Anomalous charge tunnelling in fractional quantum Hall edge states

Transcription

1 Anomalous charge tunnelling in fractional quantum Hall edge states Dario Ferraro Università di Genova A. Braggio, M. Carrega, N. Magnoli, M. Sassetti Maynooth, September 5, 2011

2 Outline Edge states tunnelling in Quantum Point Contact geometry Neutral mode dynamics: effects on transport properties Relevance of agglomerate tunnelling

3 FQHE: bulk σ xy = ν e2 h ν = N N ϕ Stormer et al., RMP 98 Incompressibility Gapped excitations with fractional charge and fractional statistics (Abelian and non-abelian)

4 FQHE: filling factors Laughlin sequence Jain sequence Laughlin, PRL83 Jain, PRL89 ν = ν = 1 2n +1 p 2np +1 ν = 5 2 Moore & Read, NPB91; Halperin et al., PRB93; Fendley et al., PRB07; Lee et al. PRL07; Levin et al., PRL07; Boyarsky et al., PRB09 ν = 7 3, 8 3, Read & Rezayi, PRB99; Bishara et al., PRB08; Bonderson & Slingerland, PRB08

5 FQHE: edge states Hall liquid Low energy sector of the incompressible fluid Quantization of the conductance Halperin, PRB82; Buttiker, PRB88; Beenakker, PRL90 Gapless excitations Same charge and statistics of bulk quasiparticles

6 Edge dynamics in the Jain sequence H up = v c 4πν dx ( x ϕ c ) 2 + v n 4π dx ( x ϕ n ) 2 charged mode ρ = 1 2π xϕ c v c v n Wen & Lee, arxiv: ; Levkivsky & Sukhorukov, PRL08; neutral mode [ϕ c (x), ϕ c (y)] = iπνsign(x y) [ϕ n (x), ϕ n (y)] = iπsign(x y) D. F., A. Braggio, M. Merlo, N. Magnoli, M. Sassetti, PRL 101, (2008)

7 Multiple-quasiparticle excitations quasiparticle m-agglomerate Charge e m = me 1 = m ν p e Statistics Ψ (m) (x)ψ (m) (y) =Ψ (m) (y)ψ (m) (x)e iθ msign(x y) θ m = m 2 π ν p 2 1 p 1 mod(2π) Monodromy: quasiparticles acquire no phase in a loop around electrons Ino, PRL 81; D. F., A. Braggio, N. Magnoli, M. Sassetti, Physica E 42, 580 (2010)

8 Multiple-quasiparticle operators Bosonization Ψ (m) (x) = 1 2πa e i[α mϕ c (x)+β m ϕ n (x)] Single quasiparticle Ψ (1) (x) = 1 2πa e i 1 p ϕ c(x)+ p -agglomerate Ψ ( p ) (x) = 1 2πa e iϕ c(x) 1+ 1 p ϕ n(x)

9 Quantum Point Contact geometry I = ν e2 h V I B Out of equilibrium system Schwinger-Keldysh contour formalism Schwinger, J. Math. Phys 61; Keldysh, Zk. Ekso. Teor. Fiz. 64

10 Extremely weak backscattering: current change in the slope Chung et al. PRL03

11 Extremely weak backscattering: noise S B = S B =2e coth + e V 2k B T dt{δi B (t), δi B (0)} I B q = e p = pe 1!! bunching of qp q = e 1 = ν p e single-qp S B 2e I B k B T e V S B 4k B TG B k B T e V Chung et al PRL03

12 H B = t 1 Ψ (1) R (0)Ψ(1) (0) + h.c. Ψ (1) (x) = 1 2πa e i L 1 p ϕ c(x) p ϕ n(x) weak backscattering and V 0 only charged modes T 2 [νg c /p 2 +g n (1+1/ p ) 1] charged and neutral modes k B T T 2 [νg c /p 2 1] Renormalization parameters Dalla Torre et al. Nat. Phys.10; Rosenow & Halperin, PRL02; Papa & MacDonald, PRL04; Yang, PRL03.

13 Comparison with experiments T 2 [νg c /p 2 +g n (1+1/ p ) 1] charged and neutral modes Chung et al. PRL03 T 2 [νg c /p 2 1] only charged modes Finite velocity of neutral modes Change in slope

14 Noise q = e p = pe 1!! q = e 1 = ν p e Chung et al PRL03

15 Multiple-quasiparticle tunnelling t 1 Ψ (1) (1) R (0)Ψ L (0) + h.c. t mψ (m) R More relavant operators have minimal scaling dimension G m (τ) = 1 2πa Kane & Fisher, PRL92 Imaginary time Green s function (m) (0)Ψ L (0) + h.c. G m (τ) =T τ Ψ (m) (τ)ψ (m) (0) 1 1+ω c τ νgc α 2 m 1 1+ω n τ G m (τ) τ 2 m gn β 2 m

