Non-Abelian Anyons in the Quantum Hall Effect

Transcription

1 Non-Abelian Anyons in the Quantum Hall Effect Andrea Cappelli (INFN and Physics Dept., Florence) with L. Georgiev (Sofia), G. Zemba (Buenos Aires), G. Viola (Florence) Outline Incompressible Hall fluids: bulk & edge excitations CFT description Partition function Signatures of non-abelian statistics: Coulomb blockade & thermopower

2 Quantum Hall Effect 2 dim electron gas at low temperature T ~ 10 mk B and high magnetic field B ~ 10 Tesla y J y E x x Conductance tensor Plateaux: ¾ xx = 0; R xx = 0 no Ohmic conduction gap High precision & universality Uniform density ground state: J i = ¾ ij E j ; ¾ ij = R 1 ij ; Incompressible fluid i; j = x; y ¾ xy = R xy 1 = e2 h º; º = 1( 10 8 ); 2; 3; : : : 1 3 ; 2 5 ; : : : ; 5 2 ; ½ o = eb hc º

3 Laughlin's quantum incompressible fluid Electrons form a droplet of fluid: incompressible = gap fluid = ½ ½ ½(x; y) = ½ o = const: x y q N º R D A = BA= o ; # degenerate orbitals = # quantum fluxes, o = hc e filling fraction: º = N D A = 1; 2; : : : 1 3 ; 1 5 ; : : : density for quantum mech. º = 1 º = 1 3

4 Laughlin's wave function ª gs (z 1 ; z 2 ; : : : ; z N ) = Y i<j (z i z j ) 2k+1 e P jz i j 2 =2 º = 1 2k+1 = 1; 1 3 ; 1 5 ; : : : º = 1 filled Landau level: obvious gap º = 1 non-perturbative gap due to Coulomb interaction 3 effective theories quasi-hole = elementary vortex! = eb mc À kt ª = Q i ( z i) ª gs fractional charge Q = e 2k+1 & statistics µ ¼ = 1 2k+1 ª 1; 2 = ( 1 2) 1 2k+1 Q i ( 1 z i ) ( 2 z i ) ª gs Anyons vortices with long-range topological correlations

5 Chern-Simons gauge theory Special facts of 2+1 dimensions: matter current gauge field: J ¹ = (½; J i ¹ J ¹ = 0; hj ¹ i = 0 J ¹ = " º A ½ ext. source low-energy effective action, P, T: S CS = k Z " ¹º½ A º A ½ + A ¹ s ¹ + 1 4¼ M F ¹º 2 eq. of motion no local degrees of freedom F ¹º = 2¼ k " ¹º½s ½ ; B = 2¼ k ±(2) (z z 2 ) exp µ I i A z 2 = e i2¼=k Aharonov-Bohm phase

6 Conformal field theory of edge excitations The edge of the droplet can fluctuate: edge waves are massless t V ½ Fermi surface µ R / J edge ~ Fermi surface: linearize energy "(k) = v R (k k F ); k = 0; 1; : : : relativistic field theory in 1+1 dimensions, chiral (X.G.Wen '89) chiral compactified c=1 CFT (chiral Luttinger liquid)

7 CFT descriptions of QHE: bulk & edge r z = re iµ ³ = e e iµ plane (bulk excit) cylinder (edge excit) µ (z 1 z 2 ) 2 anyon wavefunction h Á 1 Á 2 i CF T (³ 1 ³ 2 ) 2 edge-excit. correlator same function by analytic continuation from the circle: both equivalent to Chern-Simons theory in 2+1 dim (Witten '89, X.G.Wen '89) simplest theory for º = 1=p is chiral Luttinger liquid (U(1) CFT): wavefunctions: spectrum of anyons and braiding edge correlators: physics of conduction experiments

8 Measure of fractional charge +V G I = G V L + R I B I T +V G electron fluid squeezed at one point: L & R edge excitations interact fluctuation of the scattered current: Shot Noise (T=0) low current I B I tunnelling of weakly interacting carriers S I = hj±i(!)j 2 i!!0 = e 3 I B Poisson statistics CFT description & integrable massive interaction: (Fendley, Ludwig, Saleur) µ G = e2 1 universality & anomalous scaling h 3 F VG T 2=3

9 (Glattli et al '97)

