Nonequilibrium current and relaxation dynamics of a charge-fluctuating quantum dot

Transcription

1 Nonequilibrium current and relaxation dynamics of a charge-fluctuating quantum dot Volker Meden Institut für Theorie der Statistischen Physik with Christoph Karrasch, Sabine Andergassen, Dirk Schuricht, Mikhail Pletyukhov, László Borda, Herbert Schoeller

2 Finite-bias current through quantum dots limiting cases spin fluctuations: nonequilibrium physics of the Kondo model charge fluctuations: the interacting resonant level model (lattice version) ul,t L ur,t R µ L µ ε R ( H =. ɛn 2 + u L n 1 1 ) ( n ( ) t. R d 2 d 3 + H.c. t α=l,r model parameters: level position: ɛ t t t t t t ) + u R ( n j=1 ( ) c j,α c j+1,α + H.c. (ɛ = : p-h symmetry) local Coulomb interaction: u α (= U α /ρ α ) local tunneling: t 2 α (= Γ α /(2πρ α )) ) ( n 3 1 ) 2 t t L ( ) d 1 d 2 + H.c. ( ) d 1 c 1,L + d 3 c 1,R + H.c. band width: B = 4t bias voltage: µ L/R = ±V/2

3 In equilibrium with L-R symmetry: scaling limit consider field theoretical limit with ɛ, u, t B: ρ(ω) ρ() relation to x-ray edge problem (Nozière & de Dominicis 69) single lead: mapping to Kondo model (Wiegmann & Finkel shtein 78) bosonization and RG methods (Schlottmann 8-82,... ) numerical RG (Borda et al. 7; Kashcheyevs et al. 9) here: functional RG and real-time RG in frequency space, small U (Karrasch, Enss & VM 6; Schoeller 9) example: susceptibility χ = d n 2 dɛ ɛ= power law.3.2 NRG FRG RTRG-FS 2U χ 1 (Γ ) 1 α χ α χ = 2U + O(U 2 ) α χ.1 RG: Γ acts as cutoff.1.2 U define scale: T K = 2 πχ

4 In equilibrium with L-R symmetry: lattice model use numerical DMRG and Kubo formula (Bohr & Schmitteckert 7) here: functional RG for small u (Karrasch, Pletyukhov, Borda & VM 1) 1 G / G.5 u/t= u/t=.3 u/t=1 u/t=5 symbols: DMRG lines: FRG t / t = ε / t

5 Finite bias: steady state at particle-hole and L-R symmetry scaling limit and V T K : I V α V, α V = 2U + O(U 2 ) (Doyon 7; Boulat, Saleur & Schmitteckert 8) does V only act as (trivial) IR cutoff? (Borda & Zawadowski 1) lattice model, td-dmrg: (Boulat, Saleur & Schmitteckert 8) hopping parameter: t /t =.5 not in scaling limit no power law! questions: compare nonequ. FRG to td-dmrg confirm power law in scaling limit go away from p-h and L-R symmetry

6 Steady-state FRG on the Keldysh contour: ɛ =, L-R symmetric (Gezzi, Pruschke & VM 7; Jakobs, VM & Schoeller 7; Jakobs, Pletyukhov & Schoeller 1) cutoff: auxiliary leads to sites 1,2,3; coupling Λ from flow equation (first order in u): Λ t Λ = i u 4π Green functions: (Γ lead (ω) = πρ(ω)t 2 ) dω {G Λ Λ [G Λ ] 1 G Λ} off-diag., t Λ= = t Keldysh ω + iγ lead + iλ t Λ [G Λ ret] 1 = t Λ ω + iλ t Λ, G Λ t Λ adv = [G Λ ret] ω + iγ lead + iλ [1 2f L ] G Λ Keldysh = 2iG Λ ret iλ[1 2f C] + Γ lead [1 2f R ] GΛ adv occupation functions: f L/R (ω) = f(ω V/2) f C (ω) = f(ω) physical leads auxiliary leads current (at Λ = ): I = 4 Γ 2 lead (ω) [f L(ω) f R (ω)] G 13 ret(ω) 2 dω

7 Comparison to td-dmrg I / e/h u/t= u/t=.3 u/t=-1 u/t=-1 t / t = V / t negative diff. cond. agreement FRG-DMRG bla bla bla bla bla bla (Boulat, Saleur & Schmitteckert 8; Karrasch, Borda, Pletyukhov & VM 1)

