Correlation effects in transport through quantum wires: a - PowerPoint PPT Presentation

Correlation effects in transport through quantum wires: a functional renormalization group approach Kurt Sch¨ onhammer Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen

Collaborators • from Stuttgart • Sabine Andergassen • Tilman Enss • Walter Metzner • from Aachen • Ulrich Schollw¨ ock • from G¨ ottingen • Volker Meden • Xavier Barnab´ e-Th´ eriault • Abdelouahab Sedeki

Outline • Experimental systems — carbon nanotubes and other quantum wires • Transport problems to be studied — correlations and impurities • The method — functional renormalization group • A single impurity — power-laws and one-parameter scaling • Resonant tunneling — “universal” and “non-univeral” behavior • Transport in more complex geometries — Y-junctions, X-junctions, . . . • Summary and outlook

Experimental systems single-walled carbon nanotube (theory ’97: Egger & Gogolin; Kane, Balents, & Fisher) other quantum wires • semiconductor heterostructures • chains of atoms on surfaces • . . . (Dekkers group)

Transport problems: lead impurity contact junction t U gate voltage ingredients: • 1d quantum wire • electron correlations — Tomonaga-Luttinger liquid • non-interacting leads • contacts • impurities, gate voltage • more complex geometries

Scattering on 1d lattices: noninteracting fermions The Hamiltonian for a single particle on the lattice is given by H = H leads + 0 V LR + H s , with 0 ∞ � � H leads = − ( | n − 1 �� n | + H.c. ) − ( | n �� n + 1 | + H.c. ) 0 n = −∞ n = N +1 V LR = t L ( | 0 �� 1 | + H.c ) + t R ( | N �� N + 1 | + H.c. ) , and H s is an arbitrary Hamiltonian in the scattering segment from site 1 to N which is connected to ideal leads with nearest neighbor hopping equal t = − 1 , corresponding to the energy dispersion ǫ k = − 2 cos k and the bandwidth B = 4 . The transmission probability | t ( ǫ k ) | 2 can be expressed in terms of the full resolvent as | t ( ǫ k ) | 2 = 4 t 2 R sin 2 k |� N | G ( ǫ k + i 0) | 1 �| 2 L t 2

Landauer-B¨ uttiker-formula For spinless fermions on the lattice the Landauer-B¨ uttiker formula for the stationary current reads � B/ 2 j � ∞ = e � ˆ [ f L ( ǫ ) − f R ( ǫ )] | t ( ǫ ) | 2 dǫ , h − B/ 2 � − 1 are the Fermi functions of the leads. e β a ( ǫ − µ a ) + 1 � where the f a ( ǫ ) = For β R = β L the linear conductance G ( T ) is given by � B/ 2 G ( T ) = e 2 e 2 ( − d f T =0 dǫ ) | t ( ǫ ) | 2 dǫ h | t ( µ ) | 2 = h − B/ 2

Example for H s : quantum dot with N D sites dot region: 1 < j l ≤ n ≤ j r < N , j r = j l + N D − 1 H s = ˜ H s + ( t l | j l − 1 �� j l | + t r | j r �� j r + 1 | + H.c. ) , where ˜ H s has no matrix elements connecting the dot to the wire. The sites neighboring the dot are given the indices ˜ j a , i.e. ˜ j l = j l − 1 , ˜ j r = j r + 1 . With ( Λ a , ∆ a real functions) a � ˜ j a | ( ǫ + i 0 − H t a =0 ) − 1 | ˜ Γ a ( ǫ + i 0) = t 2 j a � = Λ a ( ǫ ) − i ∆ a ( ǫ ) the transmission probability can then be expressed as | t ( ǫ ) | 2 = 4∆ l ( ǫ )∆ r ( ǫ ) | G j r ,j l ( ǫ + i 0) | 2

Example continued: resonant tunneling For well separated dots ( | t a | ≪ 1 ) N D narrow resonances occur near the eigenvalues ǫ α of the isolated dot. For ǫ ≈ ǫ α one obtains the generalized Breit-Wigner form 4∆ ( α ) ( ε )∆ ( α ) r ( ε ) | t ( ε ) | 2 ≈ l � 2 , � 2 � � ε − ǫ α − Λ ( α ) ( ε ) − Λ ( α ) ∆ ( α ) ( ε ) + ∆ ( α ) r ( ε ) + r ( ε ) l l where ∆ ( α ) = |� j a | ǫ α �| 2 ∆ a and correspondingly for the Λ ( α ) . a a Questions: What replaces the LB-formula when the interaction is turned on? What happens to the resonances when the interaction is turned on?

