Functional RG for interacting fermions ... Part III: Impurities in Luttinger liquids 1. Luttinger liquids 2. Impurity effects S. Andergassen, T. Enss (Stuttgart) 3. Microscopic model V. Meden, K. Sch¨onhammer (G¨ottingen) 4. Flow equations U. Schollw¨ock (Aachen) 5. Results Phys. Rev. B 65, 045318 (2002); Europhys. Lett. 64, 769 (2003); Phys. Rev. B 70, 075102 (2004); Phys. Rev. B 71, 041302 (2005); Phys. Rev. B 71, 155401 (2005); Phys. Rev. B 73, 045125 (2006) 1. Luttinger liquids One-dimensional interacting Fermi systems without correlation gaps are Luttinger liquids. (1D counterpart of Fermi liquid in 2D or 3D) One-dimensional electron systems: • Complex chemical compounds containing chains • Quantum wires (in heterostructures) • Carbon nanotubes • Edge states (Dekker’s group) Electronic structure of 1D systems: 2 k = k /2m (low carrier density) Dispersion relations: k = −2t cos k (tight binding) ”Fermi surface”: 2 points ±kF k -k F 0 k F Dispersion relation near Fermi points: εk − εF approx. linear: ξk = k − F = vF (|k| − kF ) k -k F k F Electron-electron interaction: has stronger effects than in 2D and 3D systems: no fermionic quasi-particles, Fermi liquid theory not valid. Fermi liquid replaced by Luttinger liquid: • only bosonic low-energy excitations (collective charge/spin density oscillations) • power-laws with non-universal exponents ⇒ Luttinger liquid theory Review: T. Giamarchi: Quantum physics in one dimension (2004) Bulk properties of Luttinger liquids: • Bosonic low-energy excitations with linear dispersion relation c s ξq = uc q , ξq = us q (charge and spin channel) ⇒ specific heat cV ∝ T • DOS for single-electron excitations: ρ(ε) α D() ∝ |−F | vanishes at Fermi level (α > 0) εF ε DOS in principle observable by photoemission or tunneling. • Density-density correlation function N(q): finite for q → 0 (compressibility) −α2k divergent as |q−2kF | F for q → 2kF (α > 0 for repulsive interactions) 2kF ⇒ enhanced back-scattering (2kF ) from impurity. For spin-rotation invariant (and spinless) systems all exponents can be expressed in terms of one parameter Kρ. Asymptotic low energy behavior (power-laws) of Luttinger liquids described by Luttinger model: HLM = linear k + forward scattering interactions It is exactly solvable and scale-invariant (fixed point). For spinless fermions only one coupling constant, parametrizing interaction between left- and right-movers: Z HI = g dx n+(x) n−(x) 2. Impurity effects How does a single non-magnetic impurity (potential scatterer) affect properties of a Luttinger liquid? long chain/wire impurity Non-interacting system: Impurity induces Friedel oscillations (density oscillations with wave vector 2kF ) DOS near impurity finite at Fermi level Conductance reduced by a finite factor (transmission probability) Kane, Fisher ’92: impurity in interacting system (spinless Luttinger liquid) • Weak impurity potential: Backscattering amplitude V generated by impurity grows as ΛKρ−1 2kF for decreasing energy scale Λ. (K < 1 for repulsive interactions; V is ”relevant” perturbation of pure LL) ρ 2kF ⇒ Low energy probes see high barrier even if (bare) impurity potential is weak! • Weak link: t wl α DOS at boundary of LL vanishes as |−F | B ⇒ Tunneling amplitude twl between two weakly coupled α −1 chains scales to zero as Λ B with αB = Kρ − 1 > 0 at low energy scales. (twl is ”irrelevant” perturbation of split chain) Hypothesis (Kane, Fisher): Any impurity effectively ”cuts the chain” at low energy scales and physical properties obey weak link or boundary scaling. ⇒ DOS near impurity: α Di() ∝ |−F | B for → F at T = 0 Conductance through impurity: G(T ) ∝ T 2αB for T → 0 supported within effective bosonic field theory by: refermionization (Kane, Fisher ’92) QMC (Moon et al. ’93; Egger, Grabert ’95) Bethe ansatz (Fendley, Ludwig, Saleur ’95) 3. Microscopic model Spinless fermion model: t U nearest neighbor hopping t nearest neighbor interaction U P † † P Hsf = −t j cj+1cj + cjcj+1 + U j njnj+1 Properties (without impurities): • exactly solvable by Bethe ansatz • Luttinger liquid except for |U| > 2t at half-filling • charge density wave for U > 2t at half-filling Impurity potential added to bulk hamiltonian Hsf: X † general form: Himp = Vj0j cj0cj j,j0 ”site impurity”: Himp = V nj0 (j0 impurity site) ”hopping impurity”: H = (t − t0) c† c + c† c