Fractional quantum Hall spectroscopy investigated by a resonant - PowerPoint PPT Presentation

Fractional quantum Hall spectroscopy investigated by a resonant detector Alessandro Braggio CNR-SPIN, Genoa https://sites.google.com/site/alessandrobraggio/ M. Sassetti M.Carrega D.Ferraro Genoa Genoa Geneve-Marseille New. J. Phys. 16 043018 (2014) 2012

FQHE: edge states & qps • Topological protected edge states ν = N • Fractional statistics & charges N Φ Laughlin PRL’83 • Chiral edge states with gapless modes Wen PRB90, Halperin PRB 82, Buttiker PRB 88, Beenakker PRL 90 σ xy = ν e 2 2 np + 1 = 1 , 1 1 3 , 1 • Laughlin sequence h ν = 5 , .. σ xx = 0 2 np + 1 = 2 5 , 2 p • Jain sequence ν = 3 , ... Jain, PRL’89Jain, PRL’89 Jain PRL’89, Wen & Zee PRB’92, Kane & Fisher PRB’95 Kane & Fisher PRB’95 Hierarchical models

Multiple qp excitations Ψ l ( x ) ∝ e l T · K · φ • Hierarchical theories G ( m ) ( τ ) = h Ψ ( m ) ( τ ) Ψ ( m ) † (0) i m = 1 m > 1 G ( m ) ( τ ) / | τ | − ∆ m Single-qp m-agglomerate ∆ m Scaling dimension 1 e ∗ = me ∗ Abelian 2 np + 1 + = • Fractional statistics Ψ ( m ) ( x ) Ψ ( m ) ( y ) = Ψ ( m ) ( y ) Ψ ( m ) ( x ) e − i θ m sgn( x − y ) Laughlin PRL 83, Arovas, Schrieffer & Wilczek PRL 84

QPC:Current & Noise • Weak backscattering current I = ν e 2 m-qps h V � I B I B ⌧ I • Power-law signatures in the scaling dimension ∆ m I ( m ) ∝ V 2 ∆ m − 1 G ( m ) ∝ T 2 ∆ m − 2 B B • Current noise signatures: charge measurement Z + ∞ S ( ω = 0) = h { δ I B ( t ) , δ I B (0) } + i δ I B = I B � h I B i −∞ V ∗ S ( m ) ≈ 2 k B TG B e m � T ✓ me ∗ V ◆ k B S ( m ) = I ( m ) coth B k B T ⌧ me ∗ V S ( m ) ≈ me ∗ I ( m ) 2 k B T B

Multiple-qp evidences • Fractional charges: single-qps evidences Theory:Kane & Fisher PRL 94, Fendley, Ludwig & Saleur PRL 95 Exp:De-Picciotto… Nature 97,Saminadayar.… PRL’97,Reznikov… Nature’99 Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations" • Multiple-qp. evidences Chung…PRL03, Bid PRL03,Dolev…. e e ff /e ν = 5 / 2 g n g i h n c i h n p c u q n B e p u l q g B n e i l g S n i S T (mK) M. Heiblum (2/5,3/7,2/3,5/2,..),Willet(5/2),Yacoby (2/3),…

Theoretical explanations 1 2 e ∗ e ∗ • Single-qp and multiple-qp crossover ν = 2 5 • Charge and neutral modes • Mode velocity v n ⌧ v c ω n ⌧ ω c D. Ferraro, A. B. , N. Magnoli, M. Sassetti PRL 08,PRB10,NJP10,PRL11 • Renormalization of scaling exponent ∆ m = g c ∆ c m + g n ∆ n m Coupling other degrees: Rosenow & Halperin PRL 02, Papa & MacDonald PRL 05 1/f noise + dissipation: A. B., D. Ferraro, M. Carrega, N. Magnoli, M. Sassetti NJP12

New questions • Qp. charge measurements Kou et al. PRL12,D. T. McClure et al. PRL12, Safi & Sukhorukov EPL10 • Contropropagating neutral modes evidences Bid et al., Nature 10, Gross et al. PRL12, Gurman et al. Nature 12, Shtanko et al PRB14, Takei et al. PRB11, Dolev et al. PRL11 • Heat transport & neutral modes proliferation Altimiras et al PRL12, Aita et al PRB13, Inoue et al. Nature14 • Edge reconstruction & T dependent edge coupling J. Wang et al PRL13, Karzig et al NJP12, Zhang et al 1406.7296 • Imaging of the edge structure N. Paradiso et al. PRB11,PRL12, Pasher et al. PRX14, Kozikov et al NJP13 • Edge model identification Meier et al. 1406.4517