10 Non-Abelian fractional statistics º = 5 described by Moore-Read Pfaffian state ~ Ising CFT x U(1) 2 Ising fields: I identity, Ã Majorana = electron, ¾ spin = anyon fusion rules: Ã Ã = I 2 electrons fuse into a bosonic bound state ¾ ¾ = I + Ã 2 channels of fusion = 2 conformal blocks h¾(0)¾(z)¾(1)¾(1)i = a 1 F 1 (z) + a 2 F 2 (z) hypergeometrich state of 4 anyons is two-fold degenerate (Moore, Read '91) statistics s of anyons ~ analytic continuation 2x2 matrix µ F1 F 2 ze i2¼ = µ µ F1 (z) F 2 0 z 1 1 µ F1 F 2 (z 1)e i2¼ = µ µ F1 (z) F 2 0 z 1 1 (all CFT redone for Q. Computation: M. Freedman, Kitaev, Nayak, Slingerland,..., 00'-10' )

11 Topological quantum computation qubit = two-state system jâi = j0i + j1i QC: perform U(2 n ) unitary transformations in n qubit Hilbert space Proposal: (Kitaev; M. Freedman; Nayak; Simon; Das Sarma '06) use non-abelian anyons for qubits and operate by braiding 2 n 1 4-spin system jf 1 i + jf 2 i is 1 qubit (2n-spin has dim ) anyons topologically protected from decoherence (local perturbations) more stable but more difficult to create and manipulate great opportunity new experiments and model building

12 Models of non-abelian statistics Study Rational CFTs with non-abelian excitations: best candidate: Pfaffian & its generalization, the Read-Rezayi states º = 2 + k n k + 2 ; k = 2; 3; : : : M = 1 U(1) k+2 SU(2) k U(1) 2k alternatives: other (cosets of) non-abelian affine groups U(1) G H Identify their N sectors of fractional charge and statistics Abelian (electron) & non-abelian (quasi-particles) Compute physical quantities that could be signatures of non-abelian statistics: Coulomb blockade conductance peaks thermopower & entropy

13 use partition function quantity defining Rational CFT (Cardy '86; many people) complete inventory of states (bulk & edge) modular invariance as building principle: S matrix and fusion rules further modular conditions for charge spectrum straightforward solution for any non-abelian state U(1) G H useful to compute physical quantities Inputs: G H non-abelian RCFT (i.e. ) Abelian field representing the electron simple current Output is unique

14 Annulus partition function i2¼ = v R + it; = 1 k B T i2¼³ = ( V o + i¹) Z annulus = px jµ ( ; ³)j 2 ; =1 modular invariance conditions µ ( ; ³) = Tr H ( ) he i2¼ (L 0 c=24)+i2¼³q i R geometrical properties & physical interpretation (A. C., Zemba, '97) T 2 : Z( + 2; ³) = Z( ; ³); S : Z µ 1 ; ³ = Z( ; ³); U : Z( ; ³ + 1) = Z( ; ³); V : Z( ; ³ + ) = Z( ; ³); L 0 L 0 = n 2 completeness Q Q = n Q = º µ half-integer spin excitations globally µ 1 = X 0 S 0µ 0( ) S matrix integer charge excitations globally add one flux: spectral flow µ (³ + )» µ +1 ( )

15 Disk partition function Annulus -> Disk (w. bulk q-hole ¹Q = ) Z annulus! Z disk; = µ ( ; ³) µ ( ; ³) = K ( ; ³; p) = 1 X n p e i2¼ (np+ )2 2p +³ np+ p ; º = 1 p ; c = 1 p R U : Q Q = n sectors with charge T 2 Q = p + n basic quasiparticle + n electrons : electrons have half-integer dimension (=J), and integer relative statistics with all excitations # sectors p = dim (S 0) = Wen's topological order we recover phenomenological conditions on the spectrum

16 Pfaffian & Read-Rezayi states º = 2 + k n k + 2 ; k = 2; 3; : : : M = 1 U(1) k+2 SU(2) k U(1) 2k m 6 Z 3 Z k parafermion p sectors (`; m) and characters ¾ 2 Â`m ; ` = 0; 1; : : : ; k; m mod 2k electron is Abelian ª e = e i ' à 1 ; (`; m) = (0; 2) 3 à 2 " ¾ 1 I Q = q p + electron: p = 2 + k ; k 1 µà = X =0 parity rule K a+ (k+2) Âà+2 (q; m; `)! (q + p; m + 2; `) q = m mod k 0 à 1 ¾ 2 ¾ 1 " 3 à 2 à 1 ` sectors labeled by (a; `) a = 0; : : : ; k + 1; ` = 0; : : : ; k; a = ` mod 2 3 # sectors = topological order (k + 2) k(k+1) 1 2 k = (k+2)(k+1) 2

