Numerical Renormalization Group Calculations for Impurity ...

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4. SOFT-GAP ANDERSON MODEL
4.1 Introduction
The Hamiltonian of the soft-gap Anderson model is given by
∑
H = ε f f σ † f σ + Uf † ↑ f ↑f † ↓ f ↓ + ∑ ε k c † kσ c kσ + V ∑ (f σ † c kσ + c † kσ f σ). (4.1)
σ
kσ
kσ
This model describes the coupling of electronic degrees of freedom at an impurity
site (operators f (†)
†
to a fermionic bath (operators c (†)
kσ
) via a hybridization V .
The f-electrons are subject to a local Coulomb repulsion U, while the fermionic
bath consists of a non-interacting conduction band with dispersion ε k . The model
Eq. (4.1) has the same form as the single impurity Anderson model (Hewson 1993)
but for the soft-gap model we require that the hybridization function
˜∆(ω) = πV 2 ∑ k
δ(ω − ε k ) (4.2)
has a soft-gap at the Fermi level,
˜∆(ω) = ∆|ω| r , (4.3)
with an exponent r > 0. This translates into a local conduction band density of
states ρ(ω) = ρ 0 |ω| r at low energies. The power-law density of states was first
introduced for the Kondo model (Withoff and Fradkin 1990). In contrast to the
usual Kondo model, where conduction-electrons with a non-zero density of states
at the Fermi energy form a Kondo-screening state for T → 0, a gap vanishing
at the Fermi energy brings about a non-trivial zero temperature critical point at
a finite coupling constant J c and the Kondo effect occurs only for J > J c . The
existence of the critical point was derived using a generalization of the “poorman’s-scaling”
method for the density of states given in Eq. (4.3).
J R = (D ′ /D) r J ′ ≈ J + J(JCD r − r)δE/D (4.4)
In addition to the fixed points at J = 0 and ∞, there is a new infrared unstable
fixed point at
J c = r/CD r (4.5)