Intensivas en I & D)
GRUPOS CUCI CON CCINTRA MAYOR A 50 Y COMERCIO TOTAL SUPERIOR A US$ 10 MILLONES 1992-1998-
As in the non-interacting system, excitations will be quanti- fied by the deviation of the occupation from the ground state occupation n0p
δnp = np − n0p. (69)
At low temperatures δnp ∼ 1 only for p ≈ pF where the par-
ticles are sufficiently long lived that τ ≫ t. It is important to emphasize that only δnp not n0p or np, will be physically rele-
vant. This is important since it does not make much sense to talk about quasiparticle states, described by np, far from the
Fermi surface since they are not stable. For the ideal system
E − E0 =
X
p
p2
2mδnp . (70)
For the interacting system E[np] becomes much more compli-
cated. If however δnp is small (so that the system is close to
its ground state) then we may expand: E[np] = Eo +
X
p ǫpδnp + O(δn 2
where ǫp = δE/δnp. Note that ǫp is intensive (ie. it is indepen-
dent of the system volume). If δnp = δp,p′, then E ≈ E0+ ǫp′;
i.e., the energy of the quasiparticle of momentum p′ is ǫp′.
In practice we will only need ǫp near the Fermi surface where
δnp is finite. So we may approximate
ǫp ≈ µ + (p − pF) · ∇p ǫp|p
F (72)
where ∇pǫp = vp, the group velocity of the quasiparticle. The
ground state of the N + 1 particle system is obtained by adding a particle with ǫp = ǫF = µ = ∂E∂N0 (at zero temperature); which
defines the chemical potential µ. We make learn more about ǫp by employing the symmetries of our system. If we explicitly
display the spin-dependence,
ǫp,σ = ǫ−p,−σ under time-reversal (73)
ǫp,σ = ǫ−p,σ under BZ reflection (74)
So ǫp,σ = ǫ−p,σ = ǫp,−σ; i.e. in the absence of an external
magnetic field, ǫp,σ does not depend upon σ if. Furthermore,
for an isotropic system ǫp depends only upon the magnitude of
define m∗ as the constant of proportionality at the fermi surface
vpF = pF/m∗ (75)
Using m∗ it is useful to define the density of states at the fermi surface. Recall, that in the non-interacting system,
D(EF) = 1 2π2 2m ¯ h2 3/2 EF1/2 = mpF π¯h3 (76)
where p = ¯hk, and E = p2/2m. Thus, for the interacting system at the Fermi surface
Dinteracting(EF) =
m∗pF
π¯h3 , (77)
where the m∗ (generally > m, but not always) accounts for the fact that the quasiparticle may be viewed as a dressed particle, and must “drag” this dressing along with it. I.e., the effective mass to some extent accounts for the interaction between the particles.
4 Interactions between Particles: Landau Fermi Liquid
4.1 The free energy, and interparticle interactions
The thermodynamics of the system depends upon the free en- ergy F , which at zero temperature is
F − F0 = E − E0 − µ(N − N0) . (78)
Since our quasiparticles are formed by adiabatically switching on the interaction in the N + 1 particle ideal system, adding one quasiparticle to the system adds one real particle. Thus,
N − N0 = X p δnp, (79) and since E − E0 ≈ X p ǫpδnp, (80) we get F − F0 ≈ X p (ǫp − µ) δnp. (81)
As shown in Fig. 18, we will be interested in excitations of the system which distort the Fermi surface by an amount propor- tional to δ. For our theory/expansion to remain valid, we must
δ
Figure 18: We consider small distortions of the fermi surface, proportional to δ, so that 1 N P p|δnp| ≪ 1. have 1 N X p |δnp| ≪ 1 . (82)
Where δnp 6= 0, ǫp − µ will also be of order δ. Thus,
X
p (ǫp − µ) δnp ∼ O(δ
2) , (83)
so, to be consistent we must add the next term in the Taylor series expansion of the energy to the expression for the free energy. F − F0 = X p (ǫp − µ) δnp + 1 2 X p,p′ fp,p′δnpδnp′ + O(δ3) (84)
where
fp,p′ =
δE δnpδnp′
(85) The term, proportional to fp,p′, was added (to the Sommerfeld
theory) by L.D. Landau. Since each sum over p is proportional to the volume V , as is F , it must be that fp,p′ ∼ 1/V . However,
it is also clear that fp,p′ is an interaction between quasiparticles,
each of which is spread out over the whole volume V , so the probability that they will interact is ∼ rT F3 /V , thus
fp,p′ ∼ rT F3 /V 2 (86)
In general, since δnp is only of order one near the Fermi
surface, we will only care about fp,p′ on the Fermi surface (as-
suming that it is continuous and changes slowly as we cross the Fermi surface. Interested in fp,p′ ǫp=ǫp′=µ in only! (87)
Thus, fp,p′ only depends upon the angle between p and p′.
We can also reduce the spin dependence of fp,p′ to a symmet-
field, the system should be invariant under time-reversal, so fpσ,p′σ′ = f−p−σ,−p′−σ′ , (88)
and, in a system with reflection symmetry
fpσ,p′σ′ = f−pσ,−p′σ′ . (89)
Then
fpσ,p′σ′ = fp−σ,p′−σ′ . (90)
It must be then that f depends only upon the relative orienta- tions of the spins σ and σ′, so there are only two independent components fp↑,p′↑ and fp↑,p′↓. We can split these into sym-
metric and antisymmetric parts. fp,pa ′ = 1 2 fp↑,p′↑ − fp↑,p′↓ fp,ps ′ = 1 2 fp↑,p′↑ + fp↑,p′↓ . (91) fp,pa ′ may be interpreted as an exchange interaction, or
fpσ,p′σ′ = fp,ps ′ + σ · σ′fp,pa ′ (92)
where σ and σ′ are the Pauli matrices for the spins.
Our ideal system is isotropic in momentum. Thus, fp,pa ′ and
so we may expand either fp,pa ′ and fp,ps ′ fp,pα ′ = ∞ X l=0f α l Pl(cos θ) . (93)
Conventionally these f parameters are expressed in terms of reduced units. D(EF)flα = V m∗pF π2h¯3 f α l = Flα. (94)