Solution of the Anderson impurity model via the functional renormalization group

Transcription

1 Solution of the Anderson impurity model via the functional renormalization group Simon Streib, Aldo Isidori, and Peter Kopietz Institut für Theoretische Physik, Goethe-Universität Frankfurt Meeting DFG-Forschergruppe FOR 723, Aachen, April 29, 2013 Simon Streib, Aldo Isidori, and Peter Kopietz, arxiv:

2 Anderson Impurity Model localized impurity in contact with a conduction bath (Anderson, 1961) Ĥ = (ɛ k σh) ĉ kσĉkσ + (E d σh) ˆd ˆd σ σ kσ σ +U ˆd ˆd ˆd ˆd + ( V ˆd k σĉ kσ + V k ĉ ˆd ) kσ σ kσ ɛ k : dispersion of the conduction electrons E d : atomic energy of the localized impurity state V k : hybridization energy U: on-site interaction of the impurity electrons H: Zeemann energy due to an external magnetic field Aachen, April 29, 2013 Peter Kopietz 2 / 15

3 Impurity action integrate over conduction electrons impurity action 1/T S imp = G 1 0,σ (iω) d ωσ d ωσ + U dτ n n ω σ bare Green s function (here only particle-hole symmetric case) G 0,σ (iω) = 1 iω + U/2 + σh + i sgn ω = π V 2 ρ(0) hybridization energy 0 experimental realization in quantum dots: Aachen, April 29, 2013 Peter Kopietz 3 / 15

4 Exact and approximate results on the AIM spin- and charge susceptibilities from Bethe-Ansatz (Wiegmann, Andrei, Kawakami, Okiji, ) 2 π χ s = 2 πu eπ u/8 dxe x2 /(2u) cos(πx/2) 1 x 2, π χ c = 2 πu e π 0 2 u/8 0 dxe x2 /(2u) cosh(πx/2) 1 + x 2. in strong coupling regime (u U/(π ) 2) there is only one energy scale (Kondo temperature) πu T K = 2 e π 2 u u, χ s 1/(2T K ), Z 2/(π χ s ) all previous FRG studies have so far failed to reproduce correct exponential Kondo scale in χ s and Z. Aachen, April 29, 2013 Peter Kopietz 4 / 15

5 Partial bosonization of the interaction Partial bosonization of the interaction (U = U + U ): Un (τ)n (τ) = U n (τ)n (τ) U s(τ)s(τ) s(τ) = d (τ)d (τ), and s(τ) = d (τ)d (τ) Hubbard-Stratonovich transformation: 1 ω( U D [ χ, χ] e χ ωχ ω s ωχ ω s ω χ ω) = e U ω s ωs ω, Partially bosonized action: S 0 [ d, d, χ, χ] = S 1 [ d, d, χ, χ] = ω σ G 1 0,σ (iω) d ωσ d ωσ + U 1 χ ωχ ω, ω 1/T ( s ω χ ω + s ω χ ω ) + U dτ n (τ)n (τ). ω 0 Aachen, April 29, 2013 Peter Kopietz 5 / 15

6 Skeleton equations bosonic self-energy in terms of three-legged vertex: Π (i ω) = Γ d d χ (ω, ω ω; ω) G (iω) G (iω i ω) ω three and four legged vertices are not independent: Γ d d χ (ω + ω, ω; ω) = 1 G (iω + i ω) G (iω ) ω Γ d d d d (ω + ω, ω ; ω, ω + ω) = = Aachen, April 29, 2013 Peter Kopietz 6 / 15

7 FRG with magnetic field cutoff use the external magnetic field H as an FRG flow paramater: Λ = H [Edwards and Hewson, J. Phys.: Cond. Mat. 23, (2011)] additional bosonic regulator: U 1 = U 1 + R H with R 0 = 0 and R = fermion propagator: G 1 σ (iω) = G 1 0,σ (iω) Σ σ(iω) boson propagator: F 1 1 (i ω) = U (H) Π (i ω) single scale propagators: Ġ σ (iω) = σg 2 σ (iω), F (i ω) = R H F 2 H (i ω) σ FRG flow equation: Σ (iω) H = = σ=, Aachen, April 29, 2013 Peter Kopietz 7 / 15

