Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XXIII
We continue our study of localized impurity models. The last lecture discussed potential scat-
tering, where the impurity contains no degrees of freedom, and the noninteracting Anderson model,
where the spin-up and spin-down degrees of freedom are independent. In this lecture we reproduce
Kondo’s original calculation of the resistance minimum for the case of magnetic impurities, where
the spin-up and spin-down degrees of freedom are strongly coupled.
We start with the full Anderson model, a simplified model for a single localized orbital (labeled
dsince it is typically a dor forbital of a transition metal) coupled to conduction electrons:
H=X
σ
dc†
d,σcd,σ +X
k,σ
c†
k,σck,σ +X
k,σ
(Vkc†
d,σck,σ +h.c.) + U nd↑nd↓.(1)
Consider the model with no coupling between impurity and conduction states (V= 0). Then the
impurity part contains four states: an empty state of energy 0, two states with a single electron
(spin-up or spin-down) of energy d, and a doubly occupied state of energy 2d+U. We want to
consider the situation with dnegative and Upositive, so that the lowest-energy states are the
single-electron states. For obvious reasons the specific choice d=−U/2 is known as the symmetric
Anderson model.
When we turn on V, we can use second-order perturbation theory to calculate an approximate
Hamiltonian for the low-energy states of the system with one electron on the impurity. If Uis large
and positive and dis large and negative compared to the bandwidth, we expect to have singly
occupied impurity states. The term “mixed-valence” is used to describe the situation where the
zero- and one-electron states are nearly degenerate. Here we will focus on the situation with single
occupancy, so the Kondo or s−dmodel defined below is valid.
This is nearly the same calculation we did in deriving the antiferromagnetic limit of the half-
filled Hubbard model previously, so I won’t write out the details. Again the result is an effective
spin model (known as the s−dmodel) whose interaction term is
Hs−d=X
k,k0
Jk,k0(S+c†
k,↓ck0,↑+S−c†
k,↑ck0,↓+Sz(c†
k,↑ck0,↑−c†
k,↓ck0,↓).(2)
with the spin coupling
Jk,k0=V∗
kVk0"1
U+d−0
k
+1
k−d#.(3)
The full Hamiltonian consists of Hs−dplus the conduction electron Hamiltonian. (To see the con-
nection of this to what we had before for the Hubbard model, recall that before in the intermediate
state one site became empty and one became doubly occupied, so the energy denominator was
exactly U.)
The next challenge is to figure out a way to calculate the conductivity, given a Hamiltonian
like the above. The most correct but also most time-consuming way is to work from the Kubo
formula for linear-response conductivity, which expresses the conductivity in terms of the current-
current correlation function. Then the current-current correlation function can be evaluated using
diagrammatic perturbation theory. Here we will use a quicker and perhaps more physical method
1
