Resonant Tunneling in a Dissipative Environment: Quantum Critical - PowerPoint PPT Presentation

Resonant Tunneling in a Dissipative Environment: Quantum Critical Behavior Harold Baranger, Duke University 1. Quantum mechanics + dissipation: open system 
 • resonant tunneling in a dissipative electromagnetic environment 
 • dissipation + symmetry of coupling → competition → QPT 
 2. Quantum phase transition (QPT) 
 • change in ground state upon varying a parameter 
 • exotic state of matter at the critical point 
 Source • non-equilibrium properties?? SG Gate SG Drain

Quantum Mechanics + Environment Tunneling with dissipation: � • environment as a collection of oscillators-- a “bosonic bath” 
 � � [Feynman & Vernon, 1963] • spin-boson model: 2 states + bosonic environment 
 [Leggett, Dorsey, Fisher, Garg & Zwerger, RMP 1987] Spin-boson model: 2 states + bosonic environment Classical behavior QM tunneling (2 degenerate states) QPT Environmental modes suppresses tunneling. e � � � �

� � In Quantum Transport Expt?: “Environmental Coulomb Blockade” e known energy E F � � position � � G dI/dV ~ max (eV, k T) After tunneling event, spreading of charge inhibited by environment � → Coulomb interaction leads to a charging energy � → blocks (suppresses) tunneling of electron [Reviews: Devoret,Esteve,Urbina LesHouches 95, Ingold&Nazarov 92, 
 Flensberg PhysicaScripta 91]

� � In Quantum Transport Expt?: “Environmental Coulomb Blockade” e known energy E F � � position � � G dI/dV ~ max (eV, k T) After tunneling event, spreading of charge inhibited by environment � → Coulomb interaction leads to a charging energy � → blocks (suppresses) tunneling of electron top view 
 charge localized → Coulomb barrier of structure

� � In Quantum Transport Expt?: “Environmental Coulomb Blockade” e known energy E F � � position � � G dI/dV ~ max (eV, k T) After tunneling event, spreading of charge inhibited by environment � → Coulomb interaction leads to a charging energy � → blocks (suppresses) tunneling of electron top view 
 charge localized → Coulomb barrier of structure still tunneling under Coulomb barrier

� � In Quantum Transport Expt?: “Environmental Coulomb Blockade” e known energy E F � � position � � G dI/dV ~ max (eV, k T) After tunneling event, spreading of charge inhibited by environment � → Coulomb interaction leads to a charging energy � → blocks (suppresses) tunneling of electron top view 
 charge localized → Coulomb barrier of structure still tunneling under Coulomb barrier finally comes out

� � In Quantum Transport Expt?: “Environmental Coulomb Blockade” e known energy E F � � position � � G dI/dV ~ max (eV, k T) After tunneling event, spreading of charge inhibited by environment � → Coulomb interaction leads to a charging energy � → blocks (suppresses) tunneling of electron G ≡ dI measured observable: (differential) conductance, dV r ≡ e 2 G ∝ V 2 r h R leads ( ≈ 0 . 75 here) with ⇣ I ∝ V 2 r + 1 ⌘ [Reviews: Devoret,Esteve,Urbina LesHouches 95, Ingold&Nazarov 92, 
 Flensberg PhysicaScripta 91]

Outline 1. Experiment • carbon nanotube q.dot � R ∼ h • dirty metal leads → � e 2 • B=6T ⇒ “spinless” 2. Theory of approach to quantum critical point • map to interacting 1D model-- a Luttinger liquid � • power laws from scaling at strong and weak coupling � • amazing consistency with experiment! 3. Model of quantum critical system/state • introduce Majorana fermion representation � • QCP described by a decoupled zero-mode Majorana � • indirect experimental signature of Majorana: linear T dependence

Experimental System: Carbon Nanotube Quantum Dot Gleb Finkelstein group: H. Mebrahtu, I. Borzenets, Y. Bomze, A. Smirnov, GF 
 Sample: tuning the coupling Conductance, in units of e 2 /h asymmetry by the side gate Short carbon nanotube 6 (CNT) 
 quantum dot 4 (300 nm) Source 2 connected 
 V S G (V ) to resistive SG 1.0 0 leads via 
 tunable tunnel '2 Gate barriers. SG 0.5 2 '4 Drain 0.0 '6 B=6 T (spinless case) 0.3 0.4 0.5 0.6 V gate 2(V )

Resonant Tunneling Γ L Γ R Ignore environment for now: 
 Tunneling through a double barrier → ¡ resonances (sharp) 4 Γ L Γ R can tune T = ( ∆ ✏ ) 2 + ( Γ L + Γ R ) 2 ∆ ✏ , Γ L , Γ R Symmetric coupling + on resonance → ¡perfect ¡transmission G = e 2 Conductance is Transmission! 
 h T (Landauer viewpoint) Now connect the environment-- what happens? is T suppressed? B=6 T (spinless case)

