Part III: Impurities in Luttinger liquids

Transcription

1 Functional RG for interacting fermions... Part III: Impurities in Luttinger liquids 1. Luttinger liquids 2. Impurity effects 3. Microscopic model 4. Flow equations 5. Results S. Andergassen, T. Enss (Stuttgart) V. Meden, K. Schönhammer (Göttingen) U. Schollwöck (Aachen) Phys. Rev. B 65, (2002); Europhys. Lett. 64, 769 (2003); Phys. Rev. B 70, (2004); Phys. Rev. B 71, (2005); Phys. Rev. B 71, (2005); Phys. Rev. B 73, (2006)

2 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)

3 Electronic structure of 1D systems: Dispersion relations: ɛ k = k 2 /2m ɛ k = 2t cos k (low carrier density) (tight binding) Fermi surface : 2 points ±k F -k F 0 k F k Dispersion relation near Fermi points: ε k ε F -k F k F k approx. linear: ξ k = ɛ k ɛ F = v F ( k k F )

4 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)

5 Bulk properties of Luttinger liquids: Bosonic low-energy excitations with linear dispersion relation ξq c = u c q, ξq s = u s q (charge and spin channel) specific heat c V T DOS for single-electron excitations: D(ɛ) ɛ ɛ F α ρ(ε) vanishes at Fermi level (α > 0) DOS in principle observable by photoemission or tunneling. ε F ε

6 Density-density correlation function N(q): finite for q 0 (compressibility) divergent as q 2k F α 2k F for q 2k F (α 2kF > 0 for repulsive interactions) enhanced back-scattering (2k F ) from impurity. For spin-rotation invariant (and spinless) systems all exponents can be expressed in terms of one parameter K ρ.

7 Asymptotic low energy behavior (power-laws) of Luttinger liquids described by Luttinger model: H LM = linear ɛ 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: H I = g dx n + (x) n (x)

8 2. Impurity effects How does a single non-magnetic impurity (potential scatterer) affect properties of a Luttinger liquid? impurity long chain/wire Non-interacting system: Impurity induces Friedel oscillations (density oscillations with wave vector 2k F ) DOS near impurity finite at Fermi level Conductance reduced by a finite factor (transmission probability)

9 Kane, Fisher 92: impurity in interacting system (spinless Luttinger liquid) Weak impurity potential: Backscattering amplitude V 2kF generated by impurity grows as Λ K ρ 1 for decreasing energy scale Λ. (K ρ < 1 for repulsive interactions; V 2kF is relevant perturbation of pure LL) Low energy probes see high barrier even if (bare) impurity potential is weak! Weak link: t wl DOS at boundary of LL vanishes as ɛ ɛ F α B Tunneling amplitude t wl between two weakly coupled chains scales to zero as Λ α B with α B = K 1 ρ 1 > 0 at low energy scales. (t wl is irrelevant perturbation of split chain)

10 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: D i (ɛ) ɛ ɛ F α B for ɛ ɛ F at T = 0 Conductance through impurity: G(T ) T 2α B 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)

11 3. Microscopic model Spinless fermion model: t U nearest neighbor hopping t nearest neighbor interaction U H sf = t j ( c j+1 c j + c j c ) j+1 + U j n jn j+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

12 Impurity potential added to bulk hamiltonian H sf : general form: H imp = j,j V j j c j c j site impurity : H imp = V n j0 (j 0 impurity site) hopping impurity : H imp = (t t ) ( c j 0 +1 c j 0 + c ) j 0 c j0 +1 Later also double barrier (two site or hopping impurities)

13 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 d Λ Σ Λ = Γ Λ S Λ where d Γ d Λ Λ = G Λ + Γ Λ 3 etc. for Γ Λ 3, Γ Λ 4,... G Λ = [ (G Λ 0 ) 1 Σ Λ] 1 S Λ = [ 1 G Λ 0 Σ Λ] 1 dg Λ 0 dλ [ 1 Σ Λ G Λ 0 ] 1

