Functional Renormalization-Group Analysis of Luttinger Liquids with - PowerPoint PPT Presentation

Functional Renormalization-Group Analysis of Luttinger Liquids with Impurities S. Andergassen, T. Enss, W. Metzner (MPI Stuttgart) V. Meden, K. Sch¨ onhammer (Universit¨ at G¨ ottingen) Grenoble, 1.6.2006

Outline − Introduction: impurities in Luttinger liquids − Method: functional renormalization group (fRG) − Results: local density of states and transport → spinless fermions → spin- 1 2 fermions

Introduction: impurities in Luttinger liquids − Luttinger liquid: → effective low-energy model of correlated electrons in 1D → power laws with interaction-dependent exponents ( K ρ < 1) − impurity effects: → at low energy scales impurity effectively cuts the chain → physical observables determined from open-chain fixed point local DOS: D j ∼ | ω | α B α B = ( K − 1 − 1) / 2 > 0 ρ conductance: G ∼ T 2 α B Kane, Fisher ’92 10 1 conductance through 10 0 G ( µ S) kink in carbon nanotube: power law 10 –1 Segment I Segment II Across the kink 10 –2 50 100 200 300 Yao et al. ’99 T (K)

Aim: development of quantitative theory for microscopic models of interacting Fermi systems: ◮ computation of observables on all energy scales, providing also non-universal properties ◮ determination of scale at which universal asymptotics sets in

Microscopic model V U’ U j 0 1 1 L L t j ,σ ( c † j +1 ,σ c j ,σ + c † H = − t � j ,σ c j +1 ,σ ) j n j ↑ n j ↓ + U ′ � + U � j n j n j +1 + H imp site impurity: H imp = Vn j 0 ( j 0 impurity site) H imp = ( t − t ′ )( c † hopping impurity: j 0 +1 ,σ c j 0 ,σ + h.c. )

✁ ✄ ✘ ✆ � ✆ � ☎ ✁ Method: functional renormalization group (fRG) ◮ general formulation of Wilson’s RG idea ◮ generating functional of m -particle interaction ◮ introduction of IR-cutoff Λ in G Λ 0 = Θ( | ω | − Λ) G 0 ◮ exact infinite hierachy of coupled flow equations: G Λ = [( G Λ 0 ) − 1 − Σ Λ ] − 1 �✂✁ S Λ = G Λ [ ∂ Λ ( G Λ 0 ) − 1 ] G Λ ✠☛✡ ☞✍✌ ✎✑✏ ✒✔✓ �✂✁☎✄ ✕✗✖ + ✝✟✞ Σ Λ 0 = bare impurity potential ◮ initial conditions: Γ Λ 0 = bare interaction Γ Λ 3 = Γ Λ 0 ◮ truncation of hierarchy: 3 = 0 Wetterich ’93, Morris ’94, Metzner ’99, Salmhofer and Honerkamp ’01

Results: Local DOS at impurity impurity induces long-range 0.4 2 k F oscillations U = 0 0.3 U = 1 ⇓ D j 0 − 1 0.2 strong suppression of DOS at Fermi energy: 0.1 0 D j 0 − 1 ∼ | ω | α B -3 -2 -1 0 1 2 3 ω n = 1 / 2 , V = 1 . 5 , L = 1025 boundary exponent α B spinless fermions independent of impurity potential

Results: Spinless fermions: effective exponents Dependence on impurity potential → convergence to 0.2 universal boundary exponent in general 0.1 very slow 0 α boundary 1 1 L ∼ V 1 − K ρ V = 3 -0.1 V = 1 V = 0 . 3 V = 0 . 1 → non-universal -0.2 behavior relevant! 10 2 10 3 10 4 10 5 10 6 L

� ✁ ✖ � ✁ ✖ ✄ ✄ Results: Spinless fermions: effective exponents Effect of vertex renormalization (VR) ✄✍✌✏✎ ✄✍✌✏✎ ✄✍✌✒✑ ✄✍✌✒✑ ✄✍✌✔✓ ✄✍✌✔✓ ✄✍✌✕✂ ✄✍✌✕✂ ✄✍✌✕✂ ✄✍✌✕✂ without VR with VR ✂☎✄✟☞ ✂☎✄✝☛ ✂☎✄✡✠ ✂☎✄✟✞ ✂☎✄✝✆ ✂☎✄✟☞ ✂☎✄✝☛ ✂☎✄✡✠ ✂☎✄✟✞ ✂☎✄✝✆ boundary n = 1 / 4 impurity: V = 1 . 5 horizontal lines: BA U = 0 . 5 (circles) filled symbols: DMRG U = 1 . 5 (diamonds)

Results: Spin- 1 2 fermions: local DOS at boundary 0.4 parameters: U ′ = 0 0.3 n = 1 / 4 L = 4096 0.4 D 1 0.2 U = 0 0.1 U = 0 . 5 L = 10 6 0.35 U = 1 U = 2 -0.01 0 0.01 0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ω − clear increase instead of expected suppression in contrast to low-energy description! − effect of 2-particle backscattering

