A unified description of static and dynamic properties of Fermi liquids

13 Pages·0.27 MB·English

by  N. Dupuis

A unified description of static and dynamic properties of Fermi liquids
N. Dupuis
Department of Physics, University of Maryland, College Park, MD 20742-4111, USA
Laboratoire de Physique des Solides, Universit´e Paris-Sud, 91405 Orsay, France∗
(January 12, 1999)
InLandau’sphenomenologicalFermi-liquidtheory(FLT),mostphysicalquantitiesarederivedfrom
theknowledgeoftheenergyvariationδE[δn]correspondingtoachangeδnofthequasi-particle(QP)
distribution function n ≡ {nkσ}. We show that the internal energy E[n] (or, more precisely, the
thermodynamicpotentialΦ[n]),expressedasafunctionoftheQPdistributionn,canbeinterpreted
as an effective potential (in the sense of field theory), which is obtained from the free energy by a
Legendre transformation. This allows to obtain explicitly δΦ (or δE) starting from a microscopic
Hamiltonian and to relate the Landau f function to the forward-scattering two-particle vertex
9
9 without considering the collective modes as in the standard diagrammatic derivation of FLT. Out-
9 of-equilibriumpropertiesareobtainedbyextendingthedefinitionoftheeffectivepotentialtospace-
1 and time-dependent configurations. Φ[n] is then a functional of the Wigner distribution function
n n≡{nkσ(r,t)}. Itcontainsinformation aboutboththestaticanddynamicpropertiesof theFermi
a liquid. In particular, it yields the quantum Boltzmann equation satisfied by nkσ(r,t). Finally, we
J showhowδΦ[δn]canbederived(inthestaticcase)usingafinite-temperaturerenormalization-group
approach. In agreement with previous results based on this technique, we find that the Landau f
2
1 function is defined by the fixed-point value of the Ω-limit of the forward-scattering two-particle
vertex.

1
v PACS Numbers: 05.30.Fk, 71.10.Ay, 71.10.Ca
0
I. INTRODUCTION equilibrium QP distribution function is determined by
0
1 minimizing the thermodynamic potential. The excita-
1 Landau’s original approach to Fermi-liquid theory tion energies are defined as functional derivatives of the
0 (FLT) is phenomenological.1 The main assumption is energy with respect to the distribution function. They
9 the existence of a one-to-one correspondence between are usually referred to as statistical QP energies to dis-
9
the low-energy elementary excitations (quasi-particles tinguish them from the QP energies obtained from the
/
t (QP’s)) of the Fermi liquid and the elementary exci- poles of the single-particle Green’s function.5 Although
a
m tations (‘particles’ and ‘holes’) of the non-interacting this formulation provides a natural bridge between mi-
fermiongas. Thermodynamicquantitiesarederivedfrom croscopicmodelsandphenomenologicalFLT,fewexplicit
-
d theknowledgeoftheenergyvariationδE[δn]correspond- calculations have been done in this framework. For in-
n ing to a change δn of the QP distribution function n = stance, the Landau f function (which can be obtained
o {n }. [k and σ denote the QP momentum and spin.] as the second functional derivative of the energy with
kσ
c δE[δn] is parameterized by the Landau f function (or, respect to the QP distribution function) has not been
:
v equivalently,theLandauparameters)whichdescribesthe related to the forward-scattering two-particle vertex as
i interaction between two QP’s. Non-equilibrium proper- was done within the standard FLT.2,3 Besides, it is not
X
ties are obtained by generalizing the energy functional clearhowthecollectivemodesofthe Fermiliquidcanbe
ar δE[δn]tospace-andtime-dependentconfigurations,and studied within this approach.
by simultaneously considering the Boltzmann transport Inthis paperweproposeaunifieddescriptionofstatic
equation satisfied by the QP distribution. and dynamic properties of Fermi liquids. The main idea
Much effort has been dedicated to justify Landau’s is to interpret the thermodynamic potential Φ[n], ex-
phenomenological FLT from a microscopic Hamiltonian. pressed in terms of the QP distribution function n, as
ThestandardderivationconsistsinshowingthattheQP aneffective potential (in the sense of field theory),which
dynamicsobtainedfromthemicroscopictwo-particlever- is obtained from the free energy by a Legendre trans-
tex agrees with the conclusion of the phenomenological formation. A similar definition of the thermodynamic
FLT.2,3 Although this demonstration provides a micro- potentialΦ[n]canbe foundinthe statisticalFLT andin
scopicdefinitionoftheLandauf function,itdoesnotaim particularintheworkofBalianandDeDominicis.4 This
at calculating explicitly the energy functional δE[δn]. allowstoobtainexplicitlyδΦ(orδE)startingfromami-
The so-called statistical FLT provides an alternative croscopic Hamiltonian and to relate the Landau f func-
approach which is in spirit much closer to the phe- tion to the forward-scattering two-particle vertex with-
nomenological theory of Landau. The main idea is to out considering the collective modes as in the standard
expressthethermodynamicpotentialΦ[n]obtainedfrom diagrammaticderivationofFLT.Dynamicpropertiesare
a microscopic Hamiltonian as a function of QP occupa- obtained by extending the definition of the effective po-
tionnumbers(see Refs.4–6andreferencestherein). The tentialtospace-andtime-dependentconfigurations. Φ[n]
1
is then a functional of the Wigner distribution function into account. We use a grand canonical formalism at fi-
n ≡ {n (r,t)}. We show how we can extract from Φ nite temperature T, the limit T → 0 being taken at the
kσ
both the response functions of the Fermi liquid and the end of the calculations. We take h¯ =k =1 throughout
B
quantum Boltzmann equation satisfied by n (r,t). the paper.
kσ
Outline of the paper II. PHENOMENOLOGICAL FLT
InsectionII,webrieflyreviewsomebasicresultsofthe In this section, we summarize the basic results of the
phenomenologicalFLTthatareusefulinsubsequentsec- phenomenologicalFLT which are useful for our purpose.