14 Cutoff: At T = 0 sharp frequency cutoff: G Λ 0 = Θ( ω Λ) G 0 At finite T (discrete Matsubara frequencies) soft cutoff with width 2πT G 0 bare propagator without impurities and interaction

15 Approximations: Scheme 1 (first order): Approximate Γ Λ Γ Λ0 (ignore flow of 2-particle vertex) S Λ Σ Λ tridiagonal matrix in real space Flow equation very simple; at T = 0: d dλ ΣΛ j,j = U 2π s=±1 ω=±λ G Λ j+s,j+s(iω) d dλ ΣΛ j,j±1 = U 2π d Σ Λ = Γ dλ 0 ω=±λ G Λ j,j±1(iω) where G Λ (iω) = [G 1 0 (iω) ΣΛ ] 1. Kane/Fisher physics already qualitatively captured!

16 Scheme 2 (second order): Neglect Γ Λ 3 ; approx. Γ Λ by flowing nearest neighbor interaction U Λ 1-loop flow for U Λ ; flow of Σ Λ as in scheme 1 with renormalized U Λ d dλ ΣΛ j,j = U Λ 2π s=±1 ω=±λ G Λ j+s,j+s(iω) d dλ ΣΛ j,j±1 = U Λ 2π ω=±λ G Λ j,j±1(iω) Works quantitatively even for rather big U

17 Derivation of flow equation (scheme 1): Flow equation for self-energy: d dλ ΣΛ (1, 1) = T e iω + 20 S Λ (2, 2 ) Γ 0 (1, 2 ; 1, 2) 2,2 Single-scale propagator S Λ = G Λ [ Λ (G Λ 0 ) 1 ]G Λ 1 G Λ 0 = 1 G Λ 0 ΣΛ Λ 1 1 Σ Λ G Λ 0 S Λ d Σ Λ = Γ dλ 0 Self-energy and propagator diagonal in frequency: ω 1 = ω 1 and ω 2 = ω 2. Γ 0 freqency-independent Σ frequency-independent.

18 Sharp frequency cutoff (T = 0): G Λ 0 (iω) = Θ( ω Λ) G 0 (iω) S Λ (iω) = 1 1 Θ( ω Λ)G 0 (iω)σ Λ δ( ω Λ)G 0(iω) 1 1 Θ( ω Λ)Σ Λ G 0 (iω) δ(.) meets Θ(.): ill defined! Consider regularized (smeared) step functions Θ ɛ with δ ɛ = Θ ɛ, then take limit ɛ 0, using dx δ ɛ (x Λ) f[x, Θ ɛ (x Λ)] ɛ 0 1 Integration can be done analytically, yielding 0 dt f(λ, t) proof: substitution t = Θ ɛ d dλ ΣΛ j 1,j = 1 1 2π ω=±λ j 2,j 2 e iω0+ GΛ j 2,j 2(iω) Γ 0 j 1,j 2 ;j 1,j 2 where G Λ (iω) = [G 1 0 (iω) ΣΛ ] 1

19 Insert real space structure of bare vertex for spinless fermions with nearest neighbor interaction U: Γ 0 j 1,j 2 ;j 1,j 2 = U j1,j 2 (δ j1,j 1 δ j 2,j 2 δ j 1,j 2 δ j 2,j 1 ) U j1,j 2 = U (δ j1,j δ j1,j 2 +1) Flow equations d dλ ΣΛ j,j = U 2π d dλ ΣΛ j,j±1 = U 2π s=±1 ω=±λ ω=±λ e iω0+ e iω0+ GΛ j+s,j+s (iω) GΛ j,j±1 (iω) Convergence factor e iω0+ matters only for Λ

20 Initial condition at Λ = Λ 0 : Σ Λ 0 j 1,j 1 = V j1,j j 2 Γ 0 j 1,j 2;j 1,j 2 where V j1,j 1 is the bare impurity potential and the second term is due to the flow from to Λ 0 (!) Flow equations at finite temperatures T > 0: Replace ω = ±Λ by ω = ± ω Λ n in flow equations, where ω Λ n is the Matsubara frequency most close to Λ.

