Local moment approach to the multi - orbital single impurity Anderson and Hubbard models Anna Kauch Institute of Theoretical Physics Warsaw University PIPT/Les Houches Summer School on Quantum Magnetism Les Houches, 6th - 23rd June 26
Outline 1. Kondo effect 2. Single impurity Anderson model single orbital Anderson model mean field local moment approach (LMA) multiorbital Anderson model and generalization of LMA 3. Hubbard model and DMFT 4. LMA + DMFT in multiorbital Hubbard model
Kondo effect Example: resistivity minimum in metals with rare earth impurities Universal in rare earth impurities, quantum dots, carbon nanotubes
Kondo effect below the temperature characteristic for a given material Kondo temperature T K (crossover) spin fluctuations spin-flip processes screening of local moments
Single impurity Anderson model (SIAM) H SIAM = k,σ ǫ k c kσ c kσ+ k,σ V k (d σc kσ + c kσ d σ ) + σ (ǫ d Un d σ )n dσ Hybridization function (ω) = k V k 2 ω ǫ k SIAM model describes charge fluctuations spin fluctuations screening of local magnetic moments
Mean field (Hartree-Fock) approximation Un d σ n dσ U < n d σ > n dσ + Un d σ < n dσ > +U < n d σ >< n dσ > Energies depend on fillings < n > and < n > E σ = ǫ d + U < n σ > Non-vanishing local magnetic moment µ =< n > < n >,2 Total density of states Spectral function,1-9 -6-3 3 6 9 Energy Figure: (Anderson 1961)
Energy scales ǫ d,u T atomic limit U non-magnetic regime ǫ d,u, T > T K local moment regime ǫ d,u, T < T K strong coupling regime (Kondo effect) ǫ d mixed valence regime
Local moment approach (Logan) LMA Unrestricted Hartree-Fock solution dependent explicitly on local moment µ: G HF σ (µ, ω) RPA perturbation series around the UHF solution: Gσ A,B HF
Local moment approach (LMA) Using RPA we obtain two selfenergies Σ A σ and Σ B σ and two LMA Green functions: G A σ (Σ A ) LMA, G B σ (Σ B ) LMA Symmetry restoration G σ (ω) LMA = 1 2 (G A σ (ω)lma + G B σ (ω)lma) Minimization of the ground state energy with respect to the local moment value µ and n E phys. = min µ,n E( µ,n).
Local moment approach in SIAM,3 Reproduces Hubbard bands (charge fluctuations) Describes well Kondo resonance near the Fermi level (spin fluctuations), reproducing its exponential scale Conservative (there exists Luttinger - Ward functional for this theory) Numericaly efficient Spectral function Spectral function,2,1,1-9 -6-3 3 6 9 Energy asymmetric -9-6 -3 3 6 9 Energy
Multiorbital Anderson model and LMA generalization H SIAM = ǫ k c kσ c kσ + V kα (d ασ c kσ + c kσ d ασ k,σ k,σ,α + (ǫ α U α n α, σ ) n α,σ + ( ) U αβ Jδ σσ nασ n βσ α,σ σ,σ α β RPA perturbation series around the UHF solutions, that are now dependent on local moments on the orbitals µ α Symmetry restoration G αβ σ (ω) = 1 2N orb ± µ α,± µ β G αβ σ (µ α µ β ;ω) Energy minimization with respect to µ α and n α E phys. = min µ α,n α E(µ α,n α ). )
Multiorbital SIAM and LMA n=1 J= Spectral functions A αα σ (ω) = 1 αα ImGσ π The influence of Hund s rule J suppresses Kondo peak Differences in spectral functions between different orbials Possible investigation of transport in multiorbital quantum dots Spectral function Spectral function,3,2,1,3,2,1-9 -6-3 3 6 9 Energy n=1 J=.1 U -9-6 -3 3 6 9 Energy
Hubbard model interaction U (repulsion) between electrons on each site H Hubbard = ijσ t ij c iσ c jσ+u i n i n i solution (approx.) within the scheme of dynamical mean field theory
Dynamical mean field theory (DMFT) Hubbard model is mapped onto a model of a single impurity in mean field G (iω n ) this mean field is frequency dependent (dynamical) DMFT Green function G(k,iω n ) 1 = iω n +µ ǫ k Σ LOC (iω n ) self-consistency condition Σ LOC (iω n ) = G (iω n ) 1 +G LOC (iω n ) 1 oraz G = G [G LOC ]
Dynamical mean field theory (DMFT) Thermodynamicaly consistent and conservative Exact in the non-trivial limit of d = Frequency dependent mean field ( Weiss field G (iω n ) ) describes spin and charge fluctuations on sites Exact in the atomic (t i,j = ) and non-interacting (U = ) limits Requires an impurity solver to solve self-consistent equations the solver must work for arbitrary hybridization function (e.g. NRG, QMC, IPT, NCA) LMA can serve as an impurity solver
DMFT + LMA Mott-Hubbard metal - insulator transition Spectral function,6,3,6,3,6,3 U/D=. -4-2 2 4 U/D=1.2-4 -2 2 4 U/D=3.4-4 -2 2 4 Energy Suppression of Kondo peak when J (Hund s rule) Orbital selective metal - insulator transition function Spectral 2 1,5 1,5,3,2,1-4 -3-2 -1 1 2 3 4-4 -3-2 -1 1 2 3 4 Energy (Bethe lattice at half-filling)
Summary Local moment approach (LMA): Conservative, numericaly efficient method to treat multiorbital Anderson and Hubbard models Can be used to invesigate: Mott- Hubbard metal - insulator transition in multiorbital Hubbard model transport in multilevel quantum dots that are described by Anderson impurity model