tions. The microscopic definition of the thermodynamic
potential Φ[n] is given in section III. We first consider
thefreeenergyinthepresenceofanexternalsourcefield A. Landau’s functional δΦ[δn]
h that couples to the QP number operator. The QP
kσ
distribution function n is then obtained by taking the In the ground state of a Fermi liquid, the QP distri-
functional derivative of the free energy with respect to bution function corresponds to a filled Fermi sea: n(0) =
the source field. By performing a Legendre transforma- kσ
Θ(k −k), where k is the Fermi momentum and Θ(x)
F F
tion, we obtain a functional Φ[n] of the QP distribution
the step function. In the phenomenological FLT, one
n. In field theory, Φ[n] is known as an effective poten- postulates that a change δn = n −n(0) of the QP
tial. The energy variation δE[δn], or equivalently the distribution induces the enerkgσy variaktσion1–k3σ
Landauf function,isobtainedfromthevariationδΦ[δn]
of the thermodynamic potential Φ[n] around its equilib-
δE[δn]= ǫ δn
riumvalue. As afirstexample,we calculateΦ[n]exactly k kσ
for non-interacting fermions. In the case of interacting Xk,σ
1
fermions, we use general properties of Legendre trans- + fσσ′(k,k′)δnkσδnk′σ′, (2.1)
forms to express δΦ[δn] as a function of the (linear) re- 2ν
k,k′,σ,σ′
sponse function χσσ′(k,k′) to the externalfield hkσ. We X
giveamicroscopicdefinitionofthe Landauf functionin neglectingtermsoforderO(δn3). ǫ istheenergyofaQP
k
terms of the susceptibility χ. Using the usual assump- in the absence of other excited QP’s (for simplicity, we
tions of FLT, we calculate χ and recover the standard consideronlycaseswhereǫ doesnotdependonspin). In
k
microscopic definition of f. thevicinityoftheFermisurface,itcanbewrittenasǫ ≃
k
In section IV, we extend the definition of the effec- v∗(k−k )+µ, where v∗ is the QP Fermi velocity and
F F F
tive potential Φ[n] to space- and time-dependent config- µ the chemical potential. The term of order δn2 in (2.1)
urationsbyconsideringthe Wignerdistributionfunction comes from the interaction between quasi-particles. For
n≡{nkσ(r,t)}. Restrictingourselvestosmalldeviations states near the Fermi surface, k ≃ kF and k′ ≃ kF, the
δnkσ(r,t) from equilibrium, we are able to compute the function fσσ′(k,k′)=fσσ′(θ) depends only on the angle
correspondingvariation δΦ[δn] of the effective potential. θ between k and k′. The Landau parameters Fs and
l
In the static and uniform limit (nkσ(r,t)=nkσ), δΦ re- Fa are defined by expanding f on the basis of Legendre
l
duces to the Landau’s functional studied in section III. polynomials:
Inout-of-equilibriumcases,byrequiringδΦtobestation-
ary with respectto variationsof the Wigner distribution ∞
function δnkσ(r,t), we obtain the quantum Boltzmann 2N(0)fσσ′(θ)= (Fls+σσ′Fla)Pl(cosθ), (2.2)
equation satisfied by δn (r,t). We also show how the l=0
kσ X
charge and spin response functions are related to the ef-
whereN(0)=k2/2π2v∗ is the density ofstates per spin
fective potential. F F
at the Fermi level.
In section V, we use a finite-temperature renormaliza-
Inthegrand-canonicalensembleatfinite temperature,
tion-group(RG)approachtoreproducetheresultsofsec-
the physical quantity of interest is the thermodynamic
tion III. This approach provides an alternative method
potential Φ[n] = E[n] − µN[n] − β−1S[n]. N[n] =
for computing the Landau’s functional δΦ[δn] and con-
firms previous results based on this technique.7–10 We k,σnkσ is the total QP number, S[n] the entropy, and
β = 1/T the inverse temperature. Because of the cor-
find that the f function is givenby the fixed-point value
P
respondence between excitations of the non-interacting
of the Ω-limit of the forward-scatteringtwo-particle ver-
tex (ΓΩ∗). fermiongasandQPexcitations,theentropyhasthesame
expression as in a perfect Fermi gas:1–3
Unless otherwise specified, we consider a three-
dimensional spin-1 fermion gas (of volume ν) with a
2 S[n]=− [n lnn +(1−n )ln(1−n )]. (2.3)
spherical Fermi surface, and assume isotropy in spin kσ kσ kσ kσ
space. Only short-range repulsive interactions are taken Xk,σ
2
At equilibrium, the QP distribution function n¯ is ob- C. Dynamic properties
tained from the condition δΦ[n]/δn | =0:
kσ n¯
TheLandau’sfunctionalδΦ[δn]allowstocompute the
n¯ =n (ǫ˜ −µ), (2.4)
kσ F k thermodynamic properties of the Fermi liquid, but does
notcontainanyinformationaboutthe QPdynamics. To
wheren (x)=(eβx+1)−1 istheFermi-Diracfactor,and
F obtain the latter, one has to extend the definition of δE
to out-of-equilibrium configurations:2,3
δE[n]
ǫ˜ =
k
δnkσ (cid:12)(cid:12)n¯ δE[δncl]= d3rǫkδncklσ(r,t)
1(cid:12)
=ǫk+ ν(cid:12)(cid:12) fσσ′(k,k′)[n¯k′σ′ −n(k0′)σ′] (2.5) Xk,σ1Z
kX′,σ′ + d3rfσσ′(k,k′)
2ν
is the QP energy corresponding to the equilibrium dis- k,kX′,σ,σ′Z
tribution function n¯. Thus, if we expand Φ[n¯ +δn] = ×δncklσ(r,t)δnckl′σ′(r,t). (2.9)
Φ[n¯]+δΦ[δn] around its equilibrium value, we obtain to
lowest order in δn ncl (r,t)=n(0)+δncl (r,t) givesthe probability offind-
kσ kσ kσ
ing a QP with momentum k and spin σ at point r in
δΦ[δn]= 1 − δσ,σ′δk,k′ space and time t. We use the notation ncl to emphasize
2 n′ (ǫ˜ −µ)
k,kX′,σ,σ′h F k that this approach is semiclassical, since it is assumed
that one can simultaneously specify the momentum and
1
+ fσσ′(k,k′) δnkσδnk′σ′, (2.6) position of the QP. For low-energy excitations, the in-
ν
teraction between QP’s (last term of Eq. (2.9)) can be
i
where n′ (x) = −β/4cosh2(βx/2). There is no linear assumed to be local in space. The time-dependent QP
F
term in δn since Φ[n] is stationary at equilibrium. energy is defined by
δE[ncl]
ǫ (r,t)=
B. Deformation of the Fermi surface k δncklσ(r,t)(cid:12)(cid:12)ncl
1 (cid:12)
Because of the thermal factor 1/n′F(ǫk −µ) in (2.6), =ǫk+ ν (cid:12)(cid:12) fσσ′(k,k′)δnckl′σ′(r,t). (2.10)
small variations of the thermodynamic potential corre- k′,σ′
X
spondtoQPexcitationslyinginthe thermalbroadening
of the Fermi surface (|ǫ˜k−µ|<∼T). When T →0, these Eq. (2.9) is supplemented with the Boltzmann trans-
excitationshavevanishingenergiesandcanbe viewedas port equation2,3
resultingfromadisplacementu (kˆ)oftheFermisurface.
