In the second stage, we consider the spin fluctuation at low energies of *D < D*_{1}. For the purpose, we make the Kondo Hamiltonian via the Schrieffer-Wolff transformation,

where

,

and

are the spin operators in the quantum dot. The density of states in the lead is given by Equation 6 and half of the band width is now

.

*H*_{J }represents the exchange coupling between spin 1/2 in the dot and spin of conduction electrons, whereas

*H*_{K }represents the potential scattering of the conduction electrons by the quantum dot. The coupling constants are given by

By changing the bandwidth, we renormalize the coupling constants *J *and *K *so as not to change the low-energy physics within the second-order perturbation with respect to *H*_{J }and *H*_{K}. Then we obtain the scaling equations of

The energy scale *D *where the fixed point (*J *→ ∞) is reached yields the Kondo temperature.

Scaling equations (15) and (16) are analyzed in two extreme cases. In the case of

*D * *ε*_{T}, the oscillating part of the density of states

*ν*(

*ε*_{k}) is averaged out in the integration [

22]. Then the scaling equations are effectively rewritten as

In the case of

*D * *ε*_{T}, the expansion around the fixed point [

23] yields

where *ξ *= *D*/*T*_{K }- 1 and

Now we evaluate the Kondo temperature in situations (i)

*L*_{c } *L*_{K } *L*, (ii)

*L*_{c } *L * *L*_{K}, and (iii)

*L * *L*_{c } *L*_{K}, where

*L*_{K }=

*ν*_{F}*ħ*/

*T*_{K}. In situation (i),

*ε*_{T } *T*_{K }and thus

*J *and

*K *follow Equations 17 and 18 until the scaling ends at

*D * *T*_{K}. Integration of Equation 17 from

*D*_{1 }to

*T*_{K }yields

where

.

In situation (iii),

*D*_{1 } *ε*_{T}. Then the scaling equations (19) and (20) are valid in the whole scaling region (

*T*_{K }<

*D *<

*D*_{1}). From the equations, we obtain

where *f*(*ϕ*) = [1 - *f*(*k*_{F}*L *+ *π*/2, *ϕ*)]^{-1}.

In situation (ii),

*T*_{K } *ε*_{T } *D*_{1}. The coupling constants,

*J *and

*K*, are renormalized following Equations 17 and 18 when

*D *is reduced from

*D*_{1 }to

*ε*_{T }and following Equations 19 and 20 when

*D *is reduced from

*ε*_{T }to

*T*_{K}. We match the solutions of the respective equations around

*D *=

*ε*_{T }and obtain

where *γ *≈ 0.57721 is the Euler constant.

The different expressions of

*T*_{K}(

*ϕ*) in the three situations can be explained intuitively. In situation (i),

*ε*_{T } *T*_{K}. Then the oscillating part of the density of states

*ν*(

*ε*_{k}) with period

*ε*_{T }is averaged out in the scaling procedure (Figure ). As a result, the magnetic-flux dependence of

*T*_{K }disappears. In situation (iii),

*T*_{K } *ε*_{T}. Then

*ν*(

*ε*_{k}) is almost constant in the scaling (Figure ). The Kondo temperature significantly depends on the magnetic flux through the constant value of

*ν*(0) at the Fermi level.