16 Relevance Different energy scales create two regimes E ω c, ω n ω n E ω c m=p-agglomerate dominates Kane & Fisher, PRL92; Wen Adv. Phys. 95 Ψ ( p ) (x) = 1 2πa e iϕ c(x) Ψ (1) (x) = 1 2πa e i single-qp dominates 1 p ϕ c(x) p ϕ n(x) Possible crossover between the two excitations D. F., A. Braggio, M. Merlo, N. Magnoli, M. Sassetti PRL08

17 Current I (tot) B = I (1) B k B T e V + I(p) B I (tot) B quasiparticle T 2 [νg c /p 2 1] p-agglomerate T 2 [νg c /p 2 +g n (1+1/ p ) 1] T 2[νg c 1] ω n k B T

18 Fitting of the experimental data ν =2/5,p=2 ω n = 50mK ω n /ω c = 10 2 g c =3,g n =4 t 1 2 / t 2 2 =1.66 e V =1.16mK D. F., A. Braggio, M. Merlo, N. Magnoli, M. Sassetti PRL08

19 S (tot) =2e 1 I (1) B Shot noise + pi(p) B k B T e V F = S(tot) 2eI (tot) B e 1 = ν p e single qp e p = pe 1 p-agglomerate D. F. et al. PRL 08 Chung et al PRL 03

20 Effective charge Excess noise S (tot) B,ex = S(tot) B S (tot) B,ex =2e eff coth Effective charge eeff V 2k B T e eff /e 4k B TG (tot) B I (tot) B ν =2/5,p=2 ω n = 50mK ω n /ω c = 10 2 g c =3,g n =4 D. F., A. Braggio, N. Magnoli, M. Sassetti, PRB 82, (2010) 4k B TG (tot) B

21 Edge dynamics for ν = 5 2 Anti-Pfaffian model Lee et al. PRL07; Levin et al. PRL07 L = 1 2π xϕ c ( t + v c x ) ϕ c 1 4π xϕ n ( t + v n x ) ϕ n iψ ( t + v n x ) ψ [ϕ c (x), ϕ c (y)] = i π 2 sign(x y) [ϕ n (x), ϕ n (y)] = iπsign(x y) ψ Majorana fermion v c v n Hu et al. PRB09.

22 Multiple-quasiparticle operators Ψ (m) (x) χ(x)e i [( m 2 )ϕ c (x)+( n 2 )ϕ n (x)] χ = I,ψ, σ m, n even m, n σ σ = I + ψ Non-Abalian statistics single-qp Ψ (1) (x) σ(x)e i [( 1 2)ϕ c (x)+( 1 2)ϕ n (x)] 2-agglomerate q = e 4 q = e 2 Ψ (2) (x) e iϕ c(x) odd

23 Comparison with experimental data Exp. Data from Dolev et al., PRB10 Raw data with the courtesy of M. Heiblum g c =2.8 g n =8.5 ω n = 150mK ω c = 500mK M. Carrega, D. F., A. Braggio, N. Magnoli, M. Sassetti, arxiv: (to appear on PRL)

24 Conclusions Tunnelling in the composite edge states of the FQHE Effects of the neutral modes dynamics Relevance of agglomerate tunnelling D. F., A. Braggio, M. Merlo, N. Magnoli, M. Sassetti, PRL (2008); D. F., A. Braggio, N. Magnoli, M. Sassetti, NJP (2010); D. F., A. Braggio, N. Magnoli, M. Sassetti, Physica E (2010); D. F., A. Braggio, N. Magnoli, M. Sassetti, PRB 82, (2010); M. Carrega, D. F., A. Braggio, N. Magnoli, M. Sassetti, arxiv: (to appear on PRL)

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26 Fano: temperature effect D. F., A. Braggio, N. Magnoli, M. Sassetti, NJP (2010)

27 Skewness More stable against thermal effect D. F. et al. NJP 10

28 (ρ, η) = Edge- phonon interaction Chiral Luttinger Liquid coupled with 1D phonons S χll = 1 4πν S ph = 1 8π νu + β 0 β S int = λ dx π dτ dτ 0 β 0 + Rosenow & Halperin, PRL02 + dτ + dx x ϕ (i τ + v x ) ϕ dxξ 2 τ u 2 2 x ξ dx x ξ x ϕ Imaginary time Green s function D(0, 0, Ω n )=g(ρ, η) 2πν Renormalization Ω n (1 + ρ 2 x 2 )(1 + x 2 ρ(ρ η 2 )) 1+x 2 (2ρ 2 + 1) + x 4 ρ(ρ 3 + 2(ρ η 2 )) + x 6 ρ 2 (ρ η 2 ) 2

29 S = 1 4πν 1/f noise and dissipation (1) S 1/f = i + Dalla Torre et al., Nature Physics 10 + dx dx β 0 β 0 dτ x ϕ(i τ + v x )ϕ dτρ(x, τ)f(x, τ) Out of equilibrium noise with spectral density f(ω,q) f (ω,q) = F ω Absorbed energy dissipated by the cooling setup S diss = γ 4π + dx β 0 dτdτ π2 (ϕ(x, τ) ϕ(x, τ )) 2 β 2 sin 2 [π(τ τ )/β]