Why not at finite frequency ? ω m = me ∗ V/ ~ • Josephson resonances Blanter&Buettiker Phys.Rep.00, Rogovin&Scalapino Ann. Phys 74 • Rich theoretical tools & interesting non-equilibrium phys. Chamon..PRB95; Chamon..PRB96; Dolcini..PRB05; Bena..PRB06; Bena..PRB07; Sukhorukov..PRB01; Sukhorukov..EPL10; Schoelkopf…03; Deblock…Science ’03; Engel…’04;Hekking….06;…… • Interesting questions: how to measure it? Lesovik..JETP97;Gavish U..PRB00;Gavish U.. arXiv:0211646; Bednorz& Belzig PRL13; Aguado..PRL00; Symmetrized or non-symmetrized ? [ I ( t ) , I ( t 0 )] 6 = 0 • Symmetrized noise (Landau docet) Z + ∞ S ( m ) S ( m ) ( ω ) = X e i ω t h { δ I B ( t ) , δ I B (0) } + i = ( ω ) i −∞ i = ± • Non-symmetrized (Emission/absorption from QPC) Aguado PRL00, Blanter 05, Martin&Crepieux 04-05-06,….. Z + ∞ S ( m ) e ± i ω t h δ I ( m ) ( t ) δ I ( m ) + / − ( ω ) = (0) i B B −∞

Finite frequency detection Lesovik G B and Loosen R JETP 65 295 (1997); Gavish U,….arXiv:0211646 � ( ω ) Emission S ( m ) 50 Ω Resonant + T p 1 /LC ω = T c � � Cold detector � T c ⌧ T Hot detector S ( m ) ( ω ) 25 k Ω Absorption − T c � T • Impedance matched resonant detection scheme Altimiras et al. APL13, PRL14 ω ≈ 5 GHz T ≈ 15 mK δ h x 2 i • Output power proportional to variation of LC energy n h io S ( m ) S ( m ) ( ω ) − S ( m ) S ( m ) meas ( ω ) = K ( ω ) + n B ( ω ) ( ω ) + + − ⌘ 2 1 ⇣ α 1 n B ( ω ) = K = 2 η ⌧ 1 h i G ( m ) e ω /T C − 1 � ω < e ac ( ω ) 2 L

Noise properties in QPC-LC � k B T c ⌧ ω • Detector quantum limit (Cold detector) � � � meas ( ω ) ≈ KS ( m ) S ( m ) ( ω ) + O ( e − ~ ω /k B T c ) + � • Absorbitive QPC limit (Hot detector) k B T c � ω � � � n h io S ( m ) S ( m ) G ( m ) meas ( ω ) ⇡ K ( ω ) � k B T c < e ac ( ω ) + • Is it measurable? ω 0 = e ∗ V/ ~ S meas ≡ S ex T = T c • | t m | 2 • Lowest order in the tunnelling (purely additive) X X S ( m ) S ( m ) S sym ( ω ) = sym ( ω ) S meas ( ω ) = meas ( ω ) m m Γ ( m ) ( E ) • Keldysh formalism blow up in Fermi’s rule: rate

Non-interacting result ν = 1 S meas ( ω , ω 0 ) /K S sym ( ω , ω 0 ) Electron T c = 15mK ω = 7 . 9GHz(60mK) c ) a ) ω c = 660GHz(5K) ∝ V ∝ V T = 0 . 1 , 5 , 15 , 30[mK] ˜ S 0 = e 2 | t 1 | 2 1 S 0 ˜ S sym ( ω , ω 0 ) = 2 [ θ ( ω 0 − ω ) ω 0 + θ ( ω − ω 0 ) ω ] 2 2 πα 2 ω c ω c ✓ ◆ S meas ( ω , ω 0 ) ⇡ KS + ( ω , ω 0 ) = K ω S sym ( ω , ω 0 ) � 2 ˜ S 0 k B T c ⌧ ω 2 ω c Γ (1) ( E ) ∝ θ ( E ) E Lesovik G B and Loosen R JETP 65 295 (1997)