imp j0+1 j0 j0 j0+1 Later also double barrier (two site or hopping impurities) 4. Flow equations Starting point (for approximations): Exact hierarchy of differential flow equations for 1-particle irreducible vertex functions with infrared cutoff Λ: S Λ d Λ Σ = Γ Λ d Λ S Λ d Γ Λ = + d Λ Λ Λ Γ Λ Λ G 3 etc. for Γ3 , Γ4 , ... where dGΛ GΛ = (GΛ)−1 − ΣΛ −1 SΛ = 1 − GΛΣΛ −1 0 1 − ΣΛGΛ −1 0 0 dΛ 0 Cutoff: Λ At T = 0 sharp frequency cutoff: G0 = Θ(|ω| − Λ) G0 At finite T (discrete Matsubara frequencies) soft cutoff with width 2πT G0 bare propagator without impurities and interaction Approximations: Scheme 1 (first order): Λ 0 S Approximate ΓΛ ≈ ΓΛ (ignore flow of 2-particle vertex) d ⇒ ΣΛ tridiagonal matrix in real space Σ Λ = Γ dΛ 0 Flow equation very simple; at T = 0: d U X X d U X ΣΛ = − G˜Λ (iω) ΣΛ = G˜Λ (iω) dΛ j,j 2π j+s,j+s dΛ j,j±1 2π j,j±1 s=±1 ω=±Λ ω=±Λ ˜Λ −1 Λ −1 where G (iω) = [G0 (iω) − Σ ] . Kane/Fisher physics already qualitatively captured ! Scheme 2 (second order): Λ Λ Λ Neglect Γ3 ; approx. Γ by flowing nearest neighbor interaction U ⇒ 1-loop flow for U Λ; flow of ΣΛ as in scheme 1 with renormalized U Λ d U Λ X X d U Λ X ΣΛ = − G˜Λ (iω) ΣΛ = G˜Λ (iω) dΛ j,j 2π j+s,j+s dΛ j,j±1 2π j,j±1 s=±1 ω=±Λ ω=±Λ Works quantitatively even for rather big U Derivation of flow equation (scheme 1): Flow equation for self-energy: S Λ X d Λ 0 iω 0+ Λ 0 0 0 Σ (1 , 1) = −T e 2 S (2, 2 )Γ (1 , 2 ; 1, 2) d Λ dΛ 0 Σ = Γ 2,20 dΛ 0 Single-scale propagator 1 ∂GΛ 1 SΛ = GΛ[∂ (GΛ)−1]GΛ = − 0 Λ 0 Λ Λ Λ Λ 1 − G0 Σ ∂Λ 1 − Σ G0 Self-energy and propagator diagonal in frequency: ω1 = ω10 and ω2 = ω20. Γ0 freqency-independent ⇒ Σ frequency-independent. Λ Sharp frequency cutoff (T = 0): G0 (iω) = Θ(|ω|−Λ) G0(iω) ⇒ 1 1 SΛ(iω) = δ(|ω|−Λ)G (iω) Λ 0 Λ 1 − Θ(|ω|−Λ)G0(iω)Σ 1 − Θ(|ω|−Λ)Σ G0(iω) δ(.) meets Θ(.): ill defined! 0 Consider regularized (smeared) step functions Θ with δ = Θ , then take limit → 0, using Z Z 1 →0 proof: dx δ(x − Λ) f[x, Θ(x − Λ)] −→ dt f(Λ, t) 0 substitution t = Θ Integration can be done analytically, yielding X X d Λ 1 iω0+ ˜Λ 0 Σj0 ,j = − e Gj ,j0 (iω) Γj0 ,j0 ;j ,j dΛ 1 1 2π 2 2 1 2 1 2 ω=±Λ 0 j2,j2 ˜Λ −1 Λ −1 where G (iω) = [G0 (iω) − Σ ] Insert real space structure of bare vertex for spinless fermions with nearest neighbor interaction U: 0 0 0 0 0 Γ 0 0 = Uj1,j2 (δj ,j δj ,j − δj ,j δj ,j ) j1,j2;j1,j2 1 1 2 2 1 2 2 1 Uj1,j2 = U (δj1,j2−1 + δj1,j2+1) ⇒ Flow equations d U X X + ΣΛ = − eiω0 G˜Λ (iω) dΛ j,j 2π j+s,j+s s=±1 ω=±Λ d U X + ΣΛ = eiω0 G˜Λ (iω) dΛ j,j±1 2π j,j±1 ω=±Λ Convergence factor eiω0+ matters only for Λ → ∞ Initial condition at Λ = Λ0 → ∞: 1 X Λ0 0 Σ = V 0 + Γ 0 j1,j j0 ,j ;j ,j j1,j1 1 2 1 2 1 2 j2 where V 0 is the bare impurity potential and j1,j1 the second term is due to the flow from ∞ to Λ0 (!) Flow equations at finite temperatures T > 0: Λ Replace ω = ±Λ by ω = ± ωn in flow equations, Λ where ωn is the Matsubara frequency most close to Λ. Calculation of conductance: Interacting chain connected to semi-infinite non-interacting leads via smooth or abrupt contacts leadinteracting chain/wire lead contactimpurity contact Z e2 Conductance G(T ) = − d f 0() |t()|2 with |t()|2 ∝ |G ()|2 h 1,N Propagator G1,N () calculated in presence of leads, which affect the interacting region only via boundary contributions Σ1,1() and ΣN,N () to the self-energy Vertex corrections vanish within our approximation (no inelastic scattering) (see Oguri ’01) fRG features: • perturbative in U (weak coupling) • non-perturbative in impurity strength • arbitrary bare impurity potential (any shape) • full effective impurity potential (cf. Matveev, Yue, Glazman ’93: only V ) 2kF • cheap numerics up to 105 sites for T > 0 and 107 sites at T = 0. • captures all scales, not just asymptotics. 5. Results Renormalized impurity potential (from self-energy Σjj at Λ = 0): 0.4 0.2 Vj 0 -0.2 -0.4 450 500 550 j long-range 2kF -oscillations ! (associated with Friedel oscillations of density) 2kF -oscillations also in renormalized hopping amplitude around impurity Results for local DOS near impurity site: (half-filling, ground state, U = 1, V = 1.5, 1000 sites) 0.4 0.3 0.2 red: Lutt. liquid green: Fermi gas local DOS 0.1 0 -3 -2 -1 0 1 2 3 ω Strong suppression of DOS near Fermi level Power law with boundary exponent αB for ω → 0, N → ∞ Spectral weight at ω = 0 in good agreement with DMRG for U < 2.