Download MAGNETIC SUSCEPTIBILITY IN LINEAR RESPONSE

MAGNETIC SUSCEPTIBILITY IN
LINEAR RESPONSE
notes by L. G. Molinari
revised JAN 2011
1
The magnetic moment of the electron
~ with
~ of an electron carries a magnetic moment ~µs = −g e S,
The spin S
2mc
Landé factor g = 2 resulting from Dirac’s equation. In the representation
~ = (~/2)~σ it is ~µs = −µB ~σ , with Bohr’s magneton:
S
µB =
e~
= 0.5788 × 10−8 eV/gauss
2mc
(1)
or µB = 0.9274 × 10−20 erg/gauss. The potential energy of a single spin in
~ = µB ~σ · B.
~ The Boltzmann factor e−βE (E is the
a (local) field is −~µ · B
energy of the electron) is enhanced if the magnetic moment is aligned with
the magnetic induction field (paramagnetism), i.e. the spin is antiparallel.
For an assembly of electrons, one defines the magnetization density operators:
X
X
†
~i )δ3 (~x − X
~ i ) = −µB
m
~ (~x) =
~µs (S
(2)
ψm
(~x)~σmm′ ψm′ (~x)
i
mm′
Exercise: prove that [mi (~x), mj (~x′ )] = i~δ3 (~x − ~x′ )
2
P
x).
k ǫijk mk (~
Spin paramagnetism
R
~ x, t).
We study the gran canonical Hamiltonian K ′ = K − d3 x m(~
~ x) · H(~
According to linear response theory, if the equilibrium magnetization is null,
for a weak field it is:
Z
1
d4 x′ Dijret (x, x′ )Hj (x′ )
(3)
mi (x) = −
~
1
with the response function iDijR (x.x′ ) = θ(t − t′ )h[mi (x) , mj (x′ )]i. Since the
system at equilibrium is invariant under time-translations, we may shift to
ω space:
Z
1
mi (~x, ω) = −
d3 x′ Dijret (~x, ~x′ , ω)Hj (~x′ , ω)
~
Going back to time:
Z
1
dω 3 ′ R
mi (~x, t) = −
d x Dij (~x, ~x′ , ω)Hj (~x′ , ω)e−iωt
~
2π
The retarded correlator can be evaluated via the Lehmann representation
from the τ - ordered correlator
−DijT (~xτ, ~x′ τ ′ ) = hT δmi (~xτ )δmj (~x′ τ ′ )i,
where δmi (~xτ ) = mi (~xτ ) − hmi (~x)i. Its expansion in Matsubara frequencies,
with even frequencies, yields coefficients DijT (~x, ~x′ , iωk ). The retarded correlator is obtained from it by the replacement iωk with ω + iη.
In detail:
j
′
†
′
†
i
x)ihmj (~x′ )i
−DijT (x, x′ ) = µ2B σmm
′ σnn′ hT ψm (x)ψm′ (x)ψn (x )ψn′ (x )i − hmi (~
j
′
i
= −~µ2B σmm
′ σnn′ Πmm′ ,nn′ (x, x )
(4)
where the full polarization of the system at equilibrium appears.
2.1
The polarization
For independent particles the evaluation of the polarization is simple:
Πmm′ ,nn′ (~x, ~x′ , iωk ) = −2
X
hm′~x|aiha|m~x′ ihn′~x|bihb|n~x′ i
ab
na − nb
i~ωk − (ǫa − ǫb )
For interacting electrons, the polarization is the sum of diagrams with two
electron lines joining in x, (one leaving with spin m and one arriving with
spin m′ ) and two lines joining in x′ with spin n (outgoing) and n′ (ingoing).
Since an electron line leaving from x must either return to it or join x′ , if the
~ = 0 has Green functions and interactions diagonal in spin, it
system with H
is:
Πmm′ ,nn′ (x, x′ ) = δmn′ δm′ n Π1 (x, x′ ) + δmm′ δnn′ Π2 (x, x′ )
where in Π1 the points x and x′ are joined by two lines, and Π2 has diagrams
where a line closes on x and another closes on x′ . All Π1 diagrams are in
2
Π∗ , but there are also diagrams in Π2 that belong to Π∗ (also for Π∗ one can
write a decomposition as above). Then:
j
i j
i
j
i
σmm
′ σnn′ Πmm′ ,nn′ = tr(σ σ )Π1 + trσ trσ Π2 = 2δij Π1
where we used the useful relation σ i σ j = δij I + iǫijk σk and the fact that Pauli
matrices are traceless. In this situation: DijR (~x, ~x′ , ω) = 2~µ2B δij Π1 (~x, ~x′ , ω +
iη).
2.2
Homogeneous systems
If the system at equilibrium is invariant for space translations, the response
function is local in 4D Fourier space: mi (k) = − ~1 Dijret (k)Hi (k). The coefficient is the magnetic susceptibility:
1
χij (k) = − Dijret (k)
~
and the magnetization density in space time is:
Z 3
d kdω i~k·~x−iωt
mi (x) =
e
χij (~k, ω)Hj (~k, ω)
(2π)4
(5)
(6)
Moreover, if the field is homogeneous and constant, Hj (~k) = Hj (2π)4 δ4 (k),
the resulting magnetization is homogeneous and constant: mi = χij Hj , with
j
i
~
χij = −µ2B σmn
σm
′ n′ Πmn,m′ n′ (k → 0)
2.3
(7)
Ideal electron gas
For the free electron gas, it is
1
Πmn,m′ n′ = (−1/~)(−1)δmn′ δm′ n G 0 (x, x′ )G 0 (x′ , x) = δmn′ δm′ n Π(0) (x, x′ )
2
where Π(0) is the polarization bubble diagram. In Fourier space:
Z
n~k+~q − nq~
d3 q
(0) ~
Π (k, iωn ) = −2
3
(2π) i~ωn − (ǫ~k+~q − ǫq~)
(8)
We replace iωn with ω + iη and let ω = 0 without any harm. Then take k to
0:
Z
n~k+~q − nq~
d3 q
2
(0)
2
χij = −µB δij lim Π (k, 0) = −2µB δij
lim
3
k→0
(2π) k→0 ǫ~k+~q − ǫq~
Z
Z
d3 q ∂n
∂n
= −2µ2B δij
(~q) = −2µ2B δij dǫρ(ǫ)
(9)
3
(2π) ∂ǫ
∂ǫ
3
Therefore χ = −µ2B
R
dǫρ(ǫ) ∂n
with limit behaviours: χ = µ2B ρ(ǫF ) for T = 0
∂ǫ
µ2 n
(Pauli paramagnetism) and, for large T, χ = kBBT (Curie’s law) where n =
R
dǫρ(ǫ)nǫ is the electronic density. The same result could be obtained in few
lines by other means.
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