Anyon Physics. Andrea Cappelli (INFN and Physics Dept., Florence)

Transcription

1 Anyon Physics Andrea Cappelli (INFN and Physics Dept., Florence) Outline Anyons & topology in 2+ dimensions Chern-Simons gauge theory: Aharonov-Bohm phases Quantum Hall effect: bulk & edge excitations measure of fractional charge & statistics non-abelian fractional statistics & topological quantum computation

2 Fractional statistics in 2+ dimensions Exchange Monodromy 2 e iµ i2¼ ª (z z2 )e ; z2 = ei2µ ª [z ; z2 ] µ = ¼º; e.g. º = =3 fractional 6= exchange of identical particles described by the braid group eiµ 6= e iµ violates P and T symmetries If excitation is described by multiplet of m states: ªa [z ; z2 ]! Uab ªb [z ; z2 ] a,b =,.., m m-dim unitary repres. of braid group = Non-Abelian statistics

3 Chern-Simons gauge theory C Special facts of 2+ dimensions: matter current gauge field: J¹ = (½; Ji ); m J¹ = A½ low-energy effective action, P, T: SCS k = 4¼ eq. of motion F¹º = J¹ = 0; hj¹ i = 0 ext. source 2 "¹º½ A½ + A¹ s + F¹º M ¹ no local degrees of freedom 2¼ "¹º½ s½ ; k B= µ I exp i A = ei2¼=k z2 2¼ (2) ± (z z2 ) k Aharonov-Bohm phase

4 Quantum Hall Effect 2 dim electron gas at low temperature T ~ 0 mk 2 and high magnetic field B ~ 0 Tesla a B y Jy Ex x Conductance tensor Plateaux: Ji = ¾ij Ej ; ¾ij = Rij ; ¾xx = 0; Rxx = 0 e2 ¾xy = Rxy = º; h High precision & universality Uniform density ground state: i; j = x; y no Ohmic conduction g gap 2 º = ( 0 8); 2; 3; : : : ; ( 0 6) 3 5 eb ½o = º hc Incompressible fluid

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6 Laughlin's quantum incompressible fluid Electrons form a droplet of fluid: incompressible = gap fluid = ½(x; y) = ½o = const: ½ ½ y x q N º R DA = BA= o ; # degenerate orbitals = # quantum fluxes, o = filling fraction: º = hc e N º= = ; 2; : : : ; ; : : : density for quantum mech. DA 3 5 º= 3

7 Laughlin's trial wave function ªº (z ; z2 ; : : : ; zn ) = Y i<j º = º= 3 2k+ (zi zj ) e P jzi j2 =2 filled Landau level: obvious gap º=!= 2n+ eb mc = ; 3 ; 5 ; : : : À kt non-perturbative gap due to Coulomb interaction ground state w. vortex condensation, like QCD but chiral Q quasi-hole excitation = elementary vortex ª / i ( zi ) fractional charge Q= ª ; 2 Anyons & statistics ¼µ = 2k+ Q 2k+ / ( 2 ) i ( zi ) ( 2 zi ) e 2k+ vortices w. long-range topological correlations long-distance physics reproduced by effective field theory

8 Conformal field theory of edge excitations The edge of the droplet can fluctuate: edge waves are massless t Fermi surface V ½ µ R/J edge ~ Fermi surface: linearize energy relativistic field theory in + dimensions, chiral (X.G.Wen) "(k) = v R (k kf ); k 2 Z + conformal field theory here compactified boson (c=) = chiral Luttinger liquid vortex in the bulk charged excitation at the edge

9 CFT descriptions of QHE r z = reiµ plane (bulk excit) ³ = e eiµ cylinder (edge excit) µ i ' i '2 e vertex operators e CF T 2 anyon wave function 2 edge-excit. correlator (z z2 ) (³ ³2 ) same function by analytic continuation from the circle: both equivalent to Chern-Simons theory in 2+ dim (Witten) µ spectrum of chiral boson CFT proofs Laughlin's fractional Q and ¼ wave functions: spectrum of anyons and braiding edge correlators: conduction experiments (low V and small I)