8 Low-energy expansion neglect the frequency-dependence of the vertices: Γ d σ dσ d σ d σ 4 (ω 1,..., ω 4 ) Γ d σ dσ d σ d σ 4 (0) Γ d σd σ χχ 4 (ω 1,..., ω 4 ) Γ d σd σ χχ 4 (0) Γ d d χ 3 (ω 1, ω 2, ω 3 ) Γ d d χ 3 (0) γ expand the fermionic self-energy: Σ σ (iω) = U/2 σm + (1 Z 1 )iω + O(ω 2 ) flowing parameters M and Z for H, how to close the FRG hierarchy? M U/2 and Z 1 (Hartree-Fock) bosonic self-energy Π (i ω) skeleton equation [Bartosch et al. (2009)] flow equations for the vertices Ward identities Aachen, April 29, 2013 Peter Kopietz 8 / 15

9 Ward identities Koyama-Tachiki Ward identities [Prog. Theor. Phys. Supp. 80, 108 (1984)] Γ d σd σχ σ (0, 0; 0) γ = H + M H + U s, χ Π (i0) = 1 U Π (i0) = s H, where magnetic moment s = n n 2 satisfies Friedel sum rule: s = 1 ( ) H + M π arctan Yamada-Yosida Ward identities [Yamada-Yosida, Prog. Theor. Phys. (1975); Kopietz et al., J. Phys. A: Math. Theor. 43, (2010)] Z 1 = χ c + χ 2, ργ (4) = χ c χ 2 where χ c = χ c /ρ, χ = χ /ρ and Γ (4) = Γ d σ dσ d σ d σ (0, 0; 0, 0) is the full four-point vertex (without bosonization) Aachen, April 29, 2013 Peter Kopietz 9 / 15

10 Restoring rotational invariance in FRG flow flow equation for Z: ln Z ln H = ZH lim ω 0 flow equation for M: M H = σ H Σ σ (i0) Σ σ(iω) (iω) H problem: (truncated) FRG flow equation for M does not restore the rotational invariance for H 0: lim M 0! H 0 solution: use Yamada-Yosida Ward identities to derive an alternative flow equation which restores rotational invariance for H 0: M (4) = ργ H + 1 Z 1 1 [ b b 1 + b 2 arctan b h 1 ] + 1 Z Z 1 where b = (H + M)/ and h = H/. lim H 0 M = 0 without fine-tuning. Aachen, April 29, 2013 Peter Kopietz 10 / 15

11 Initial conditions and bosonic regulator start FRG flow for H first order perturbation theory becomes exact; small parameter: U/H: Σ σ (1) (iω) = U 2 σ U 2 Z 0 = 1, M 0 = U 2, for H bosonization depends on the bosonic regulator: { U 1 = U 1 + R H use two-slope function: r = π 2 αh for H R H = βh for H, β1u ansatz: α(u) = α0 α 2+u, β(u) = 2 β 2+u, with u = U 2 π need four constraints to fix four parameters α 0, α 2, β 1, β 2. two constraints: weak coupling limit for χ s and Z (e.g., at u = 0.1) two more constraints: strong coupling limit of Wilson ratio R W = 2 for two strong coupling values (e.g., u = 4, u = 8). R W = χ s Z = 2 χ s /( χ s + χ c ) 2, u 2 Aachen, April 29, 2013 Peter Kopietz 11 / 15

12 Our bosonic regulator π R H two-slope function: π 2 R H = (α β) 1+h + βh U/(π ) = 1 U/(π ) = U/(π ) H/ h α β α(u) = α 0 α 2 + u 2 β(u) = β 1u β 2 + u 2 u U/(π ) matching exact weak- and strong-coupling limits α , α β , β Aachen, April 29, 2013 Peter Kopietz 12 / 15

13 Quasiparticle residue and spin-susceptibility at H = Bethe ansatz FRG U/(π ) 1.5 R 1 Bethe ansatz π χ s Bethe ansatz U/(π ) Z -1 FRG π χ s FRG Z -1 Z -1 2nd order PT Aachen, April 29, 2013 Peter Kopietz 13 / 15

14 Magnetic field behavior at strong-coupling, U/(π ) = γ s s (NRG) Z H/ Aachen, April 29, 2013 Peter Kopietz 14 / 15

15 Summary and Outlook FRG approach with external magnetic field as physical flow parameter bosonization of transverse spin fluctuations interaction vertices constrained by Ward identities correct Kondo scale in the quasiparticle residue and spin susceptibility magnetic field dependence of physical observables in agreement with Bethe-ansatz next:generalization to non-equilibrium [Yamada-Yoshida Ward identities: Oguri, Phys. Rev. B 64, (2001)] Aachen, April 29, 2013 Peter Kopietz 15 / 15