphere Braid group B N ( S 2 ), constraint equation: 2 ... σ 2 σ 1 = 1 σ 1 σ 2 ... σ N − 1 Number N of anyons related to statistics angle θ : ( ) θ = n π N − 1 n = integer e ff ects of curvature: relation modi fi ed if the anyons carry spin Explicit construction: Haldane 1983 Quantum Hall e ff ect on a sphere D. Li 1993 søndag 10. april 16

Non-abelian anyons matrix representations of the braid group B N ( ℝ 2 ) Meaningful generalization? consistent with the indistinguishable of the particles? New degrees of freedom: anyons as excitations in a topological fl uid, the quantum state of the fl 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 fl uid vortex excitations in topological super fl 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: fi ber bundles abstract vectors section of fi 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 fi eld søndag 10. april 16

Braids and geometry fl at fi ber bundles simply connected space: path independent globally de fi ned parallel basis multiply connected space: depends on homotopy classes of paths unitary representation of braid group closed loops: indistinguishable particles in ℝ 2 : n= winding number parallel basis multivalued (2 anyons) wave function singlevalued basis Aharonov-Bohm type of gauge fi eld connection 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 1 S 2 S 1 S 2 transposition P e undetermined sign individual spins unchanged under parallel transport! søndag 10. april 16

Spin-statistics theorem speci fi es the sign change basis to total spin symmetry of CG coe ff .: M=m 1 +m 2 gives symmetries ⇒ 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 µ ≡ eg  c = n Dirac condition n integer S 2 e charge and monopole minimal value n=1 -> spin s = 1/ 2 as spinless bosons g is it a fermion? Calculation of the exchange e ff ect by deforming the path λ = 0 : exchange of of tightly bound eg pairs g λ → ∞ : monopoles g removed far from the charges e d g phase angle determined by surface integral r ∂τ × ∂  ∂   e � ∞ 1 ⎛ ⎞ r r r ∫ ∫ θ = d λ d τ ⎟ ⋅ r 3 = −π ⎜ it is a fermion! ⎝ ⎠ ∂λ e 0 0 d r = 2 ρ cos( πτ )   e 1 + sin( πτ )  ) + λ  ( Leinaas 1978 e 2 e 3 søndag 10. april 16

Spin and statistics in 2D Rotation group SO(2) continuous spin s Spin-statistics relation: (mod 1) ? Example: charge - fl uxtube composite, s-s relation satis fi ed Wilczek 1982 generalized spin-statistics relation a = anyon, a = anti-anyon, no long range e ff ects 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 fi eld Quasiholes in a quantum Hall fl uid: vortex like excitations - physical space like a phase space Use a semiclassical decription multi-vortex state  = 1 quantum Lagrangian restricted to manifold of vortex states Berry connection potential Two-vortex system Arovas, Schrie ff er and Wilczek 1984 Integral determines: 1) Berry phase associated with the the loop Hansson, Isakov, Leinaas and Lindström 2001 2) phase space area within the loop søndag 10. april 16

Anyons in the lowest landau level boson Two anyons with relative coordinate z (complex): anyon ω = − f zz d z ∧ dz symplectic form fermion ds 2 = − 2 i f zz d z dz R metric f zz = ∂ z A z − ∂ z A z with N+1 particles: available phase space area smoothened of last particle added within relative space radius R Reduction in number of quantum states g = θ exclusion statistics with statistics parameter π Haldane 1991 søndag 10. april 16

An alternative description of identical particles Heisenberg quantization (in 1d) Observables are symmetric in particle indices Basic observables for two particles relative coordinates 4 ( p 2 + x 2 ), 4 ( p 2 − x 2 ), A = 1 B = 1 C = 1 4 ( px + xp ) A The fundamental algebra [ ] = − iC , [ ] = − iA , [ ] = iB A , B B , C C , A su(1,1) B - B + Quantization as irreducible representations A a = a a , a = a 0 + n n = integer B ± a = b ± a ± 1 , B ± = B ± iC a 0 1/4 a 0 > 0 1/4 1/2 3/4 statistics parameter Bose Fermi a 0 = 1/ 4 fermions: a 0 = 3/ 4 bosons: Myrheim and Leinaas 1988 søndag 10. april 16

Heisenberg quantization deriving the Berry connection s(1,1) coherent states ∞ β n 2 a 0 − 1 ∑ β B − β = β β ⇒ β = N β N β = n , a 0 n ! Γ ( n + 2 a 0 ) I 2 a 0 − 1 (2 β ) n = 0 Berry connection A φ = 2 β ( β ∂ β − β ∂ β ) β = 2 β I 2 a 0 (2 β ) I 2 a 0 − 1 (2 β ) ( ) = 1 4 p 2 + x 2 β ≈ 1 4 R 2 A φ = 2 β − 2( a 0 − 1 for β >> 1 4)) 4) = θ 2( a 0 − 1 π = g statistics parameters three di ff erent 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 fl 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: quantum computing ➟ .... søndag 10. april 16