30 G = 1/f noise and dissipation (2) Green s functions in the Keldysh picture q 2iγ ω +2i 2 (2πν) 2 F γ 1 1 (2πν) 2 (ω+vq)2 q 2 +γ 2 ω 2 1 2πν (ω+vq)q+iγω πν (ω+vq)q iγω G K = ϕ cl (0,t)ϕ cl (0, 0) = νg ln (1 + iω ρ t) Renormalization g =1+ 1 F (2π) 2 γ 1/f and dissipation are relevant perturbations with massive coupling constants, it is possible to extend this approach to the disorder-dominated phase of the composite edge states

31 Noise Chung et al PRL 03 Chung et al PRL 03

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33 Extremely weak backscattering: noise Chung et al PRL 03 Chung et al PRL 03

34 extremely weak backscattering general solutions at any order in mode dynamics Moon, Yi, Kane, Fisher PRL 93 (MC simulations) Yue, Matveev, Glazman PRB 94 (weak interaction expansion ) Fendley, Ludwig, Saleur PRL 95, 96 (thermodinamic Bethe Ansatz) Weiss, Egger, Sassetti PRB 95 (real time P.I. ) Aristov, Woelfle EPL 08 (fermionic representation, RG equation)

35 extremely weak backscattering V theory B Roddaro et al. PRL 04 Chung, Heiblum, Umansky PRL 03 non-universal power law exponent! minimum! Other experimental deviations Chang et al., PRL 96; Grayson et al. PRL 98; Glattli et al. Physica E 00; Chang et al. PRL 01; Grayson et al. PRL 01; Hilke PRL negative slope!

36 most relevant operators for tunneling processes

37 extremely weak backscattering theory experiments B V maximum Roddaro et al. PRL 03, PRL 04 minimum Chung et al. PRL 03 theory: negative slope

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39 Skewness D. F., A. Braggio, N. Magnoli, M. Sassetti, NJP (2010)

40 Models for the Jain sequence A single mode model can only describe states of the Laughlin sequence We need to introduce additional neutral modes Hierarchical model Fradkin-Lopez model Wen, Zee, PRB 92 Lopez, Fradkin, PRB 99 p-1 neutral fields two neutral fields We consider a minimal model with only one neutral mode

41 Neutral mode dynamical topological Wen, Zee, PRB 92; Kane, Fisher, Polchinski,PRL 94; Lee, Wen, cond-mat ; Levkivskyi, Sukhorukov,PRB 08; D. F. et al. PRL 08; D. F. et al. NJP 10 Lopez, Fradkin, PRB 99, PRB 01; Chamon, Fradkin, Lopez, PRL 07

42 Skewness More stable against thermal effect D. F. et al. NJP 10

43 Quantum Point Contact geometry

44 Edge states in the Laughlin sequence chiral Luttinger liquids Halperin PRB 82; Wen PRL 90, PRB 90,91; MacDonald PRL 90; Lopez, Fradkin PRB 99...

45 Quasiparticles excitations Fractional charge Fractional statistics Laughlin PRL 83; Arovas, Schrieffer, Wilczek PRL 84. Bosonization

46 Extremely weak backscattering theory experiments B V Chung et al. PRL 03 Roddaro et al. PRL 03, PRL 04 maximum minimum

47 Non-universal power law exponents! deviations also for electron tunneling between edge-metal and edge-edge: Chang et al., PRL 96; Grayson et al. PRL 98; Glattli et al. Physica E 00; Chang et al. PRL 01; Grayson et al. PRL 01; Hilke PRL 01, Grayson SSC Several proposals e-ph coupling (Heinonen & Eggert PRL 96, Rosenow & Halperin PRL 02, Klhlebnikov PRB 06) e-e interaction (Imura EPL 99, Mandal & Jain PRL 02, Papa & MacDonald PRL 05, D Agosta et al. PRB 03) edge reconstruction (MacDonald et al. J. Phys 93, Chamon & Wen PRB 94, Wan et al. PRL 02, Yang PRL 03) local filling factor (Sandler et al PRB 98, Roddaro et al. PRL 04, 05, Lal EPL 07, Rosenow & Halperin cond-mat ) Renormalization of the Luttinger parameter

48 Zero frequency noise weak backscattering Poissonian process for! Kane, Fisher PRL 94; Fendley, Ludwig, Saleur PRL 95 De-Picciotto et al. Nature 97 Saminadayar et al. PRL 97 Reznikov et al. Nature Chung et al PRL 03 De-Picciotto et al. Nature 97

49 Noise bunching of qp single qp Chung et al PRL 03

50 Current: temperature effects Finite temperature correction p-agglomerates contribution D. F. et al. NJP 10

51 Multiple-quasiparticle tunnelling More relavant operators single quasiparticle p-agglomerates Tunnelling hamiltonian

52 Tunneling of p-agglomerates: most relevant process at low energy for