Meir-Wingreen-formula Using the Keldysh nonequilibrium Green function technique Meir and Wingreen (PRL 68 ,2512 (1992)) derived a formula for the stationary current for the transport through an interacting region connected to noninteracting leads. For our geometry the (nonsymmetrized) stationary current (on the link 0 → 1 ) is given by � B/ 2 j � ∞ = e � ˆ f L ( ǫ ) ( G r 11 ( ǫ ) − G a 11 ( ǫ )) + G < � � 2 i ∆ L ( ǫ ) 11 ( ǫ ) dǫ, h − B/ 2 which can easily shown to reduce to the LB-formula in the noninteracting case. The simplification under the special assumption discussed by MW cannot be used for our geometry (in the interacting model). The correction to the LB-formula as presented vanishes if Σ r − Σ a = 0 and Σ < = 0 holds (approximately).

Luttinger liquids with impurities much is known from bosonization and exactly solved models • density response of homogeneous system: (Luther & Peschel ’74; Mattis ’74) χ ( q ≈ 2 k F ) ∝ | q − 2 k F | 2 K − 2 • spinless fermions, nearest neighbor interaction, half-filling: (Haldane ’80) � � K − 1 = 2 − U π arccos 2 • local sine-Gordon model : (e.g. Apel & Rice ’82) • perturbative RG for K < 1 : (Kane & Fisher ’92) V k F , − k F relevant; hopping between open ends irrelevant • numerics: no intermediate fixed point (Moon et al. ’93; Egger & Grabert ’95) • Bethe ansatz: no intermediate fixed point (Saleur et al. ’95) alternative fermionic approach for weak two body interaction: • leading-log resummation for transmission (Glazman et al. ’93)

Functional renormalization group (Polchinski ’84, Wetterich ’93, Morris ’94, Salmhofer ’98, . . . ) idea • generating functional Γ of grand can. pot., self-energy, m -particle interaction • cutoff Λ in free propagator G 0 , Λ , cuts out infrared divergencies (if necessary) • take ∂ Λ , expand in sources ⇒ exact hierachy of differential equations formalism • “interacting” part of action: ψ k ′ ψ k + 1 { ¯ V k ′ ,k ¯ 2 ,k 1 ,k 2 ¯ 1 ¯ � � � � S int ψ } , { ψ } = u k ′ ¯ ψ k ′ ψ k ′ 2 ψ k 2 ψ k 1 1 ,k ′ 4 k ′ ,k k ′ 1 ,k ′ 2 ,k 1 ,k 2 • generating functional of connected Green functions: � 1 − 1 ψ � � � � ψ, [ G 0 , Λ ] ¯ − S int ( { ¯ ψ } , { ψ } ) − ( ¯ ψ,η ) − (¯ W c, Λ ( { ¯ η,ψ ) D ¯ η } , { η } ) = ln ψψ e Z Λ 0 • Legendre transformation: G 0 , Λ � − 1 φ � � { ¯ � ¯ ¯ Γ Λ � = −W c, Λ � η Λ } , { η Λ } φ, η Λ � η Λ , φ � � � � � φ } , { φ } { ¯ − − ¯ + φ,