21 Calculation of conductance: Interacting chain connected to semi-infinite non-interacting leads via smooth or abrupt contacts lead interacting chain/wire lead contact impurity contact Conductance G(T ) = e2 h dɛ f (ɛ) t(ɛ) 2 with t(ɛ) 2 G 1,N (ɛ) 2 Propagator G 1,N (ɛ) calculated in presence of leads, which affect the interacting region only via boundary contributions Σ 1,1 (ɛ) and Σ N,N (ɛ) to the self-energy Vertex corrections vanish within our approximation (no inelastic scattering) (see Oguri 01)

22 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 10 5 sites for T > 0 and 10 7 sites at T = 0. captures all scales, not just asymptotics.

23 5. Results Renormalized impurity potential (from self-energy Σ jj at Λ = 0): V j j long-range 2k F -oscillations! (associated with Friedel oscillations of density) 2k F -oscillations also in renormalized hopping amplitude around impurity

24 Results for local DOS near impurity site: (half-filling, ground state, U = 1, V = 1.5, 1000 sites) local DOS red: Lutt. liquid green: Fermi gas ω Strong suppression of DOS near Fermi level Power law with boundary exponent α B for ω 0, N Spectral weight at ω = 0 in good agreement with DMRG for U < 2.

25 0.4 Log. derivative of spectral weight at Fermi level as fct. of system size: near boundary (solid lines) near hopping impurity (dashed lines) α circles: quarter-filling, U = 0.5 squares: quarter-filling, U = L open symbols: frg filled symbols: DMRG top panel: without vertex renorm. bottom panel: with vertex renorm. α horizontal lines: exact boundary exponents L

26 Friedel oscillations from open boundaries: (half-filling, ground state) N=128, U=1 DMRG frg <n j > j Excellent agreement between frg and DMRG

27 One parameter scaling of conductance (T = 0): Single impurity, smooth contacts: G(N) = e2 h G K (x), x = [N/N 0 (U, V )] 1 K 10 0 U= U=2.23 G/(e 2 /h) /K x 1-x 2 G/(e 2 /h) x=[n/n 0 (U,V)] (1-K) x=[n/n 0 (U,V)] 1/2 Crossover size as function of bare reflection amplitude N 0 (U,V) U=0.5 U=1 U= R

28 Conductance at T > 0 smooth contacts V=10, N= U=0.5, N=10 4 G/(e 2 /h) T 2α B U=0.5 U= T G/(e 2 /h) T 2α B V=10 V=1 V= T Asymptotic power law G(T ) T 2α reached on accessible scales only for sufficiently strong impurities

29 Resonant tunneling through double barrier: Treated theoretically by many groups; controversial results! (Dekker s group 01) Model setup: smooth gate voltage

30 Resonance peaks in conductance as a function of gate voltage: G/(e 2 /h) N D =6, V l/r =10, U=0.5, N=10 4 T=0.3 T=0.1 T=0.03 T= V g At T = 0, width w N K 1 T -dependence of t(ɛ) 2 important

31 frg results for G p (T ) (symmetric double barrier): G p (T)/(e 2 /h) exponent V l/r =10, U=0.5, N=10 4 N D =2 N D =100 2α B 0 2α B α -1 B T G p (T)/(e 2 /h) exponent V l/r =0.8, U=1.5, N=10 4 N D =10 N D = T 2α B 2α B -1 α B -1 Various distinctive power laws, in particular (Furusaki, Nagaosa 93, 98): exponent 2α B (looks like independent impurities in series) exponent α B 1 ( uncorrelated sequential tunneling ) No indications of exponent 2α B 1 ( correlated sequential tunneling )

32 Summary... frg is reliable and flexible tool to study Luttinger liquids with impurities can be applied to microscopic models, restricted to weak coupling provides simple physical picture interplay of contacts, impurities, and correlations method covers all energy scales resonant tunneling: universal behavior and crossover captured

33 ... and outlook include spin (extended Hubbard model: Andergassen et al., PRB 73, (2006)) more complex geometries (Y-junctions: Barnabé-Thériault et al., PRL 94, (2005)) include bulk anomalous dimension include inelastic processes extend to non-linear transport