Here kˆ =k/k is a unit vector inσ the direction of k. It is ∂ncklσ(r,t) −∇ ncl (r,t)·∇ ǫ (r,t)
∂t k kσ r k
sometimes convenient to express δΦ directly in terms of
uσ(kˆ). Writing11 +∇rncklσ(r,t)·∇kǫk(r,t)=0, (2.11)
δn =v∗u (kˆ)n′ (ǫ˜ −µ), (2.7) wherewehaveusedthequasi-classicalequationsdr/dt=
kσ F σ F k ∇ ǫ (r,t) and dk/dt = −∇ ǫ (r,t). To first order in
k k r k
we obtain δncl, Eq. (2.11) reduces to
δΦ[u]= νvF∗22N(0) δσ,σ′ d4Ωπkˆu2σ(kˆ) ∂δnck∂lσt(r,t) +vF∗kˆ·∇rδncklσ(r,t)
Xσ,σ′n Z 1
+N(0) d4Ωπkˆ d4Ωπkˆ′fσσ′(kˆ,kˆ′)uσ(kˆ)uσ′(kˆ′) +δ(ǫk−µ)ν k′,σ′fσσ′(k,k′)vF∗kˆ·∇rδnckl′σ′(r,t)=0.
X
Z (2o.8) (2.12)
in the limit T → 0. We have used ǫ˜ → ǫ and The solution of this equation can be written as
k k
n′ (x) → −δ(x) when T → 0. Ω denotes the solid an-
glFe in the direction of kˆ. Eq. (2.kˆ8) was first derived by δncklσ(r,t)=vF∗uσ(kˆ,q,Ω)δ(ǫk−µ)ei(q·r−Ωt) (2.13)
PomeranchukconsideringthechangeinenergyδEresult-
ing from a Fermisurface displacementu (kˆ) at T =0.12 whereuσ(kˆ,q,Ω)isadynamicdisplacementoftheFermi
σ surface. The solutions of Eq. (2.12) correspond to zero-
By requiring Φ[u] to be minimum for u = 0, we easily
sound (u =u ) and spin-wave (u =−u ) modes.
deducefrom(2.8)the stabilityconditionsfora3DFermi ↑ ↓ ↑ ↓
liquid: Fs >−2l−1 and Fa >−2l−1.12
l l
3
III. LANDAU’S FUNCTIONAL Φ AS AN B. Non-interacting fermions
EFFECTIVE POTENTIAL
As a first example, we consider non-interacting
In statistical mechanics and in the theory of phase fermions. Using nˆ (τ)=ψ∗(k,τ)ψ (k,τ), we obtain:
kσ σ σ
transitions, it is natural to consider the Legendre trans-
form of the free energy. In magnetism for instance, the Z[h]= Dψ∗Dψe k,σ,ωψσ∗(k,ω)(iω−ǫk−hkσ+µ)ψσ(k,ω)
Gibbs thermodynamic potential, whichis a functionalof
Z P
the magnetization M(r), is obtained from the free en- = 1+e−β(ǫk+hkσ−µ) , (3.6)
ergyby performinga Legendretransformation.13 Infield
theory, such Legendre transforms are known as effective Yk,σ(cid:16) (cid:17)
potentials.14 In this section we show that the Landau’s where ǫ is the energy of a fermion with momentum
k
functional Φ[n] can be naturally interpreted as an effec- k. We have introduced the Fourier transformed field
tive potential. ψ(∗)(k,ω) where ω = πT(2m + 1) (m integer) is a
σ
fermionic Matsubara frequency. Eq. (3.3) yields
A. Effective potential Φ n =n (ǫ +h −µ). (3.7)
kσ F k kσ
Inverting (3.7) and using (3.6), we eventually obtain
InordertodefinetheeffectivepotentialΦ,wefirstcal-
culatethefreeenergy−β−1lnZ[h]inpresenceofa(real)
1
externalsource field hkσ that couples to the QP occupa- Φ[n]= (ǫk−µ)nkσ+ β [nkσlnnkσ
tionnumberoperatornˆkσ. ThenweobtaintheLandau’s Xk,σ Xk,σ
functional Φ[n] by performing a Legendre transforma- +(1−n )ln(1−n )]. (3.8)
kσ kσ
tion.
We write the partition function in the Matsubara for- Eq. (3.8) is the expected result for non-interacting
malismasafunctionalintegraloverGrassmannvariables fermions.