• differentiate: � δ 2 Γ Λ � �� ∂ Λ Γ Λ = Tr G 0 , Λ � − 1 G 0 , Λ � G 0 , Λ � − 1 V � G Λ ∂ Λ δφδφ, G Λ � � ∂ Λ − Tr • expand in sources: ∞ ( − 1) m { ¯ m ; k 1 , . . . , k m ) ¯ 1 . . . ¯ � � Γ Λ � γ Λ m ( k ′ 1 , . . . , k ′ � φ } , { φ } = φ k ′ φ k ′ m φ k m . . . φ k 1 ( m !) 2 m =0 k ′ 1 ,...,k m • exact hierachy of flow equations: � − 1 G 0 , Λ � − 1 + γ Λ • with G Λ = �� S Λ = G Λ ∂ Λ G 0 , Λ � − 1 G Λ � , 1 • and G 0 , Λ 0 = 0 , G 0 , Λ=0 = G 0 , Γ Λ 0 � { ¯ { ¯ � � � φ } , { φ } = S int φ } , { φ } m c +1 = γ Λ 0 • approximation: γ Λ m c +1 = 0 for m c ≥ 2 ; γ Λ=0 correct to order m c m

Transport in a quantum wire – fRG approach (linear response) set up: t U • cutoff in Matsubara frequency G 0 , Λ ( iω ) = Θ ( | ω | − Λ) G 0 ( iω ) , for T = 0 • approximations: • γ Λ 3 = γ Λ= ∞ = 0 3 • no self-energy corrections in ∂ Λ γ Λ 2 2 frequency independent ⇒ Σ Λ freq. indep. ⇒ no bulk power-laws • γ Λ • interaction remains nearest neighbor j,j = − 1 equations: ∂ Λ Σ Λ 2 πU Λ � � G Λ j +1 ,j +1 ( i Λ) + G Λ j − 1 ,j − 1 ( i Λ) + ( i Λ → − i Λ) j,j ± 1 = 1 2 πU Λ � � ∂ Λ Σ Λ G Λ j,j ± 1 ( i Λ) + G Λ j,j ± 1 ( − i Λ) � − 1 �� � − 1 − Σ Λ − Σ leads ( iω ) U Λ = f ( U, Λ) G Λ ( iω ) = G 0 ( iω ) , � � t 2 l | 1 � � 1 | + t 2 G 0 , leads Σ leads ( z ) = r | N � � N | boundary ( z ) initial condition: Σ Λ= ∞ j,j ( ± 1) = V j,j ( ± 1) conductance: G ( N ) = e 2 h | t (0 , N ) | 2 , | t ( ε, N ) | 2 ∼ | G 1 ,N ( ε ) | 2

Two examples for the effective scattering potential 2 2 V j,j Σ j,j+1 1 ⇒ 1 0 Σ j,j 400 450 500 550 600 400 450 500 550 600 j j 1 Σ j,j+1 0.34 V j,j+1 0.8 0.6 0.32 ⇒ 0.4 0.3 0.2 400 450 500 550 600 400 450 500 550 600 j j long range oscillatory potentials → power law behaviour in | t ( ǫ ) | 2

Comparison with DMRG for “small” N • G , no impurity, abrupt contacts: • Friedel oscillations, open boundaries: N=128, U=1 N=12 0.7 1 DMRG fRG DMRG 0.8 fRG 0.6 2 /h) 0.6 <n j > G/(e 0.5 0.4 0.2 0.4 0 0 1 2 3 4 5 0 50 100 U j • no impurity, smooth contacts: G = e 2 • � n j � requires its own flow-equation h (composite operator)

Limiting cases • weak link from j imp to j imp + 1 , • infinitesimal imp., smooth contacts: smooth contacts: e 2 h − G ∝ N 2(1 − K RG ) ρ j ≈ j imp ∝ N − α RG G ∝ N − 2 α RG ⇒ B B 1 1 exact exact leading order leading order fRG fRG 0.5 0.5 1-K α B 0 0 -0.5 -0.5 -1 -1 -2 -1 0 1 2 -2 -1 0 1 2 U U • K RG and α RG “almost” fulfill the Tomonaga-Luttinger relation α B = 1 /K − 1 B • we can distinguish between 1 − K and 1 /K − 1