10 CFT modelling of fractional QHE CFTs exactly describe nonperturbative quantum effects Big zoo of interacting theories (& integrable massive FT) experimental confirmations: tunneling of edge excitations sophisticated technical tools all relevant: repres. theory (affine and W algebras) (A.C.,Trugenberger,Zemba) fusion rules (& modular invariance & boundaries) n-point correlators (braid & fusion relations) nice spin-off of string theory of '85-'95(-'05)

11 Measure of fractional charge +VG IT L I = G V + IB R +VG electron fluid squeezed at one point: L & R edge excitations interact fluctuation of the scattered current: Shot Noise (T=0) low current IB I tunnelling of weakly interacting carriers 2 SI = hj±i(!)j i!!0 e = IB 3 Poisson statistics CFT description & integrable massive interaction: (Fendley, Ludwig, Saleur) µ 2 e VG universality & anomalous scaling G= F h 3 T 2=3

12 (Milliken et al '95)

13 (Glattli et al '97)

14 (Fendley et al '97)

15 Remark. Can we prove the Laughlin state? Effective theories but no microscopic theory Exact eigenstate of model interactions Gap is nonperturbative NR fermions + extra Chern-Simons interaction (Fradkin et al.; Halperin Matrix gauge theory et al.; Shankar et al.) (Susskind '0; Polychronakos; A.C., I. Rodriguez) electrons D0 branes non commutative: NxN matrices ~ ab (t); X [X ; X2 ] = iµ ½o = (Haldane,...) need Non Relativistic effective theory ~ xa (t); a = ; : : : ; N numerics 2¼µ ; º= minimal area +Bµ = +2k predicts Laughlin's states and more general Jain's states = + 2k º n composite fermion

16 Remark 2. W-infinity symmetry one CFT for each plateau: which CFT? area-preserving diffeomorphisms of incompressible fluid: Z d2 x ½(x) = N = ½o A A = constant A A W-infinity symmetry can be implemented in CFT representations completely known minimal models of W match Jain's states (A.C., Trugenberger, Zemba) [ \ U ()2k+ SU (n) SU (n) CFT (V.Kac, A. Radul) º= n 2k n plateaux

17 Measure of fractional statistics C A e 3 B D need interference like double slit experiment 4-point function of edge states induce anyon(s) in the central cell first experiment has side effects and instabilities (Goldman et al. '05) can manufacture better interference geometries (cf. Stern review '07) no doubts by low-energy effective theory Aharonov-Bohm phase

18 Non-Abelian fractional statistics 5 2 described by Moore-Read Pfaffian state ~ Ising CFT x boson º= Ising fields: I fusion rules: identity, à Majorana = electron, ¾ spin = anyon 2 electrons fuse into bosonic bound state à à = I ¾ ¾ =I +à 2 channels of fusion = 2 conformal blocks h¾(0)¾(z)¾()¾()i = a F (z) + a2 F2 (z) Hypergeometric functions state of 4 anyons is two-fold degenerate statistics of anyons ~ analytic continuation µ F F2 µ F F2 i2¼ ze = µ 0 i2¼ (z )e = 0 µ 0 µ 0 F F2 µ F F2 (z) (z) (Moore, Read '9) 2x2 matrix 0 z 0 z (CFT tech: Verlinde; Moore, Seiberg; Alvarez-Gaume, Gomez, Sierra)

19 Quantum computation qubit = two-state quantum system, e.g. spin ½: jâi = j0i + ji boolean gates unitary transformations on qubits discrete subgroup of U (2n ) transformations in n qubit Hilbert space minimal set of generators: 2x2 Pauli matrices + one specific 4x4 matrix Universal Quantum Computation many proposals of systems for QC: excitement & money quantum computer is unavoidable & useful (e.g. for war, electronic) big problem: decoherence by the environment

20 Topological quantum computation Proposal: use non-abelian anyons for qubits and operate by braiding e.g. in Ising-like state º = 5 2 (Kitaev; M. Freedman; Nayak; Das Sarma) anyons topologically protected from decoherence (local perturbations): decay due to finite size thermal pair creation P» exp( L=»); P» exp( =T ); (system size)=` = O(04 ) =T = O(02 ) use 4-spin system jf i + jf2 i as qubit ( 2n spin has dim = 2n ) consider multi-gate bar geometry of before: perform anyon exchanges by tuning the various gate voltages Ising is not universal QC; Z3 parafermions º = study other anyonic media, e.g. array of Josephson junctions many ideas & open problems 2 5 are OK & others

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