ψ(∗)(k,τ) (τ is an imaginary time):
σ
Z[h]= Dψ∗Dψe−S[ψ∗,ψ]−Sh[ψ∗,ψ], (3.1) C. Interacting fermions
Z
where S[ψ∗,ψ] is the action when h = 0. The source For interacting fermions, it is not possible to calculate
kσ
exactly the thermodynamic potential Φ[n]. However, we
field contributes to the action a term
do not require the whole knowledge of Φ[n], but only
β its variationδΦ[δn]whenthe QPdistributionfunctionn
S [ψ∗,ψ]= h dτnˆ (τ). (3.2)
h kσ kσ variesfromitsequilibriumvaluen¯ =n| byanamount
Xk,σ Z0 δn. [For T → 0, n¯kσ = Θ(kF −k) cohr=r0esponds to the
Note that we do not specify at this point the expres- ground-statedistributionfunction.] Thisturns outto be
sion of nˆ (τ) as a function of the ψ field. Only for a much easier task than obtaining Φ[n].
kσ
non-interacting fermions do we have the simple relation Expanding Φ[n¯+δn] to second order in δn, we obtain
nˆ (τ) = ψ∗(k,τ)ψ (k,τ). The QP occupation number
kσ σ σ
n =hnˆ iisobtainedfromthefunctionalderivativeof 1 δ(2)Φ[n]
thkeσfree eknσergy: δΦ[δn]= 2k,kX′,σ,σ′ δnkσδnk′σ′(cid:12)(cid:12)n¯δnkσδnk′σ′. (3.9)
1δlnZ[h] (cid:12)
(cid:12)
nkσ =− . (3.3) ThereisnolineartermsinceΦ[n]is(cid:12)stationaryatequilib-
β δh
kσ rium[Eq.(3.5)]. Toproceedfurther,weusethefollowing
Now we introduce the Legendre transform15 relationbetweenthefunctionalderivativesofthefreeen-
ergy −β−1lnZ[h] and those of the Legendre transform
Φ[n]=−1 lnZ[h]− h n , (3.4) Φ[n]:14
kσ kσ
β
k,σ
X β−1δ(2)lnZ[h] δ(2)Φ[n]
=δ δ .
wofhsetraethekfσo[rnt]hiesotbhtearminoeddybnyaimnvicerptiontgen(t3i.a3l).ΦT[nh]ereeaqdusation kX3,σ3 δhk1σ1δhk3σ3 (cid:12)(cid:12)h=0δnk3σ3δnk2σ2(cid:12)(cid:12)n¯ σ1,σ2 k1,k2
(cid:12) (cid:12)
(cid:12) (cid:12) (3.10)
δΦ[n] (cid:12) (cid:12)
=−h . (3.5)
kσ
δnkσ Introducing the matrix
At equilibrium, i.e., in the absence of source field, Φ[n]
β−1δ(2)lnZ[h]
is stationary with respect to small variations of the QP χσσ′(k,k′)= , (3.11)
distribution function. δhkσδhk′σ′ (cid:12)
(cid:12)h=0
(cid:12)
(cid:12)
(cid:12)
4
we thus obtain Ω/q → 0 and q/Ω → 0, respectively. Since a static and
uniformexternalfieldcannotcreatequasi-particle-quasi-
1
δΦ[δn]= 2k,k′,σ,σ′χ−σσ1′(k,k′)δnkσδnk′σ′. (3.12) hQoPledeisxtcriitbautitoionns f(uanncdtitohne)reinfotrheemΩo-dliimfyitt,h16efgriosuonbdt-asitnaetde
X
from the q-limit of the function χ. It turns out to be
Notethatχσσ′(k,k′)isnothingbutthe(linear)response moreconvenienttokeepafiniteq˜atintermediatestages,
function to the external source field hkσ. Comparing the limit q˜→0 (with Ω/q →0) being taken only at the
(2.6)and(3.12)(andassumingthatthesystemisaFermi end of the calculations. Besides, the function χ (with q˜
liquid), we obtaina microscopic definition of the Landau finite) will also have to be considered in the analysis of
f function: the dynamic properties of Fermi liquids (section IV).
χ˜ satisfies the Dyson equation
ν1fσσ′(k,k′)= nδ′Fσ(,σǫk′δ−k,kµ′) +χ−σσ1′(k,k′). (3.13) χ˜σσ′(k˜,k˜′;q˜)=χ˜(σ0σ)′(k˜,k˜′;q˜)
1
Fǫrosmincneowweoanr,ewueltiimgnaotreelytihnetedrieffseterednicnetbheetwlimeeint Tǫ˜k→an0d. −βν χ˜(σ0σ)1(k˜,k˜1;q˜)
k k˜1,k˜X2,σ1,σ2
Thus, the calculation of the Landau f function reduces
tothecalculationofthesusceptibilityχ. Inthenextsec- ×Γ¯iσr1rσ2(k˜1,k˜2;q˜)χ˜σ2σ′(k˜2,k˜′;q˜), (3.16)
tion,weshowhowχcanbecalculatedusingthestandard
where
assumptions of FLT.
Γ¯iσrσr′(k˜,k˜′;q˜)≡Γ¯iσrσr′,σ′σ(k˜+,k˜−′ ,k˜+′ ,k˜−) (3.17)
D. Calculation of the Landau f function
is the irreducible QP vertex in the Landau channel.17
The non-interacting part
In this section, we first express the Landau f function
ibnarteerfmersmoiofnthperoQpPertpireosp.erties, and then relate it to the χ˜(σ0σ)′(k˜,k˜′;q˜)=−δσ,σ′δk˜,k˜′G¯(k˜+)G¯(k˜−) (3.18)
is easily expressed in terms of the QP propagators
1. f in terms of QP properties G¯(k˜ )=−hψ¯ (k˜ )ψ¯∗(k˜ )i
± σ ± σ ±
Ω q −1
= iω±i −ǫ ∓v · +µ , (3.19)
Since the external field hkσ couples to the QP’s, we 2 k k 2
have to distinguish between the QP field (that we de- h i
note by ψ¯(∗)) and the field ψ(∗) corresponding to the where ǫk is the QP energy and vk = ∇kǫk the QP
bare fermions. With these notations, we have nˆkσ(τ) = group velocity. Approximating ǫk by vF∗(k − kF)+ µ,
ψ¯σ∗(k,τ)ψ¯σ(k,τ), wheretheoperatornˆkσ(τ)hasbeenin- we have vk = vF∗kˆ (with kˆ = k/k). The quantity χ˜(0)
troduced in (3.2). From (3.2) and (3.11), we obtain becomes singular in the forward-scattering limit q˜ → 0
since the poles of the two QP propagators coalesce. In
χσσ′(k,k′;q˜)= β1 χ˜σσ′(k˜,k˜′;q˜), (3.14) iFnLTth,isonliemaists.2u,m3 eTshtihsaitmχ˜p(l0ie)sisththaetothnelyisrirnegduulcairblqeuvaenrttietxy
ωX,ω′ Γ¯irr,whichdoes notcontainχ˜(0),is anon-singularquan-
where χ˜ is the two-particle Green’s function tity that has a well-defined limit when q˜→0. We there-
fore set q˜=0 in Γ¯irr. Furthermore, since the singularity
χ˜σσ′(k˜,k˜′;q˜)=hψ¯σ∗(k˜−)ψ¯σ(k˜+)ψ¯σ∗′(k˜+′ )ψ¯σ′(k˜−′ )ic ofχ˜(0)isduetoQPstatesinthevicinityoftheFermisur-
≡hψ¯σ∗(k˜−)ψ¯σ(k˜+)ψ¯σ∗′(k˜+′ )ψ¯σ′(k˜−′ )i fsaectek, w=ekc′an=igknoraendthωe k=aωnd′ =ω0d)e,pwenhdicehncteheonf Γ¯bierrco(mi.ee.s,
F
−δq˜,0hψ¯σ∗(k˜)ψ¯σ(k˜)ihψ¯σ∗′(k˜′)ψ¯σ′(k˜′)i. a function of kˆ and kˆ′. [The variables |k−kF|,|k′−kF|
(3.15) and ω,ω′ are irrelevantin the RG sense (see sectionV).]