8 FRG and RTRG-FS in scaling limit: ɛ =, L-R symmetric flow equation for renormalized rate Γ Λ : Λ Γ Λ = 2UΓ Λ Λ + Γ Λ (V/2) 2 + (Λ + Γ Λ ) 2, ( ) 2U Γ Γ Λ, Λ B max{v/2, Γ} V and Γ act as IR cutoffs ΓΛ= = Γ observables: susceptibility at V = : χ 1 = πγ/2 = πt K /2 (Γ ) 1 α χ, α χ = 2U + O(U 2 ) current for V Γ = T K : (agrees with Doyon 7) I = 2Γ π arctan V Γ Γ V α V, α V = 2U + O(U 2 ) no interesting nonequilibrium physics??

9 FRG and RTRG-FS in scaling limit: general case flow of level positions is of order U 2 and can be neglected flow equation for renormalized rates Γ Λ α: (Γ Λ = α ΓΛ α) Λ Γ Λ α = 2U α Γ Λ Λ + Γ Λ /2 α (µ α ɛ) 2 + (Λ + Γ Λ /2), ΓΛ= α ( ) 2Uα Γ α Γ Λ α, Λ B max{ µ α ɛ, Γ/2} = Γ α current (at Λ = ): I = 1 π Γ L Γ R Γ [ arctan V 2ɛ Γ + arctan V + 2ɛ ] Γ introduce T α K : T α K = Γ α (2Λ /T K ) 2U α, T K = α T α K ( Γ) introduce asymmetry parameter: c 2 = T L K /T R K

10 Away from p-h symmetry, still L-R symmetric G/G V/T K = V/T K =.6 V/T K =1 V/T K =2 V/T K =5 V/T K = ε/t K L-R symmetric U =.1/π lines: RTRG-FS symbols: FRG maximum of G = di/dv at ɛ = V/2: resonance!

11 On-resonance current ε= L-R symmetric I/T K 1-1 ε=v/2 log. deriv. U =.1/π lines: RTRG-FS V/T K current for V T K : I(V ) T K 2 power law only for V T K -.6 ( TKΓ ) 2UL ( TK 2V ) 2UR symbols: FRG c( TKΓ ) 2UL+ 1 c ( TK 2V ) 2UR V is not a simple IR cutoff c 1 I V 2U R (does not agree with Doyon 7) c 1+c 2.

12 Left-right asymmetry, off-resonance (1+c 2 ) 2 I/ 4c 2 T K.2.1 γ =.75, c 2 =21.4 γ =.5, c 2 =7.9 γ =.25, c 2 =2.8 γ =, c 2 = V/T K L-R asymmetric U L/R = (1 ± γ).1/π ɛ = lines: RTRG-FS ( TKV ) 2UL ( TKV ) 2UR current for V T K, V ɛ: I(V ) T K c ( TKV ) 2UL+ ( c 1 TKV ) 2UR. 1+c 2 c V is not a simple IR cutoff for generic cases power law only for U L = U R I V 2U U L U R, c 1 I V 2U L U L U R, c 1 I V 2U R

13 Relaxation dynamics into the steady state: RTRG-FS long-time behavior, off-resonance (ɛ, V, ɛ V/2 T K, 1/t) n 2 (t) ( 1 e Γ 1 t) n 2 1 [ sin ( (ɛ+ V 2 )t) (ɛ+ V 2 )2 t 2 πu 4 2π e Γ 2 t (T K t) 1+2U cos ( (ɛ+ V 2 )t) ] + (V V ) (ɛ+ V 2 )2 t 2 <n 2 (t)> I(t)/T K V/T K =1 V/T K =2 V/T K =5 V/T K = T K t L-R symmetric n 2 (t = ) = U =.1/π, ɛ = 1T K Γ 1 Γ (charge relaxation) Γ 2 Γ/2 (level broadening) oscillation freqs. ɛ ± V/2 exponential and power-law decay, two decay rates

14 Summary finite bias IRLM contains interesting nonequilibrium physics... which doesn t show up for left-right symmetry and off-resonance interesting relaxation dynamics FRG and RTRG-FS are suitable tools to study the nonequ. IRLM td-dmrg results (not in scaling limit) confirmed Refs.: Karrasch, Andergassen, Pletyukhov, Schuricht, Borda, VM & Schoeller, EPL 9, 33 (21) Karrasch, Pletyukhov, Borda & VM, PRB 81, (21)