Finite-frequency Matsubara FRG for the SIAM Final status report Christoph Karrasch & Volker Meden Ralf Hedden & Kurt Schönhammer Numerical RG: Robert Peters & Thomas Pruschke Experiments on superconducting SIAM : Alexander Eichler, Markus Weiss & Christian Schönenberger Richard Deblock & Hélène Bouchiat Ladenburg, November 28
The Stage Single Impurity Anderson Model ( Severin) P-irreducible Matsubara FRG with sharp multiplicative Θ-cutoff usual flow equation hierachy: truncation procedure usually employed for SIAM-like models: (a) neglect the contribution of γ 3 to the flow of γ 2 (b) neglect the frequency dependence of γ 2 zero/finite-frequency properties are described well/badly: 2 U/Γ=4π U/Γ=, ε= Bethe ansatz FRG NRG FRG G/G πργ - ε/u -5 5 ω/γ
Finite-frequency FRG Straight-forward way of implementing frequency-dependence: regard γ 2 as a function of three indep. bosonic frequencies ν i preserves symmetries automatically parametrize the self-energy Σ Λ (iω) and the two-particle vertex γ Λ 2 (iν, iν 2, iν 3 ) using a discrete mesh of N Matsubara frequencies: ω n = ω a n, n =... N Important numerical aspect: verify that physical properties are independent of the actual choice of the discretization! Technical issues: compute spectral function from (ill-controlled) Padé approximation replace S Λ G Λ G Λ G Λ (Katanin 24)
Results: small to intermediate U πργ U/Γ=5 ε=b=t= NRG FRG U/Γ=4 ε/γ= B=T= -5 5-5 5 πργ U/Γ=4 T/Γ=.4 ε=b= U/Γ=4 B/Γ=.2 ε=t= -5 5 ω/γ -5 5 ω/γ
Numerical efficiency The FRG works well for arbitrary parameters and intermediate U. It is, however, numerically demanding. Approximation to increase efficiency: γ 2 = PP-term (ν, ν 2 =, ν 3 = ) PH-term (ν =, ν 2, ν 3 = ) + HP-term (ν =, ν 2 =, ν 3 ) one-dimensional frequency meshes only πργ U/Γ=5 ε=b=t= NRG FRG RHF At intermediate U, reliable results can be obtained with minor numerical effort! -5 5 ω/γ
Large U: the Kondo scale quantities governed by T K : spin susceptibility, effective mass, width of the Kondo resonance,... NRG frequency-independent FRG frequency-independent FRG χ/χ 5 U/Γ Frequency-independent FRG shows Frequency- exponential behavior. dependent FRG shows no exponential behavior.
Discussion Frequency-dependent FRG: gives better results at small to intermediate U there is no exponential energy scale BUT: there are numerical (discretization) issues! choose N large enough so that results are converged different ways to parametrize γ 2 do not give coinciding results in the strong coupling regime (limitation of num. resources?!) why does the non-katanin scheme break down for large U? frequency discretization? fundamental reasons (neglection of γ 3 )?
So what? Consider SIAM with BCS leads: low-energy physics: governed by an interplay of the Kondo effect and induced superconductivity (ratio T K / ) interesting quantity: supercurrent as a function of the gate voltage advantage: interesting physics at intermediate U Zero-frequency FRG works fine for zero temperature!. theory exp..5 /Γ=.5 U/Γ=9 Γ R /Γ L =6 6 /Γ=2, t d /Γ=.2, φ/π=.2 J/J.2. /Γ=.5 U/Γ=5 Γ R /Γ L =3 U crit /Γ 4 2 doublet singlet -.8 -.4.4.8 ε/u -2-2 ε/γ Finite-frequency FRG needed to treat finite temperatures!
Frequency-dependent FRG can be used to fast compute finite-energy properties of the SIAM at small to intermediate U. There is no exponential energy scale. Thank you for your attention!
Single Impurity Anderson Model The SIAM describes an impurity of interacting spin up and down electrons coupled to a bath of Fermi-liquid leads. The low-energy physics of this model is dominated by the Kondo effect. The Hamiltonian consists of three parts, H = H dot + H leads + H coup, where H dot = σ ɛ σ d σd σ + Ud d d d H leads = ɛ sk c skσ c skσ s=l,r kσ H coup = ( t s c sσ d σ + d ) σc sσ s=l,r Single-particle energy: ɛ σ = ɛ U/2 ± B/2 Local electron operators at the impurity site: c sσ = k c skσ/ N Hybridisation energy: Γ = Γ L + Γ R, where Γ s = πt 2 sρ s = const. (wide-band limit)
The QD Josephson junction Model Hamiltonian: H dot = (ɛ U/2) σ d σd σ + Ud d d d quantum dot H lead s=l,r = kσ ɛ sk c skσ c skσ k [e iφ s c sk c s k + H.c. ] BCS leads H coup s=l,r = t s c sσd σ + H.c. σ H direct = t d σ c Lσ c Rσ + H.c. coupling QD-leads direct coupling
Kondo scale: the effective mass Effective mass: /T K m = Im Σ(iω )/ω NRG frequency-dependent FRG m* 5 U/Γ FRG does not show exponential behaviour!
Spin susceptibility.8 NRG frequency-independent FRG frequency-dependent FRG χ/χ.6.4.2.4.6 U/Γ