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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XXV
In the previous lecture we claimed that at low enough temperature, the s − d model (an inter- acting spin coupled to mobile conduction electrons) had behavior controlled by its singlet ground state. The temperature scale of this behavior is the famed Kondo temperature
TK ∼ De−^1 /Jρ^0. (1)
This was obtained as the temperature where the two terms in Kondo’s resistivity formula become comparable in magnitude.
We also mentioned the existence of a Kondo resonance. The Kondo effect results in the creation of another peak in the spectral function, called the Kondo resonance. This resonance is sometimes described as representing the singlet pair of the impurity spin, but this is somewhat imprecise. A detailed calculation of the spectral function using the numerical RG or other methods predicts a resonance centered on the Fermi level, with width of order TK. Although the spectral weight in this resonance can be quite small, it is seen strongly in experiments because its height is large and it moves around with the Fermi level. The Fermi-liquid theory for the strong-coupling fixed point described below is based on these assumptions about the Kondo resonance. Note that, just as with superconductivity, the important Kondo action is on quite low energy scales and involves states close to the Fermi level. We continue to follow the treatment of Hewson; the original paper of Nozieres used a slightly different approach based on phase shifts.
Going back to the Anderson model
H =
∑ σ
dc† d,σcd,σ +
∑
k,σ
c† k,σck,σ +
∑
k,σ
(Vkc† d,σck,σ + h.c.) + U nd↑nd↓, (2)
we expect that an STM measurement on top of the impurity, say, would reveal one peak at addition energy d and another peak at energy d + U. These peaks would have widths Γ which we could estimate using the noninteracting Anderson model studied before. The “nanosystem” realizations of the Kondo effect work by tunneling into this Kondo resonance. Its existence shows up as an increase of conductance through a dot when there is an odd number of electrons (and hence an unpaired spin). This leads to a pairing of peaks in the curve of linear-response conductance versus voltage: the conductance is increased with decreasing temperature between some peaks but not between others, since between some peaks there is an odd number of electrons on the dot.
Now we try to develop a Fermi liquid theory for the strong-coupling limit of the problem, based on the assumption of a Kondo resonance that is tied to the Fermi energy. We start from Landau’s form for the energy of an excited state with quasiparticle occupancies δnl,σ:
Eex =
∑
l,σ
l,σδnl,σ +
∑
l,l′,σ,σ′
f l,l
′ σ,σ′^ δnl,σδnl′,σ′^.^ (3)
Recall that this was obtained by neglecting terms of order (δn)^3. Here is a “bare” quasiparticle energy of a single quasiparticle, and f is the interaction between quasiparticles that modifies the to- tal energy when there are many excitations. The effective energy of a quasiparticle in a background
of other quasiparticles is written
˜l,σ = l,σ +
∑
l′,σ′
f l,l
′ σ,σ′^ δnl′,σ′^.^ (4)
The fundamental assumption of the existence of a Kondo resonance means that, in the basis before where only site 1 is coupled to the impurity (I think that before I called this site 0), we can write a low-energy effective Hamiltonian
Hef f =
∑
k,σ
k,σc† k,σck,σ +
∑
k,σ
( V˜kc† 1 ,σck,σ + V˜ (^) k∗ c† k,σc 1 ,σ
)
∑ σ
˜ 1 ,σc† 1 ,σc 1 ,σ + U n˜ 1 ,↑n 1 ,↓. (5)
This looks just like a new Anderson model for site 1 instead of the impurity site. However, the effective parameters V˜ , U˜ , and ˜ are related in some complicated way to the original parameters of the original problem. All we know is that, if we assume particle-hole symmetry, then ˜ = 0 (the resonance is centered exactly on the Fermi energy), and that the width of the resonance
∆ =˜ π
∑
k
| V˜k|^2 δ( − k) (6)
is of the order of the Kondo temperature.
The above Fermi liquid theory works well for the single-impurity, s = 1/2 Kondo problem, coupled to one “channel” of conduction electrons (here this means that we can write the system in a way that a single orbital couples to the impurity spin, as mentioned before in the discussion of the linear chain). This problem is fully screened in the sense that the ground state winds up as a spin singlet, and no impurity spin survives. If we took the same problem but with a spin- impurity, so that now a passing conduction electron only partly raises or lowers the spin, we wind up with an “underscreened” situation: at low energy the spin gets screened down to s = 1/2. (As a result of the spin degeneracy of the ground-state, there is a residual T = 0 entropy.)
A quite complex case is the so-called two-channel Kondo problem. This is very difficult to realize experimentally (although there are somewhat controversial claims that it has been seen). The two-channel model applies as an effective description if there are two different “types” of electrons coupled to the impurity. That is, the two-channel model looks schematically like (using continuum field operators ψ 1 and ψ 2 for the two channels)
H 2 CH = S ·
( ψ 1 (0)†σψ 1 (0)†^ + ψ 2 (0)†σψ 2 (0)†
)
. (7)
Here we have written only the interaction part; there is also the conduction electron Hamiltonian for ψ 1 and ψ 2. To see why this is nontrivial, note that it is very different from a combination like
H 1 CH = S · (aψ 1 (0) + bψ 2 (0))†^ σ (aψ 1 (0) + bψ 2 (0)) (8)
In the second case, we can introduce a new basis ψ˜ 1 = aψ 1 + bψ 2 , ψ˜ 2 = bψ 1 − aψ 2 so that the impurity is only coupled to ψ 1. So, with the cross terms, we have just a one-channel model plus another decoupled channel.
The two-channel model is unstable to even small cross terms, so that it may seem quite difficult to generate exactly. The reason for the interest in the two-channel model is that it shows exotic non- Fermi-liquid behavior, which is sometimes referred to as “overscreened.” Some very elegant work on the strong-coupling critical point of this model was carried out in the 1980s and 1990s by Affleck,
