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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture XXII
For the last few lectures we have concentrated on models like the quantum Heisenberg model where the only low-energy degrees of freedom were local spins. Now we consider the case of local spins interacting with mobile conduction electrons. As we will show, this is an appropriate model for magnetic impurities in a metal (here an impurity is “magnetic” if, in the metallic environment, it has a local electronic magnetic moment) and can be used to explain the puzzling “resistance minimum” seen in some alloys. For example, copper with some iron impurities shows a marked rise in the resistivity as the temperature decreases below a few tens of Kelvins, which is difficult to understand from a weak-interaction picture. A reference for the next few lectures is A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge U.P.).
This lecture starts by considering how to describe impurity scattering in our second-quantization language, then looking at the difference between nonmagnetic and magnetic impurity scattering. Let us start by considering just potential scattering, which you have probably seen before. We assume that we have a Hamiltonian describing Bloch eigenstates of energies k, plus a localized potential around one point that scatters different eigenstates into each other:
H =
∑
k,σ
c† k,σck,σ +
∑
k,k′,σ
Vk,k′^ c† k,σck′,σ. (1)
Here Vk,k′^ = 〈k|V |k′〉 is just the matrix element of the potential V. Since the above Hamiltonian is quadratic, it can be diagonalized into a new single-particle Hamiltonian with some slightly different eigenstates, which we write as H =
∑ α,σ
αc† α,σcα,σ. (2)
Here the notation α is used because these new states are no longer momentum eigenstates. We would like to start by answering the question of how the density of states is modified from its value without the impurity, ρ 0 () =
∑ k δ(^ −^ bk).^ The density of states can then be used to express physical quantities like the specific heat and spin susceptibility. We solve this using a Green’s function method even though it can also be done directly. Consider an operator G(t − t′) that Hewson calls the resolvent Green’s function: the idea of this operator G is that taking its matrix element between 〈r| and |r′〉 gives the ordinary equation. We define G(t − t′) as the solution of the equation (^) (
i¯h
∂t
− H
) G(t − t′) = δ(t − t′). (3)
To be precise we should define boundary conditions on the solution of this equation. The retarded or causal Green’s function satisfies G−(t) = 0 if t < 0, and the advanced Green’s function satisfies G+(t) = 0 if t > 0. Their Fourier transforms are
G±() =
∫ (^) ∞
−∞
G±(t)ei(±iη)/¯h^ dt. (4)
where, as usual, η is a positive infinitesimal. Then the Fourier transform of the evolution equation, with a necessary convergence factor, is G±()
( ± iη − H)G±() = I. (5)
Again, this is an operator equation, but formally we can write
G±() =
I
± iη − H
The matrix element of G±() between two exact eigenstates is then
〈α|G±()|α′〉 = δαα′ ± iη − α
This shows that G is indeed useful to get the density of states:
∓
π
Im T r G±() = ∓
π
Im
∑ α
± iη − α
∑ α
δ( − α) = ρ(). (8)
Recall that the trace is basis-independent, so we could just as well have chosen a different basis in which to evaluate the above.
We now write out a simple perturbation theory to express the full G in terms of G 0 , defined as the Green’s function of the problem without the V terms, i.e., H = H 0 + V and G 0 is the Green’s function of G 0. We can write, suppressing the argument of all the G,
( ± iη − H 0 − V )G±^ = I (9)
and therefore
G±^ =
I
± iη − H 0
± iη − H 0 − V + V ± iη − H 0 − V =
I
± iη − H 0
I
± iη − H 0
V
I
± iη − H 0 − V
= G± 0 + G± 0 V G±. (10)
A compact way to write this is in terms of the T -matrix, defined so that
G±^ = G± 0 + G± 0 T ( ± iη)G± 0. (11)
The change in the density of states because of the impurity is
∆ρ() = ρ() − ρ 0 () = −
π
Im T r (G+() − G+ 0 ()). (12)
It turns out that this can be expressed in terms of the generalized phase shift. First recall that trace of log = log of det, so, using (6)
T r G+() = T r G+()
∂(G+())−^1
log det G+(). (13)
Finally
∆ρ() = ρ() − ρ 0 () = −
π Im
log
[ det(G+()/G 0 ())
] = −
π Im
log [det(T ( ± iη))]. (14)
The generalized phase shift is defined as the complex arg of the T -matrix:
η() = arg det T (+), +^ = + iη. (15)
This means that the T -matrix, which was previously related to the phase shift, has simple matrix elements 〈k|T ()|k′〉 = V (^) k∗ G+ d,d()Vk′. (25)
We can calculate the change in the density of states from the above T matrix through the connection to phase shifts mentioned above. Another way is just to calculate the trace of G+,
T r G+() =
∑
k
G+ k,k() + G+ d,d()
∑
k
+ iη − k
log
( + iη − d −
∑
k
|Vk|^2 + iη − k
)
. (26)
We want the imaginary part of this multiplied by −π−^1 , or
δρ() =
π
∂η() ∂
with the phase shift being equal to the complex angle, which we can find from the arctangent of the ratio of real and imaginary parts of the argument of the logarithm:
η() = π 2 − tan−^1
( d + Λ() − ∆()
)
. (28)
Here the π 2 comes from the assumption of no phase shift at = −∞ and Λ() and ∆() are the real and imaginary parts of the sum,
Λ() = P
∑
k
|Vk|^2 − k
, ∆() = π
∑
k
|Vk|^2 δ( − k). (29)
These two terms can be simply interpreted: Λ represents a shift in the energy d because of the hybridization V , and ∆ is the linewidth induced by the hybridization.
On the last problem set you are asked to calculate these forms exactly for the standard simple model. Assuming that d is well within a flat conduction band of constant density of states ρ 0 and ranging over ( = −D, = D), and that V is k−independent, then
∆() = πρ 0
∑
k
|V |^2 = ∆ (30)
if || < D, and zero otherwise.
The phase shift for this case is given by
η() = π 2 − arctan ˜d − ∆
where ˜d is the solution of the self-consistent equation
˜d = d + Λ(˜d). (32)
Our expectation is that, if the conduction electrons do not modify the state of the impurity, then the earlier potential scattering model is correct. The noninteracting Anderson model describes physics when the conduction electrons can change the occupancy of the spin-up and spin-down impurity states independently. Turning on an interaction energy HI dramatically changes the physics at low temperature, and explains an experimental puzzle that dates to the 1930s or earlier.
