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 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: e e n pp e e 3 He 4 He bosonfermion np n p ++ 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. Path integral description QM propagator: (manifold) Paths on M are classified by homotopy : andhomotopic if { with smoothly deformable YES NO The propagator is expressed as a sum over homotopy classes : Path composition group structure : 1 ( M ) = fundamental group weight for class In order that the composition rule be preserved, the weights () must form a unitary representation of 1 ( M ) : Think about the Aharonov-Bohm effect : 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 : == generated by : - unitary one-dimensional representations of: - absorb into Lagrangian: - topological phase : change in relative angle with 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 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 away from the Bose and Fermi points. What happens at higher densities?? 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 -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 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! (?) 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) 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 doesnt hinge on broken U(1) symmetry Unresolved issues (not much work since early 1990s) 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 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 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) 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 Numerical calculations of e * and Sang, Graham and Jain ( ) Kjo nsberg and Myrheim (1999) Laughlin quasielectronsJain quasielectrons - good convergence for quasihole states - quasielectrons much trickier ; convergence better for Jains WFs - must be careful in defining center of quasielectron charge statistics 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 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, 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 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 Exclusion statistics Haldane (1991) = # of quasiparticles of species = # of states available to qp Model for exclusion statistics : FQHE quasiparticles obey fractional exclusion statistics : Key Points In d=2, a one-parameter () family of quantum statistics exists between Bose (=0) and Fermi (=), 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