17 Z RR annulus = kx `=0 X^p 1 a=0 µà( ; ³) 2 ; Z RR disk = µà = k 1 X =0 K a+ (k+2) Âà+2 Ex: Pfaffian (k=2) ground state + electrons Z Pfa±an annulus = jk 0 I + K 4 Ãj 2 + jk 0 à + K 4 Ij 2 + j(k 1 + K 3 ) ¾j 2 + jk 2 I + K 2 Ãj 2 + jk 2 à + K 2 Ij 2 + j(k 3 + K 1 ) ¾j 2 K charge parts Q = 4 + 2n I; Ã; ¾ Ising parts (Majorana fermion) non-abelian quasiparticle 6 sectors Q = 0; 1 2 also Abelian excitations (Milavanovich, Read '96; AC, Zemba '97)

18 Z RR annulus = kx `=0 X^p 1 a=0 µà( ; ³) 2 ; Z RR disk = µà = k 1 X =0 K a+ (k+2) Âà+2 charge and neutral q. #'s are coupled by parity rule but S-matrix for µà is factorized: S a`;a0`0» e i2¼aa0 N=M s``0 generalization to other N-A models: (A.C, G. Viola, '10) Wen's non-abelian Fluids Anti-Read-Rezayi Bonderson-Slingerland N-A Spin Singlet state U(1) SU(2) k U(1) SU(2) k U(1) Ising SU(n) 1 U(1) q U(1) s SU(3) k U(1) 2 unique result once N-A CFT and electron field have been chosen

19 Experiments on non-abelian statistics e 3 e 3 (a) (b) (a) interference of edge waves (Chamon et al. '97; Kitaev et al 06) Aharonov-Bohm phase, checks fractional statistics experiment is hard (Goldman et al. '05; Willett et al '09) (b) electron tunneling into the droplet Coulomb blockade conductance peaks check quasi-particle sectors (Stern, Halperin '06) (Ilan, Grosfeld, Schoutens, Stern '08 ) (Stern et al.; A.C. et al. '09 - '10) Thermopower (Cooper, Stern; Yang, Halperin '09; Chickering et al. '10)

20 Thermopower n ` fusion of -type quasiparticles: multiplicity» (d`) n ; n! 1 ` 2¼R Entropy put temperature T and potential V o gradients between two edges at equilibrium: S(T = 0)» n log(d`); d` = s`0 s 00 > 1 d = SdT QdV o = 0 (quantum dimension) thermopower Q = V o T = S Q (Cooper,Stern;Yang, Halperin '09) entropy from Z: S = µ 1 d d log µà( + ; ³ + ³) µ 0 0 ( ; ³)» log s`0 s 00 ;» R! 0 it could be observable by varying B off the plateau center Q = B B o (Chickering et al '10) e B 0 log(d 1)

21 Coulomb blockade S Droplet capacity stops the electron Bias & T ~ 0: needs energy matching E(n + 1; S) = E(n; S) current peak energy deformation by S» Q bkg e 3 E(n; S) = v R ( + pn ¾) 2 2p / (Q Q bkg ) 2 E ¾ (n) ¾ = B S o = 1 º ; S = e n o U(1) equidistant peaks ¾ U(1) G H ¾`m = 1 º + v n v modulated pattern h`m+2 2h`m + h`m 2 v n v» 1 10 ¾

22 compares states in the same sector spectroscopy of lowest CFT states µà = k 1 X =0 K a+ (k+2) Âà+2 T = 0 : cannot distinguish NA state from parent Abelian state (Bonderson et al. '10) T > 0 corrections o log µà two scales: 0 < T n < T ch ; T n = v n R ; T ch = v R» 10 T n T < T n : ¾`m = + T µ (d`m) 2 log ; T ch d`m+2 d`m 2 T n < T < T ch : + / T T ch e h1 1 T =T n s`1 s`0 ; d`m multiplicity of neutral states in (331) & Anti-Pfaff, not in Pfaff S matrix of non-abelian part test non-abelian part of disk partition function (Stern et al., Georgiev, AC et al. '09, '10)

23 Conclusions non-abelian anyons could be seen partition function: it is simple enough it defines the CFT, its sectors, fusion rules etc. it is useful to compute observables it can be the basis for further model building