160 Dealingwith interactions
for avolume elementink-space. Landau’s approachthen writes
δU=∫
g(k)εδδf(k)dk (14.2)an equationlooking verylikethe non-interactingU=
∑
giεiiifffi(compare (14.1))but
one which is now used todefinethe excitation (quasiparticle) energyε.
In the Fermi quasiparticle gas at low temperatures, only excitations close to the
Fermisurface areimportant, so we maywrite (tofirst order)
ε(k)=μ+(
∂ε
∂k)
F(k−kkkF)The usualgroup velocitydefinition ofthe Fermivelocity gives us quasiparticle
velocity
vF=1
(
∂ε
∂k)
F
and we may also define an effective mass from the momentum relationM∗vF=kkkF.
Itis worthstressing that these values ofvFandoftheeffective masswill bedifferent
from thefreegas values,because ofthe newdefinition ofε.In addition, we can show
that the density of states at the Fermi levelg(μ)is changed; it is identical in form to
thatfor theidealgas (8.3),but now the massMmustbe replacedbyM∗.(Thisisa
result worthprovingfrom theabove, as an exercise.)
The final illumination of the Landau theory is to separate out the effect of the
interactions on these quasiparticle energies. Again, thetheory concentrates on energy
changes.Theidea nowistolookat the energyshiftofa particular quasiparticle state
labelled byk. Its energy in equilibrium atT=0isε 0 (k),and we ask what energy shift
takes place when we changethe occupation numbers ofalltheother states (labelled
byk′). The answer is
ε(k)−ε 0 (k)=∫
Fs(k,k′)
g(μ)δδf(k′)g(k′)dk′ (14.3)The‘interactionfunction’Fs(k,k′)isdefinedinthiswaybecause (i)itisdimension-
less and (ii) it is zero in the absence of interactions. For our low-temperature Fermi
liquid,itisobvious that changesin occupations ofstatesk′whichare near to the
Fermisurface are theonlyones ofanyrelevance to measuredproperties. Since allk
values are thus essentially equal tokkkF, a further simplification may be made, namely
that theinteractionfunctionFsmaybe consideredas afunction onlyofthe angle,θ
say,betweenk′andk.Thehonestperson’s mathematicaltreatmentisthen to expand
Fs(θ)in terms of the suitably orthogonal ‘Legendre polynomials’ to give
Fs(θ)=FFF 0 s+F 1 scosθ+FFF 2 s( 3 cos^2 θ− 1 )/ 2 +···