Exchange statistics. Basic concepts. University of Oxford April, Jon Magne Leinaas Department of Physics University of Oslo

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1 University of Oxford April, 2016 Exchange statistics Basic concepts Jon Magne Leinaas Department of Physics University of Oslo

2 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

3 The symmetrization postulate Landau and Lifshitz on the indistinguishability of identical particles: not the full story ->

4 Configuration space of identical particles: from permutations to topology 2 particles in 2D O r -r -r O r identify Relative coordinates r and -r representing the same configuration Elimination of double counting creates a cone Myrheim and Leinaas 1977

5 Quantum mechanics: from permutation to exchange symmetry In 2D: r r interchange of particle positions: viewed as a periodicity condition: bosons: fermions: anyons:

6 The fundamental group: from permutations to braids right-handed interchange braid representation 1 Braid group BN(R 2 ) σ k k+1 k k+1 k l k l replaces symmetric group SN = = generators σi, 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 iθ general group element: g = e inθ, n = 0,±1,±2,... n = winding number

7 In three dimensions: back to the permutations equivalence between right and left exchange e iθ = e iθ e iθ = ±1 only bosons and fermions possible braid group BN(R 3 ) = symmetric group SN

8 Anyons interpolating between bosons and fermions 0.0 bosons two-body wave function ψ l ( r,φ) r l+θ /π, r 0 (r) fermions statistics repulsion for l = 0 r Energy spectrum: Two anyons with harmonic oscillator interaction

9 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) see Jan Myrheim s talk Anyons treated as fermions with statistics interaction, but V is long range and singular! V = θ r i π ij r i r j r i r 2 j

10 Effects of topology φ 2 φ 1 Two particles on a circle coincidence points φ 1 φ 2 coincidence points identification of points gives mixing of topological effects: torus is changed to Möbius strip

11 Braid group on the torus ρ τ σ one dimensional representations: σ 2 = 1, only bosons and fermions σ 2 = τ 1 ρτρ 1 T. Einarsson 1990 matrix representations: ρ,τ noncommuting, commuting new degrees of freedom explicit construction: quantum Hall effect on a torus Haldane and Rezayi 1985

12 Anyons on a sphere 2D sphere: orientation of loop reversed by deformation over the sphere Braid group BN(S 2 ), constraint equation: 2 σ 1 σ 2...σ N 1...σ 2 σ 1 = 1 Number N of anyons related to statistics angle θ: ( N 1)θ = nπ n = integer effects of curvature: relation modified if the anyons carry spin Explicit construction: Quantum Hall effect on a sphere Haldane 1983 D. Li 1993

13 Non-abelian anyons matrix representations of the braid group BN(R 2 ) Meaningful generalization? consistent with the indistinguishable of the particles? 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 see Pacho s talk Examples: excitations of certain plateau states of a quantum Hall fluid vortex excitations in topological superfluids X-G. Wen 1991 G. Moore and N. Read 1991

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

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

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 2 S 1 S 1 transposition S 2 undetermined sign individual spins unchanged under parallel transport! P e

17 Spin-statistics theorem specifies 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

18 Charge-monopole composite example of coupling between spin and statistics eg n n integer = Dirac condition µ c 2 charge and monopole minimal value n=1 -> spin s = 1 / 2 as spinless bosons S e g is it a fermion? Calculation of the exchange effect by deforming the path g d g exchange of of tightly bound eg pairs λ : monopoles g removed far from the charges e phase angle determined by surface integral r e λ =0: d e r r r θ = d λ d τ 3 = π τ λ r 0 0 r = 2 ρ ( cos(πτ )e1 + sin(πτ )e2 ) + λ e3 1 it is a fermion! Leinaas 1978

19 Spin and statistics in 2D Rotation group SO(2) continuous spin s Spin-statistics relation: (mod 1)? Example: charge - fluxtube composite, s-s relation satisfied generalized spin-statistics relation a = anyon, a = anti-anyon, no long range effects from an aa pair Wilczek 1982 a statistics angles: a a spin: } Einarsson, Sondhi, Girvin and Arovas 1995

20 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 multi-vortex state quantum Lagrangian = 1 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 Hansson, Isakov, Leinaas and Lindström 2001 Arovas, Schrieffer and Wilczek 1984

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

22 An alternative description of identical particles Heisenberg quantization (in 1d) Observables are symmetric in particle indices Basic observables for two particles relative coordinates A = 1 4 (p2 + x 2 ), B = 1 4 (p2 x 2 ), C = 1 4 The fundamental algebra (px + xp) A [ A, B] = ic, [ B,C] = ia, [ C, A] = ib su(1,1) Quantization as irreducible representations B - B + A a = a a, a = a 0 + n n = integer B ± a = b ± a ±1, B ± = B ± ic 1/4 a 0 statistics parameter a 0 > 0 1/4 1/2 3/4 Bose Fermi bosons: a 0 = 1/ 4 fermions: a 0 = 3 / 4 Myrheim and Leinaas 1988

23 Heisenberg quantization deriving the Berry connection s(1,1) coherent states β n B β = β β β = N β n,a 0 N n!γ(n + 2a 0 ) β = n=0 β 2a 0 1 I 2a0 1(2 β ) Berry connection A φ = 2 β (β β β β ) β = 2 β I 2a 0 (2 β ) I 2a0 1(2 β ) for β >> 1 A φ = 2 β 2(a )) β 1 ( 4 p2 + x 2 ) = 1 4 R2 statistics parameters 2(a ) = θ π = g three different approaches, give the same result (but not in higher dimensions!)

24 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...