This allows to perform the frequency sums in (3.16) and
Theaveragevaluesareevaluatedintheabsenceofsource obtain a Dyson equation for χ:
field (h=0), and the meaning of h···i is defined by the
c
second line of (3.15). We use the notation k˜ = (k,iω), χσσ′(k,k′;q˜)=χ(σ0σ)′(k,k′;q˜)
q˜=(q,iΩ),andk˜± =k˜±q˜/2. ω andΩ denotefermionic 1
− χ(0) (k,k ;q˜)
and bosonic Matsubara frequencies, respectively. The ν σσ1 1
Landau f function is related to the limit q˜ → 0 of k1,kX2,σ1,σ2
χσσ′(k,k′;q˜)[Eq.(3.13)]. InaFermiliquid,theforward- ×Γ¯iσr1rσ2(kˆ1,kˆ2)χσ2σ′(k2,k′;q˜). (3.20)
scatteringlimitq˜→0isill-definedandoneshoulddistin-
guish between the q- and Ω-limits, which correspond to This can be rewritten as
5
χ−σσ1′(k,k′;q˜)=χ(σ0σ)′−1(k,k′;q˜)+ ν1Γ¯iσrσr′(kˆ,kˆ′). (3.21) G(k˜)=zG¯(k˜)+Ginc(k˜). (3.28)
Consider now the two-particle Green’s function
InEq.(3.21),χ−1(q˜),χ(0)−1(q˜)andΓ¯irr shouldbeunder-
stood as matrices in spin (σ) and momentum (k) space. GIσIσ′(k˜,k˜′;q˜)=hψσ∗(k˜−)ψσ(k˜+)ψσ∗′(k˜+′ )ψσ′(k˜−′ )ic.
Comparing (3.13) and (3.21), and noting that
(3.29)
v ·q
χ(σ0σ)′(k,k′;q˜) = δσ,σ′δk,k′n′F(ǫk−µ)iΩ−kv ·q GII can be written as
k
→−δσ,σ′δk,k′n′F(ǫk−µ) (3.22) GIσIσ′(k˜,k˜′;q˜)=GIσIσ(0′)(k˜,k˜′;q˜)
1
in the q-limit, we conclude that − GII(0)(k˜,k˜ ;q˜)
βν σσ1 1
fσσ′(k,k′)=Γ¯iσrσr′(kˆ,kˆ′). (3.23) k˜1,k˜X2,σ1,σ2
×Γ (k˜ ,k˜ ;q˜)GII(0)(k˜ ,k˜′;q˜), (3.30)
WecanproceedonestepfurtherbyrelatingΓ¯irr tothe σ1σ2 1 2 σ2σ′ 2
total QP vertex Γ¯, using the equation where
Γ¯σσ′(k˜,k˜′;q˜)=Γ¯iσrσr′(kˆ,kˆ′)− β1ν Γ¯iσrσr1(kˆ,kˆ1) Γσσ′(k˜,k˜′;q˜)≡Γσσ′,σ′,σ(k˜+,k˜−′ ,k˜+′ ,k˜−) (3.31)
k˜1,k˜X2,σ1,σ2 is the total vertex for the bare fermions, and
×χ˜(σ01)σ2(k˜1,k˜2;q˜)Γ¯σ2σ′(k˜2,k˜′;q˜). (3.24) GIσIσ(0′)(k˜,k˜′;q˜)=−δσ,σ′δk˜,k˜′G(k˜+)G(k˜−)
Since Γ¯, as Γ¯irr, is independent of frequencies, we can
perform the sum over ω and ω in (3.24), which yields =−δσ,σ′δk˜,k˜′[z2G¯(k˜+)G¯(k˜−)+ϕ(k˜)].
1 2
(3.32)
1
Γ¯σσ′(kˆ,kˆ′;q˜)=Γ¯iσrσr′(kˆ,kˆ′)− ν Γ¯iσrσr1(kˆ,kˆ1) We have separatedthe coherentpart of the particle-hole
k1,kX2,σ1,σ2 pair propagation from the incoherent part ϕ(k˜). Note
×χ(σ01)σ2(k1,k2;q˜)Γ¯σ2σ′(kˆ2,kˆ′;q˜). (3.25) thatϕ(k˜)is non-singularin the limit q˜→0and is there-
fore evaluated at q˜= 0. Inserting (3.32) into (3.30) and
χ(0) vanishing in the Ω-limit [Eq. (3.22)], we obtain retainingonlythecoherentpartoftheparticle-holeprop-
agation, we obtain an equation for the Green’s function
Γ¯iσrσr′(kˆ,kˆ′)= lim Γ¯σσ′(kˆ,kˆ′;q˜) χ˜ introduced in section IIID2:
Ω→0 q=0
≡Γ¯Ωσσ′h(kˆ,kˆ′). (cid:12)(cid:12)(cid:12) i (3.26) χ˜σσ′(k˜,k˜′;q˜)=χ˜(σ0σ)′(k˜,k˜′;q˜)
1
Eqs. (3.23,3.26) relate the Landau f function to the Ω- − χ˜(0) (k˜,k˜ ;q˜)
βν σσ1 1
limit of the forward-scatteringQP vertex.
