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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XXIV
Previously we solved for the density of states in the noninteracting Anderson model and dis- cussed Kondo’s perturbative calculation of spin-flip scattering. The T matrix was connected to the conductance via a Boltzmann equation approach, whose predictions can be confirmed by a more detailed calculation using the Kubo formula. In this lecture we will outline Wilson’s solution of the Kondo problem using the numerical renormalization group and present a simple Fermi-liquid picture due to Nozieres that explains some of the results given a few assumptions.
First we do a bit more of Kondo’s calculation. We previously calculated the T matrix to order J, which gave the first term in the resistance formula.
R =
3 πmJ^2 S(S + 1) 2 e^2 ¯hF
[ 1 − 4 Jρ 0 (F ) log
( kB T D
)]
. (1)
To calculate the second term, we would need the T matrix to order J^2 , which contains several terms. We will just calculate one diverging term to give you the idea.
The important terms turn out to be the terms where the impurity spin is flipped in the inter- mediate state. An example is the term proportional to
J^2
∑
k 1 ,k′ 1 ,k 2 ,k′ 2
〈k′, ↑ |S−c† k 1 ,↑ck′ 1 ,↓( + iη − H 0 )−^1 S+c† k 2 ,↓ck′ 2 ,↑|k, ↑〉 (2)
This term will be nonzero if k′^ = k 1 , k′ 2 = k, and k′ 1 = k 2 , and gives
J^2
∑
k 2
S−S+^
1 − f (k 2 + iη − k 2
Here the 1 − f (where f is just the Fermi distribution) appears because of the requirement that the state k 2 be initially empty for the above term to be nonzero. After adding together the contributions of many terms, the troublesome parts are proportional to the real part of the function
g() =
∑
k
f (k) k − − iη
We remark in passing that the reason why magnetic scattering comes out differently from potential scattering is essentially that, if we didn’t have the noncommutativity of the SU (2) spin operators, all the divergent terms would just cancel. For details, see Hewson.
How does the function g() behave at low temperature? For the conductivity we are interested in near the Fermi level; clearly if the density of states is finite at the Fermi level, then there is a logarithmic divergence at T = 0; the effect of a nonzero temperature is to cutoff the singular- ity (giving log(D/T ), where D is the bandwidth). Careful calculation of the prefactor gives the resistivity result quoted above.
Adding the logarithmic divergence found above to the ordinary phonon and nonmagnetic scat- tering contributions to the resistance gives a reasonable description of the location of the resistance minimum of magnetic alloys. However, we still need to find a valid way of describing the system
down to zero temperaure, and also to find a way of calculating other experimental quantities like the spin susceptibility in a magnetic field and the spectral function.
For the most part we will just be describing the phenomena associated with the Kondo effect, but will also show how a Fermi-liquid picture can be used to deduce some of these phenomena from one starting point, the existence of a “Kondo resonance.” There is a somewhat intuitive way of understanding what happens at low temperature: the impurity spin binds with the conduction electron spins to form a singlet. We will give a more precise picture of this in a moment; note that the impurity spin must bind to quite a complicated superposition of the impurity electron states in order to explain the nonanalytic behavior of the Kondo temperature.
We start with a set of simplifications of the problem introduced by Wilson. Going back to the s − d model, note that for constant J, the impurity spin is only coupled directly to one particular combination of the conduction electron orbitals: we can write the coupling as
2 S · c† 0 ,σsσ,σ′ c 0 ,σ′ (5)
where the operator c 0 ,σ ∝
∑ k ck,σ^ and^ s^ is a vector of Pauli matrices. However, the localized orbital filled by c† 0 that we have introduced is not an eigenstate of the conduction electron Hamiltonian Hc.
Let us make an orthogonal basis where Hc is tridiagonal. Consider the state Hc| 0 〉: this state has in general nonzero projection onto | 0 〉, so subtract that out and define the state | 1 〉 = (γ 0 )−^1 (Hc| 0 〉− | 0 〉〈 0 |Hc| 0 〉). The number γ 0 is chosen to give a normalized state. Continuing this orthogonalization procedure gives a basis of single-particle states where each state |n〉 is connected by Hc to |n − 1 〉 and |n + 1〉 except for the state 0〉, which is connected to | 1 〉 and the impurity spin. The general form is
|n + 1〉 =
γn (Hc|n〉 − |n〉〈n|Hc|n〉 − |n − 1 〉〈n − 1 |Hc|n − 1 〉). (6)
We can look at the resulting model as a linear chain with one end coupled to the impurity spin by the coupling J, and the other end extending out to ∞. The original conduction electron states in the problem will determine the precise energies and couplings between the sites in this new linear chain model, but we can hope that the Kondo effect is a fairly universal phenomenon. Now the conduction electron Hamiltonian looks like
Hc =
∑ n,σ
nc† n,σcn,σ +
∑ n,σ
(γnc† n,σcn+1,σ + γ∗ nc† n+1,σcn,σ). (7)
where n = 〈n|Hc|n〉. The choice of site energy n = 0 and hopping γn = D/2 corresponds to a semicircular band of states, which should be good enough. However, with this choice the numerical procedure described below is not convergent, so what Wilson did is take
γn =
D(1 + Λ−^1 )
2Λn/^2
for some parameter Λ > 1. This choice of the γi corresponds to a discrete rather than continuous spectrum of states, distributed evenly over energy scales (i.e., so that each decade of energy has the same number of states). This corresponds to the idea that a logarithmic divergence means each scale or decade of energies contribute equally in the low-energy limit.
We still have a seemingly difficult many-body problem to solve: find the eigenstates of the coupled impurity spin and linear chain. Now the procedure used by Wilson is to solve the low- energy spectrum of the linear chain by iteratively diagonalizing, keeping the lowest-energy states,