Preliminary: Environmental Coulomb Blockade Conductance far away from resonance → ¡ single barrier case c 10 d -1 10 -1 10 Y Y 2 /h) -2 10 G (e G(V,T)/G(0,T) 2 /h) -2 10 G (e Z -3 10 Z 0.1 1 -3 T (K) 10 1 -1.0 -0.5 0.0 0.5 1.0 0.1 1 10 100 V (mV) eV/kT “zero bias 
 ⌘ 2 r r ≡ e 2 anomaly” ⇣ h R leads ( ≈ 0 . 3 here) max { eV, k B T } with G ∼

Conductance Resonance: Symmetric vs. Asymmetric Coupling Asymmetric Symmetric 1 1 ((((((((T ((K ) ))))))))T )(K ) (2.0 )2.0 (0.75 )0.75 (0.18 )0.18 (0.05 )0.05 0.1 0.1 2 /h) 2 /h) G )(e G ((e 0.01 0.01 1E !3 1E !3 !30 !20 !10 0 10 20 30 !30 !20 !10 0 10 20 30 Δ V gate(mV ) Δ V gate(mV ) R expt. = 0 . 75( h/e 2 ) [Mebrahtu, et al., Nature 488, 61 (2012)] B=6 T (spinless case)

Conductance Resonance: Height and Width Power Laws Symmetric Height: Width: 1 symmetric Asymmetric asymmetric 1 F WH M$(meV ) 0.1 2 /h) G $(e 0.01 symmetric asymmetric 0.1 1 0.1 1 T emperature$(K ) T emperature$(K ) • 1 special point— symmetric & on resonance— 
 with perfect transmission, � Summary: G − → 1 • for all other parameters, conduction blocked, G − → 0

G~1 : Unstable, Strong-Coupling Fixed Point a b c d 1.0 1 Peak 1 Peak 2 0.9 0.1 1-G (e /h) 0.1 2 /h) 2 /h) 1-G (e 0.8 G (e 0.01 0.01 T (K) 0.7 0.055 0.18 0.75 0.07 0.26 1.10 0.09 0.34 1.36 0.11 0.41 1.67 T 1.2 V 1.1 0.15 0.58 1.96 1E-3 0.6 0.1 1 1 10 100 -200 -100 0 100 200 Temperature (K) V ( µ V) V ( µ V) c d • unusual cusp in conductance! � • power law approach to full transparency � • V power and T power agree (quasi-linear) [Henok Mebrahtu, et al., Nature Physics 2013]

From G~1 to G=0: Flow Toward Weak-Coupling Fixed Point a b -3.5 1 Δ V Shape of conductance resonance: � gate use V gate to tune resonant level through the chemical potential G (e 2 /h) T (mK) 340 0.1 260 Remember simple double 
 180 150 barrier result: 110 90 4 Γ L Γ R 70 T = ( ∆ ✏ ) 2 + ( Γ L + Γ R ) 2 55 0.01 0.1 1 10 Δ V gate (mV)/T r/(r+1) (K) c d [Henok Mebrahtu, et al., Nature Physics 2013]

From G~1 to G=0: Flow Toward Weak-Coupling Fixed Point a b -3.5 1 Δ V Shape of conductance resonance: � gate use V gate to tune resonant level through the chemical potential G (e 2 /h) T (mK) 340 0.1 260 Remember simple double 
 180 150 barrier result: 110 90 4 Γ L Γ R 70 T = ( ∆ ✏ ) 2 + ( Γ L + Γ R ) 2 55 0.01 Here: power in tail is 3.5 not 2 ! 0.1 1 10 Δ V gate (mV)/T r/(r+1) (K) c d [Henok Mebrahtu, et al., Nature Physics 2013]

From G~1 to G=0: Flow Toward Weak-Coupling Fixed Point a b -3.5 1 Δ V Shape of conductance resonance: � gate use V gate to tune resonant level through the chemical potential G (e 2 /h) T (mK) 340 0.1 260 Remember simple double 
 180 150 barrier result: 110 90 4 Γ L Γ R 70 T = ( ∆ ✏ ) 2 + ( Γ L + Γ R ) 2 55 0.01 Here: power in tail is 3.5 not 2 ! 0.1 1 10 Scaling collapse of data at different T Δ V gate (mV)/T r/(r+1) (K) r ≡ e 2 c d h R leads ( ≈ 0 . 65 here) [Henok Mebrahtu, et al., Nature Physics 2013]