k˜1,k˜X2,σ1,σ2
×z2Γ (k˜ ,k˜ ;q˜)χ˜(0) (k˜ ,k˜′;q˜). (3.33)
σ1σ2 1 2 σ2σ′ 2
2. f in terms of bare fermion properties
We then deduce from (3.16) and (3.33)
The last step of our derivation is to relate Γ¯irr to the Γ¯σσ′(k˜,k˜′;q˜)=z2Γσσ′(k˜,k˜′;q˜). (3.34)
vertex of the bare fermions. Consider first the single-
particle propagator in a Fermi liquid. It can be written Eq. (3.34) relates the QP vertex Γ¯ to the bare fermion
as2,3 vertexΓ. From(3.23,3.26,3.34)weobtainthewell-known
expression of the Landau f function in a Fermi liquid:
G(k˜)=−hψ (k˜)ψ∗(k˜)i
σ σ
= z +G (k˜), (3.27) fσσ′(k,k′)=z2ΓΩσσ′(kˆ,kˆ′). (3.35)
inc
iω−ǫ +µ
k
whereG (k˜)istheincoherentpart,andztheQPrenor-
inc
malizationfactor. The QPfield ψ¯(∗) canbe obtained(at IV. DYNAMIC PROPERTIES
least formally) by filtering out the incoherent part and
rescaling the field in order to ensure that the total spec- The Landau’sfunctionalΦ[n]does notcontainanyin-
tral weight is equal to one (see Nozi`eres’ book3 for a formationaboutthedynamicpropertiesoftheFermiliq-
detailed discussion). [As discussed in section V, the RG uid. In this section, we extend the definition of the ef-
approach provides an explicit realization of this proce- fective potential Φ to space- and time-dependent config-
dure.] G and G¯ are then related by urationsby consideringthe Wigner distributionfunction
6
(WDF) n ≡ {n (r,t)}. This allows to obtain the QP Evenfor non-interacting fermions,Φ[n] cannot be cal-
kσ
dynamics and the collective modes of the Fermi liquid culated exactly. We shall therefore consider only small
without introducing the semiclassical distribution func- fluctuations δn (q˜) around the equilibrium state:
kσ
tion ncl (r,t) and the corresponding Boltzmann trans-
kσ
n (q˜)=δ n¯ +δn (q˜). (4.7)
port equation as in the phenomenological FLT (see sec- kσ q˜,0 kσ kσ
tion IIC).
Eq. (3.10) can be easily generalized into
δ(2)Φ[n]
A. Generalized effective potential Φ δnkσ(−q˜)δnk′σ′(q˜′)(cid:12) =δq˜,q˜′χ−σσ1′(k,k′;q˜), (4.8)
(cid:12)n¯
(cid:12)
Thebestquantumanalogtothesemiclassicaldistribu- where (cid:12)
tion function ncl (r,t) is the WDF.18 The latter is not a (cid:12)
kσ
truedistributionfunctionsinceitis notpositivedefinite. β−1δ(2)lnZ[h]
However,asfarasitsmomentsareconcerned,itbehaves χσσ′(k,k′;q˜)= (4.9)
δhkσ(−q˜)δhk′σ′(q˜)(cid:12)
similarlytoadistributionfunction.18,19 Forourpurpose, (cid:12)h=0
(cid:12)
we define the WDF in Matsubara time as is the susceptibility introduced in(cid:12) section III
(cid:12)
n (r,τ)= d3r′e−ik·r′ [Eqs. (3.14,3.15)]. The Kronecker symbol δq˜,q˜′ in (4.8)
kσ results from translationalinvariance. To second order in
Z δn, we thus have
r′ r′
× ψ¯∗ r− ,τ ψ¯ r+ ,τ , (4.1)
σ 2 σ 2 1
and its Fourier tDrans(cid:16)form as (cid:17) (cid:16) (cid:17)E δΦ[δn]= 2k,k′,σ,σ′,q˜χ−σσ1′(k,k′;q˜)δnkσ(−q˜)δnk′σ′(q˜).
X
1 β (4.10)
n (q˜)= dτ d3re−iq·r+iΩτn (r,τ)
kσ kσ
βν
Z0 Z Using Eqs. (3.21,3.22,3.23), we can write the effective
1
= hψ¯∗(k˜ )ψ¯ (k˜ )i. (4.2) potential in terms of the Landau f function:
β σ − σ +
ω
X 1 δσ,σ′δk,k′ iΩ−vk·q
Note that these definitions involve the QP field ψ¯(∗). δΦ[δn]= 2 n′ (ǫ −µ) v ·q
Wearenowinapositiontodefineaneffectivepotential k,kX′,σ,σ′,q˜n F k k
1
Φ[n], which is a functional of the WDF n ≡ {nkσ(q˜)}. + fσσ′(k,k′) δnkσ(−q˜)δnk′σ′(q˜). (4.11)
We shall proceed along the same lines as in section ν
o
III. We consider the system in presence of a source
Eq. (4.11) generalizes the Landau’s functional to space-
field h (q˜) = h∗ (−q˜) that couples to the QP operator
kσ kσ and time-dependent configurations. As in the static and
nˆkσ(q˜)=β−1 ωψ¯σ∗(k˜−)ψ¯σ(k˜+). We writethepartition uniform case studied in section III, δΦ[δn] is essentially
function as in (3.1), with parameterizedby the Landau f function. We shall show
P
in the next section that δΦ contains all the information
S =β h (−q˜)nˆ (q˜). (4.3)
h kσ kσ about the QP dynamics. Moreover, if we consider the
kX,σ,q˜ q-limit, the WDF {nkσ(r,t)} reduces to the QP distri-
bution function {n }, and Φ[n] to the thermodynamic
TheWDFisobtainedbytakingthefunctionalderivative kσ
potentialintroducedinsectionIII. Thustheeffectivepo-
of the free energy with respect to the source field:
tential Φ describes static and dynamic properties of the
1 δlnZ[h] Fermi liquid in a unified framework. This result should
n (q˜)=hnˆ (q˜)i=− . (4.4)
kσ kσ
βδh (−q˜) be contrasted with the phenomenological FLT which re-
kσ
quiresslightlydifferentapproachestodealwithstaticand
The effective potential is then defined as
dynamicproperties(seesectionII),andstronglyrelieson
1 semiclassical arguments for the latter.
