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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!guration space with identi!cations * from permutations to braids * effects of topology * !ber bundles and parallel transport *spin and statistics * anyons in a strong magnetic !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!guration space of identical particles: 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 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 2 Braid group BN(ℝ ) k k+1 k k+1 k l k l replaces symmetric group 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 dimensions: back to the permutations equivalence between right and left exchange eiθ = e−iθ ⇒ eiθ = ±1 only bosons and fermions possible 3 braid group BN(ℝ ) = symmetric group SN 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, 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-anyon 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 matrix 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 = integer 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 quantum state 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 homotopy 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 fermion? 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 Leinaas 1978 r = 2ρ(cos(πτ )e1 + sin(πτ )e2 ) + λe3 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 phase space 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 boson 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 observables 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: quantum computing ➟ .... søndag 10. april 16.
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