Fractional Statistics of Quantum Particles

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

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

4. ++ e e e e p p p p n n n boson fermion 4He 3He ++

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

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

7. In order that the composition rule be preserved, the weights χ(μ) must form a unitary representation of π1(M) : Think about the Aharonov-Bohm effect : Path composition ⇒ group structure : π1(M) = “fundamental group” The propagator is expressed as a sum over homotopy classes μ : weight for class μ

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

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

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

11. Low density limit : F F { bosons fermions B B B F F B B B Johnson and Canright (1990) DPA (1985) 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. Integrate out the statistical gauge field via equations of motion: lazy HEP convention: linking metric So we obtain an effective action, ⇒ statistical b-field particle density Given any theory with a conserved particle current, we can transmute statistics: Wilczek and Zee (1983) minimal coupling Chern-Simons term Examples: ordinary matter, skyrmions in O(3) nonlinear σ-model, etc. Chern-Simons Field Theory and Statistical Transmutation

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

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

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

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

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

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

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

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

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

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

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

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

25. ✸ The anyon gas at ✸ Exotic nonabelian statistics at is believed to be a superconductor 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 ✸ Beautiful effective field theory description via Chern-Simons term ✸ FQHE quasiparticles have fractional charge and statistics ✸ Related to exclusion statistics (Haldane), but phases essential