Discussion: Effect of 2-particle backscattering ˜ V (2 k F ) Hartree-Fock: V (0) − 2 ˜ ˜ � � V (2 k F ) log | ω/ǫ F | + O ( ˜ D j ( ω ) = D 0 V 2 ) j ( ω ) 1 + 2 π v F − bare Hubbard model: ˜ V (0) − 2 ˜ → increase V (2 k F ) = − U < 0 suppression through O ( ˜ V 2 ) − extended Hubbard model: ˜ V (0) − 2 ˜ V (2 k F ) = 2 U ′ [1 − 2 cos(2 k F )] − U 0.3 for ˜ V (2 k F ) = 0 0.2 D 1 U = 0 → similar behavior as 0.1 U = 0 . 5 U = 1 for spinless fermions U = 2 0 -1 -0.5 0 0.5 1 ω

Results: linear conductance G ( T ) = − 2 e 2 | t ( ǫ ) | 2 ∼ | G 1 , N ( ǫ ) | 2 d ε f ′ ( ε ) | t ( ε ) | 2 � with h spin- 1 spinless fermions 2 fermions 10 0 0.05 0.04 0.03 10 − 1 0.02 V = 10 G/ ( e 2 /h ) V = 1 V = 0 . 1 U ′ = 0 0.01 10 − 2 U ′ = 0 . 25 U ′ = 0 . 5 U ′ = 0 . 75 ˜ V (2 k F )=0 U ′ = 1 10 − 3 10 − 4 10 − 3 10 − 2 10 − 1 10 0 10 1 10 − 4 10 − 3 10 − 2 10 − 1 10 0 T T development of clear power laws parameter dependent, in general non-universal behavior relevant! PRB 73 , 045125 (2006), Enss et al. ’04, Meden et al. ’04

Resonant tunneling through a quantum dot Luttinger - liquid behavior in quantum wire width w ∼ N ( K ρ − 1) / 2 Postma et al. ’01 Kondo physics in quantum dot resonance plateau w ∼ U (independent of N ) Goldahber-Gordon et al. ’98 Cronenwett et al. ’98 → interplay of correlation effects

Conductance through a single dot: Kondo physics L R U t’ t’ V g − flow equation for effective level position V Λ = V g + Σ Λ dot : UV Λ /π ∂ Λ V Λ = − U � G Λ dot ( i ω ) = − (Λ + Γ) 2 + ( V Λ ) 2 2 π ω = ± Λ − hybridization Γ = 2 π t ′ 2 ρ leads (wide-band limit) − spectral function is Lorentzian around V = V Λ=0 with width 2Γ and height 1 / ( π Γ) − solution ( v = V π/ U , v g = V g π/ U , γ = Γ π/ U ) for V Λ= ∞ = V g vY 1 ( v ) − γ Y 0 ( v ) = J 0 ( v g ) vJ 1 ( v ) − γ J 0 ( v ) Y 0 ( v g )

Conductance through a single dot: Kondo physics − for | V g | < V c small | V | − for U ≫ Γ sharp crossover to large | V | − expansion for small | v | , | v g | : V = V g exp [ − U / ( π Γ)] − exponential pinning of spectral weight at chemical potential 1 1 U/ Γ =1 2 /h) U/ Γ =10 U/ Γ =25 G/(2e 0.5 0.5 0 0 # electrons 2 1 0 0 -4 -4 -2 -2 0 0 2 2 4 4 V g /U Kondo ’64; Glazman, Raikh ’88; Ng, Lee ’88; PRB 73 , 153308 (2006)

Conductance through a single dot: Kondo physics Comparison with exact results with 2-particle vertex renormalization U/ Γ =4 π 1 1 2 /h) G/(2e 0.5 0.5 U fixed flow of U Bethe ansatz 0 0 -1 -0.5 0 0.5 1 V g /U Gerland et al. ’00, Karrasch et al. ’06

Tunneling with Luttinger - liquid leads L R U=0 t L t R Luttinger liquid V g with characteristic power laws U U’ 1 1 2 /h) 3 N=10 w ∼ N ( K ρ − 1) / 2 4 G/(2e 0.5 0.5 N=10 5 N=10 0 0 -0.8 -0.75 -0.7 K fRG = 0 . 760 V g /t ρ 0.008 K DMRG = 0 . 749 ρ w/t 0.003 Kane, Fisher ’92; Ejima et al. ’05 3 4 5 10 10 10 N

Kondo effect and Luttinger - liquid leads L R U>0 t L t R competing effects: − Luttinger liquid: w → 0 V g U − Kondo effect: w ∼ U U’ 1 1 V g on plateau: 2 /h) 3 N=10 V g = V r g : G ( V r g ) / (2 e 2 / h ) = 1 4 G/(2e 0.5 0.5 N=10 5 N=10 V g � = V r g : 1 − G ( V g ) / (2 e 2 / h ) ∼ N 1 − K ρ 0 0 -1.5 -1 -0.5 0 V g /t conductance → low-energy limit: Luttinger liquid! -3 10 3 4 5 10 10 10 PRB 73 , 153308 (2006) N

Conclusions and outlook ◮ Analysis of spectral and transport properties with fRG technique: − flexible microscopic modeling of geometries, leads and contacts − determination of relevant energy scales and non-universal behavior ◮ Results: → for moderate interaction and impurity parameters large systems required to reach low-energy asymptotics → spin- 1 2 : effects of 2-particle backscattering − deviation from low-energy description − logaritmic corrections → double barrier: Kondo effect relevant on experimentally accessible length scales Outlook − disorder − non-equilibrium phenomena