Interacting case: Laughlin ν = 1 / 3 e ∗ = e S sym ( ω , ω 0 ) S meas ( ω , ω 0 ) /K 3 Single-qp T c = 15mK ω = 7 . 9GHz(60mK) b ) d ) ω c = 660GHz(5K) ∝ V ∝ V T = 0 . 1 , 5 , 15 , 30[mK] S sym ( ω , ω 0 ) ≈ | ω − ω 0 | 4 ∆ (1) 1 / 3 − 1 Chamon, Freed & Wen PRB95,PRB96 � • Detector quatum limit k B T c ⌧ ω � � � • QPC Shot noise k B T ⌧ ω 0 ( ω ) ≈ K ( me ∗ ) 2 Γ ( m ) ( − ω + m ω 0 ) meas ( ω , ω 0 ) ≈ S ( m ) S ( m ) m = 1 ω ∼ ω 0 + 2 S ( m ) meas ( ω , ω 0 ) returns directly the rates…….

Rate detection ν = 1 / 3 , 1 / 5 , 1 / 7 T c = 10 , 30 , 60 , 90 mK S ex ( ω , ω 0 ) /K S meas ( ω , ω 0 ) /K Dashed lines T c = T theoretical rates = ν ∆ (1) T = 10mK b ) a ) ν 2 � � � � � � � � It is possible to extract the scaling dimensions without requiring an extended window in frequency and bias simplifying the experimental requirements S meas ≡ S ex T = T c Note that

Hotter is better? Safi & Sukhorukov EPL10 T c = 5 , 15 , 30 , 60 mK ∂ S meas ( ω , ω 0 ) S meas ( ω , ω 0 ) /K K ∂ω 0 ν = 1 / 3 � � � � d ) b ) T c = 15mK ∝ V ∝ V ω = 7 . 9GHz(60mK) ω c = 660GHz(5K) The QPC cannot excite detector modes only absorptive The QPC excites detector The combined effect is an enhancement of jump/peak

Multiple-qp spectroscopy: S meas ν = 2 ν = 2 T = 0 . 1 , 5 , 15 , 30[mK] 5 3 S meas ( ω , ω 0 ) /K S meas ( ω , ω 0 ) /K e ∗ = e e ∗ = e 5 3 c ) d ) S meas ≡ S ex T = T c Note that S meas ( ω , ω 0 ) ≈ α 1 Γ (1) ( ω 0 − ω ) + α 2 Γ (2) (2 ω 0 − ω ) Rates are directly fitted: scaling dimensions at finite T • Multiple-qps are observed in different window •

Conclusion • QPC+LC resonator is a powerful tool • f.f. noise resolve the presence of multiple qps • Multiple-qp spectroscopy can be done at realistic T • Information on qps by analysing bias behaviour • Changing detector temperature increases the sensibility • Validate composite edge model theories • This techniques can be used in other systems New. J. Phys. 16 043018 (2014)

Topological order in IQHE • Topological invariant 2+1D under magnetic field (Kubo) ( v y ) βα ( v x ) αβ − ( v x ) αβ ( v x ) βα X σ xy = − ie 2 ~ ( E α − E β ) 2 E α <E F <E β • Magnetic Brillouin zone (torus) xy = e 2 ✓ ∂ u α ∗ ∂ u α ∂ u α ∗ ∂ u α ◆ Z Z k 1 ,k 2 k 1 ,k 2 k 1 ,k 2 k 1 ,k 2 σ ( α ) d 2 k d 2 r − 2 π i ∂ k 2 ∂ k 1 ∂ k 1 ∂ k 2 xy = ne 2 σ ( α ) Topological invariant h Thouless, Kohmoto, Nightingale,den Nijs PRL’82; Kohmoto Ann. Phys. 160, 343 (1985)

Edge states & Multiple-qp φ i • Chiral Luttinger liquids Wen, Kane & Fisher ,…. L = 1 4 ⇡ ( K ij @ x � i @ t � j + V ij @ x � i @ x � j + 2 ✏ µ ν t j @ µ � j A ν ) Ψ l ( x ) ∝ e l T · K · φ • Multiple-qps excitations • Filling factor ν = t T · K − 1 · t 3-qps qp • Fractional charges q l = 1 2 π l T · K − 1 · t = me ∗ • Fractional statistics θ l = 2 π l T · K − 1 · l 2-qps • Monodromy: qp aquires phase in a loop 2 π around e − e − Wen & Zee PRB 92, J. Fröhlich et al JSTAT 97