Statistical Physics, Second Revised and Enlarged Edition

Liquid helium-3: a fermi liquid 159

specialistlow-temperature physicsbooksunder thebanner ofLandau Fermiliquid

theory.What Landau did was to show that for many purposes the energy of an

assemblyofN^3 He atomsintheliquidcanindeedbewritten as

U=

∑

j

nnjεεj (14.1)

where

∑

nnj=Nas usual. This looks like an ideal gas assumption. However, the
appropriate energiesεεjare nolonger thefree-particle energies. Insteadtheyare some
quasiparticle energies which allow for the fact that whenyou touch one^3 Heparticle
you touch them all! Anyone who has stirred his tea knows that this is bound to be the
caseinaliquid.Ifyou move a particular atomfromAtoBinaliquid,then there are
consequences for the ‘other’atoms also. The liquid verystronglywants to maintain a
uniform density, so that there is clearly a ‘backflow’contribution to the motion – you
needto move somebackgroundfluidout ofthewayatBandinto theholeatA.There
is also the fact that you cannot get hold of one atom without involving the interactions
withits neighbours. Soitisthat afirst approximationistotalkabout quasiparticles
withan effective masshigher than thebare mass.
The problem is, of course, truly a many-body problem, similar to that discussed
inthe previous section. The quasiparticle energiesε,being relatedto theother par-
ticlespresent, themselves mustdependon thedistribution ofthese otherparticles.
Furthermore, the influence of these other particles will depend on the property being
discussed. For example, above wediscussedasimpleAtoBmotion,but what about
reversingthespinofa particle or passinga soundwave throughtheliquid?Thus
it is not surprising that the liquid is not describable simply by a one-parameter cor-
rection suchasasingleeffective mass. Rather a range ofparametersis required.
Theclever part ofLandau’s workwas to show that onlyafew suchparameters
are needed in practice, when you are considering a Fermi liquid at low enough
temperatures.
In outline,the treatment worksasfollows. The essentialideais to convince oneself
(not obvious, unless you are called Landau!) that the only significant effect of the
interactionsistochange thedispersion relationfor the quasiparticles. Allthefitting
waves into boxes (Chapter 4) still works, so that the definition ofk-states andk-space
and the density of statesg(k)is all unchanged from the ideal gas. All that happens
isthat the energyεofa quasiparticleischangedfrom theidealgas. Allthathappens
is that the energyεof a quasiparticle is changed from the idealgasε=^2 k^2 / 2 M.
Also unchangedare the use ofthe number condition todetermine the Fermienergy
μandthe use oftheFDdistribution (8.2) togive thethermalequilibrium occupation
numbers of thequasiparticle states.
So,howisεnow workedout? Theideais really rather pretty. You concentrate
onlyon energychanges (and,after all,thatisallthat anyexperiment candetermine).
Landau considered the changeδUin the internal energy of the whole assembly when
achangeδδf(k)ismadeinthedistributionfunctionf(k).Inthis treatment, we use the
vectorkas a label forthek-states asintroducedin (4.2), andwe use the notationδk