1. Fractional Statistics of Quantum Particles D. P. Arovas, UCSD Major ideas:J. M. Leinaas, J. Myrheim, R. Jackiw, F. Wilczek, M. V. Berry, Y-S. Wu, B. I. Halperin, R. Laughlin, F. D. M. Haldane, N. Read, G. Moore Collaborators:J. R. Schrieffer, F. Wilczek, A. Zee, T. Einarsson, S. L. Sondhi, S. M. Girvin, S. B. Isakov, J. Myrheim, A. P. Polychronakos Texts: Geometric Phases in Physics Fractional Statistics and Anyon Superconductivity (both World Scientific Press) 7 Pines meeting, Stillwater MN, May 6-10 2009

3. Two classes of quantum particles: bosons gaugematter - real or complex quantum fields - symmetric wavefunctions - condensation breaks U(1) - classical limit: fermions quarksleptons - antisymmetric wavefunctions - Grassmann quantum fields - must pair to condense - no classical analog:

4. e e n pp e e 3 He 4 He bosonfermion np n p ++

5. Is that you, Gertrude? Quantum Mechanics of Identical Particles Only two one-dimensional representations of S N : Bose:Fermi: Eigenfunctions of H ̂ classified by unitary representations of S N : Hamiltonian invariant under label exchange: where i.e.

6. Path integral description QM propagator: (manifold) Paths on M are classified by homotopy : andhomotopic if { with smoothly deformable YES NO

7. The propagator is expressed as a sum over homotopy classes μ : Path composition ⇒ group structure : π 1 ( M ) = “fundamental group” weight for class μ In order that the composition rule be preserved, the weights χ(μ) must form a unitary representation of π 1 ( M ) : Think about the Aharonov-Bohm effect :

8. Laidlaw and DeWitt (1971) : quantum statistics and path integrals one-particle “base space” configuration space for N distinguishable particles ? for indistinguishables? But...not a manifold! how to fix : disconnected: simply connected: multiply connected : N-string braid group Y-S. Wu (1984) Then :

9. == generated by : - unitary one-dimensional representations of: - absorb into Lagrangian: - topological phase : change in relative angle with

10. Charged particle - flux tube composites : (Wilczek, 1982) exchange phase Particles see each other as a source of geometric flux : physicalstatistical Gauge transformation : Anyon wavefunction : single-valued multi-valued

11. Low density limit : bosons fermions { BBB FF DPA (1985)Johnson and Canright (1990) B F BB F How do anyons behave? Anyons break time reversal symmetry when i.e. for values of θ away from the Bose and Fermi points. What happens at higher densities??

12. Chern-Simons Field Theory and Statistical Transmutation So we obtain an effective action, linking Wilczek and Zee (1983) Examples: ordinary matter, skyrmions in O(3) nonlinear σ-model, etc. lazy HEP convention: metric Given any theory with a conserved particle current, we can transmute statistics: Chern-Simons termminimal coupling via equations of motion:Integrate out the statistical gauge field ⇒ statistical b-field particle density

13. Anyon Superconductivity The many body anyon Hamiltonian contains only statistical interactions: The magnetic field experienced by fermion i is fermions plus residual statistical interaction filling fraction ⇒ filled Landau levels Mean field Ansatz : ⇒ Landau levels : Total energy ⇒ sound mode : But absence of low-lying particle-hole excitations ⇒ superfluidity! (?)

14. Anyons in an external magnetic field : (n+1) th Landau level partially filled + n th Landau level partially empty + Y. Chen et al. (1989) Meissner effect confirmed by RPA calculations system prefers B=0 A. Fetter et al. (1989) ⇒

15. Signatures of anyon superconductivity - Zero field Hall effect - local orbital currents - reflection of polarized light Wen and Zee (1989) Y. Chen et al. (1989) - charge inhomogeneities at vortices - route to anyon SC doesn’t hinge on broken U(1) symmetry Unresolved issues (not much work since early 1990’s) “spontaneous violation of fact” (Chen et al.) - Pairing? BCS physics? Josephson effect? p evenp odd q odd q even B/F F B statistics of parent duality treatments of Fisher, Lee, Kane

16. Fractional Quantum Hall Effect Laughlin state at: Quasihole excitations: Quasihole charge deduced from plasma analogy (1983) The Hierarchy - Haldane / Halperin - condensation of quasiholes/quasiparticles (1983 / 1984) - Halperin : “pseudo-wavefunction” satisfying fractional statistics

17. Geometric phases Adiabatic evolution adiabatic WFsolution to SE (projected) Complete path : where M. V. Berry (1984) Evolution of degenerate levels ➙ nonabelian structure : Path : where Wilczek and Zee (1984)

18. Adiabatic quasihole statistics DPA, Schrieffer, Wilczek (1984) - Compute parameters in adiabatic effective Lagrangian quasihole chargefrom Aharonov-Bohm phase : This establishesin agreement with Laughlin For statistics, examine two quasiholes: ⇒ Exchange phase is then

19. Numerical calculations of e * and θ Sang, Graham and Jain (2003-04) Kjo ̸ nsberg and Myrheim (1999) Laughlin quasielectronsJain quasielectrons - good convergence for quasihole states - quasielectrons much trickier ; convergence better for Jain’s WFs - must be careful in defining center of quasielectron charge statistics

20. Effective field theory for the FQHE Girvin and MacDonald (1987) ; Zhang, Hansson, and Kivelson (1989) ; Read (1989) Basic idea : fermions = bosons + Extremize the action : Solution :,, incompressible quantum liquid with

21. Quasiparticle statistics in the CSGL theory - ‘duality’ transformation to quasiparticle variables reveals fractional statistics with new CS term! - quasiparticles are vortices in the bosonic field,

22. Statistics and interferometry : Stern (2008) S D Fabry-Perot relative phase : changing B will nucleate bulk quasiholes, resulting in detectable phase interference S D Mach-Zehnder relative phase : phase interference depends on number of quasiparticles which previously tunneled - dependence ⇒ fractional statistics

23. Nonabelions Moore and Read (1991) Nayak and Wilczek (1996) Read and Green (2000) Ivanov (2001) - For M Laughlin quasiholes, one state : quasihole creator - At,there are states withquasiholes : with - The degrees of freedom are essentially nonlocal, and are associated with Majorana fermions - There is a remarkable connection with vortices in (p x +ip y )-wave superconductors - These states hold promise for fault-tolerant quantum computation - This leads to a very rich braiding structure, involving higher-dimensional representations of the braid group

24. Exclusion statistics Haldane (1991) = # of quasiparticles of species = # of states available to qp Model for exclusion statistics : FQHE quasiparticles obey fractional exclusion statistics :

25. Key Points ✸ In d=2, a one-parameter (θ) family of quantum statistics exists between Bose (θ=0) and Fermi (θ=π), with broken T in between ✸ Anyons behave as charge-flux composites (phases from A-B effect) ✸ Two equivalent descriptions : (i) bosons or fermions with statistical vector potential (ii) multi-valued wavefunctions with no statistical interaction ✸ The anyon gas at is believed to be a superconductor ✸ FQHE quasiparticles have fractional charge and statistics ✸ Beautiful effective field theory description via Chern-Simons term ✸ Exotic nonabelian statistics at ✸ Related to exclusion statistics (Haldane), but phases essential