Φ[n]=− lnZ[h]− h (−q˜)n (q˜), (4.5)
kσ kσ
β
k,σ,q˜
X
and satisfies the ‘equation of state’ B. Quantum Boltzmann equation
δΦ[n]
=−h (−q˜). (4.6) In the absence of source field (h=0), the stationarity
kσ
δnkσ(q˜) condition of the effective potential [Eq. (4.6)] yields
We will show below that in the absence of source field (v ·q−iΩ)δn (q˜)−v ·qn′ (ǫ −µ)
(h = 0), the stationarity condition of the effective po- k kσ k F k
1
tential [Eq. (4.6)] yields the equation of motion for the × fσσ′(k,k′)δnk′σ′(q˜)=0. (4.12)
ν
WDF. k′,σ′
X
7
Eq. (4.12) is the quantum Boltzmann equation for the Eqs. (4.10,4.11) determine χ−1. If, for simplicity, we
WDF.[Real-timequantitiesareobtainedbytheusualan- consider the case where only the Landau parameters Fs
0
alytic continuationiΩ→Ω+i0+.] Note thatinthe sim- and Fa are non-zero, the matrix χ−1 can easily be in-
0
plecaseweareconsideringhere(noexternalelectricfield, verted, and we obtain the well-known expressions3
small fluctuations around the equilibrium state, etc), it
isidenticaltotheBoltzmannequation(2.12)satisfiedby χ (q˜)=2N(0) Ω00(iΩ/vF∗q) ,
the semiclassical distribution function ncklσ(r,t).19,20 ch 1+F0sΩ00(iΩ/vF∗q)
The solution of (4.12) can be written as Ω (iΩ/v∗q)
χ (q˜)=2N(0) 00 F , (4.17)
δn (q˜)=v∗u (kˆ,q˜)n′ (ǫ −µ), (4.13) sp 1+F0aΩ00(iΩ/vF∗q)
kσ F σ F k
where
where u (kˆ,q˜) is naturally interpreted as the dynamic
σ
Fermi surface displacement for spin σ fermions. It satis- 1 du u x x+1
fies the equation Ω00(x)= =1− ln . (4.18)
2 u−x 2 x−1
Z−1
(v ·q−iΩ)u (kˆ,q˜)
k σ
dΩ
+vk·qN(0) 4πkˆ′fσσ′(kˆ,kˆ′)uσ′(kˆ′,q˜)=0. (4.14) V. RENORMALIZATION-GROUP APPROACH
σ′ Z
X
Herewehaveusedn′F(ǫk−µ)=−δ(ǫk−µ)whenT →0. Recently, RG techniques based on low-energy fermion
Eq. (4.14) yields the zero-sound (for u↑ = u↓) and spin- effective actions have been applied to interacting
wave (for u↑ =−u↓) modes of the Fermi liquid.2,3 fermions in dimension d ≥ 2 by many authors (see
It is also possible to express the effective potential di- Refs. 7–10,21–26 and references therein). The finite-
rectly in terms of the dynamic Fermi surface displace- temperature RG approach27 was first applied to Fermi
ments: liquidsindimensiond≥2byChitovandS´en´echal.7Con-
νv∗2N(0) dΩ v ·q−iΩ trary to other works on the subject, it revealed that the
δΦ[u]= F 2 δσ,σ′ 4πkˆ kv ·q effectiveinteractionfunctionintheLandauorzero-sound
σX,σ′,q˜n Z k channel(particle-holepairsatsmalltotalmomentumand
dΩ dΩ energy) does not stay marginal under the RG transfor-
×|uσ(kˆ,q˜)|2+N(0) kˆ kˆ′fσσ′(kˆ,kˆ′) mation,since its β-function is not identically zero. From
4π 4π
Z the RG equations, the standard FLT results have been
×uσ(kˆ,−q˜)u′σ(kˆ′,q˜) . (4.15) recovered.7–10 It has been pointed out that the bare in-
teractionfunction of the low-energyfermioneffective ac-
o
Here we have used u (kˆ,−q˜) = u∗(kˆ,q˜). The equations tion cannot be identified with the Landau f function.8
σ σ
of motion(4.14)arethen directly obtainedfromthe sta- The latter, along with other observable parameters of a
tionarity condition δΦ/δu (kˆ,−q˜) = 0. Note also that Fermi liquid, is obtained at the fixed-point of the RG
σ
equations, i.e., when all degrees of freedom have been
(4.15) reduces to (2.8) in the q-limit.
integratedout.8–10 The finite temperature RG approach
has also given results that cannot be obtained within
C. Response functions the standard derivation of FLT. Chitov and S´en´echal,
taking into account the interferences between the zero-
sound (ZS) and exchange (ZS’) channels, have obtained
Since the effective potential is essentially determined
by the susceptibility χσσ′(k,k′;q˜), it also yields the re- RthGe estqaunadtiaorndsmthiacrtossactoipsfiyctdheeriPvaatuiloinproinfcFipLlTe.c9onPterrafroyrmto-
sponse functions of the Fermi liquid. For instance, the
ing a two-loop order calculation, the present author has
charge and spin response functions are given by
obtained a non trivial expression of the wave-function
1 renormalization factor,10 which was also obtained from
χch(q˜)= ν χσσ′(k,k′;q˜), 2D bosonization28 and Ward Identities.21
k,kX′,σ,σ′ In this section, we show how the response function χ
1
χsp(q˜)= σσ′χσσ′(k,k′;q˜), (4.16) introduced in section III, and therefore the Landau f
ν function, can be calculated using a a finite-temperature
k,k′,σ,σ′
X (Kadanoff-Wilson) RG approach. We follow the proce-
whereχisrelatedtotheeffectivepotentialbyEq.(4.10). dure used in Ref. 10 to obtain the compressibility of a
Note that when calculating response functions, the dif- Fermi liquid. For simplicity we consider a 2D fermion
ference between particles and QP’s can be ignored. This gas.
property is a consequence of the Wardidentities that re- We write the partition function Z[h] as a functional
sult fromparticle-numberconservation(see Ref. 21 for a integral over Grassmann variables [Eq. (3.1)], where, as-
detailed discussion). suming that the high-energy degrees of freedom have
8
been integrated out (in a functional sense), the action The meaning of the notation χ∗ is discussed below.
describes fermionic degrees of freedom with |k −k | < The constraint to have all momenta in the shell |k−
F
Λ ≪ k . We write this low-energy effective action as k |<Λ restricts the allowed scatterings to diffusion of
0 F F 0
SΛ0 +SΛ0, where particle-hole, or particle-particle, pairs with small total
h
momentum (q < Λ ).25 Consequently, only two vertex
0
SΛ0 =− ψ¯∗(k˜)(iω−ǫ +µ)ψ¯ (k˜) functions have to be considered: the forward-scattering
σ k σ
Xk˜,σ vertex function and the BCS vertex function. In the ab-
senceofBCSinstability,wecanneglectthelatter. Wede-
1
+4βν ΓΛσ10σ2,σ3σ4(k˜1,k˜2,k˜3,k˜4) note by ΓΛσi0(k˜1,k˜2;q˜) the forward-scatteringvertex. [We
k˜1X...k˜4σ1X...σ4 use the notation ΓΛσi0 = ΓΛσ10σ2,σ3σ4.] q˜ = (q,iΩ) where
×ψ¯∗ (k˜ )ψ¯∗ (k˜ )ψ¯ (k˜ )ψ¯ (k˜ ) q is the momentum transfer (q < Λ0) and the bosonic
σ4 4 σ3 3 σ2 2 σ1 1 Matsubarafrequency Ω the energytransfer. (k˜ ,σ )and
×δ δ . (5.1) 1 1
k1+k2,k3+k4 ω1+ω2,ω3+ω4 (k˜ ,σ ) correspond to the two incoming particles, and
2 2
It will be shown below that the RG procedure ensures (k˜ +q˜,σ )and(k˜ −q˜,σ )tothetwooutgoingparticles.
1 4 2 3
that the action is always expressed as a function of the Neglecting the irrelevant dependence on k and ω ,
1,2 1,2
QP field ψ¯(∗). In (5.1), the wave-vectors k satisfy |k− we consider only the coupling function ΓΛ0(kˆ ,kˆ ;q˜) ≡
σi 1 2
akrFo|u<ndΛt0h.eTFehremsiinsugrlefa-pcea:rtǫikcl−e eµx=citvaFti(okn−s akrFe)lwinheearreizveFd ΓΛσTi0(hθe1,Kθ2a;dq˜a)n.off-Wilson RG procedure consists in suc-
is the bare Fermi velocity. The bare (antisymmetrized) cessivepartialintegrationsofthe fermionfielddegreesof
two-particle vertex ΓΛ0 is assumed to be a non-singular freedom in the infinitesimal momentum shell Λ e−dt ≤
0
function of its arguments. The summation over wave |k − k | ≤ Λ where dt is the RG generator and
F 0
vectors is defined by Λ(t) = Λ e−t the effective momentum cut-off at step
0
t . Each partial integration is followed by a rescaling
1 = d2k ≡k kF+Λ0 dk 2π dθ, (5.2) of radial momenta, frequencies and fields (i.e., ω′ = sω,
ν k Z (2π)2 F ZkF−Λ0 2π Z0 2π k′ −kF = s(k −kF) and ψ¯′ = ψ¯ with s = edt) in or-
X
der to let the quadratic partof the action(5.1)invariant
ignoring irrelevant terms at tree-level. andtorestorethe initialvalueofthecut-off. Thepartial
As shown in section IIB, the Landau’s functional integration modifies the parameters of the action which
δΦ[δn] can also be expressed in terms of the Fermi sur- become functions of the flow parameter t. In the fol-
face displacements uσ(kˆ) ≡ uσ(θ), or equivalently the lowing, we note Γ the running (i.e., cut-off dependent)
‘density’ variations δρσ(θ) = (kF/2π)uσ(θ). [Here and vertex (we do not write its t-dependence explicitly). For
in the following, we denote by θ the direction of a given the purpose of our calculation, it is sufficient to consider
momentum k, i.e., kˆ = (cosθ,sinθ).] For purely techni- the q- and Ω-limits of the forward-scatteringvertex:
cal convenience, we therefore consider an external field
hσ(θ) that couples directly to the QP density operator Γqσi(θ1−θ2)=qli→m0 Γσi(θ1,θ2;q˜) Ω=0 ,
ρˆσ(θ,τ) and write the action ShΛ0 as ΓΩ(θ −θ )= limhΓ (θ ,θ ;q˜)(cid:12)(cid:12) i. (5.6)
ShΛ0 = 2dπθhσ(θ) βdτρˆσ(θ,τ), (5.3) The σRiG1proc2ess Ωa→lso0hgσeiner1ate2s c(cid:12)o(cid:12)(cid:12)(cid:12)qr=re0citions to the
σ Z Z0 source field h (θ) along with higher-order terms in h.
X σ
At step t, the source term in the action can be written
where
as29 (ignoring terms of order h3)
β kF+Λ0
dτρˆ (θ,τ)=v N(0) dk dθ dθ′ β
Z0 σ F ω ZkF−Λ0 ShΛ = 2πhσ(θ) 2πzσ(hσ)′(θ−θ′) dτρˆσ′(θ′,τ)
X σ,σ′Z Z Z0
×lim ψ¯σ∗(k˜+q˜)ψ¯σ(k˜) . (5.4) X1 dθ dθ′
q→0h (cid:12)Ω=0i −2 2π 2πhσ(θ)βχσσ′(θ−θ′)hσ′(θ′). (5.7)
We take the q-limit as discussed in sectio(cid:12)(cid:12)n III. N(0) = Xσ,σ′Z
kF/2πvF is the bare 2D density of states per spin. We do not write explicitly the dependence of z(h) and χ
It is straightforwardto show that the microscopicdef- on the flow parameter t.
inition of the Landau f function [Eqs. (3.11,3.13)] be- Notethatz(h)andχarenotphysicalobservables,since
comes: they do not result from the integration over all degrees
ν of freedom. Only their fixed-point values z(h)∗ and χ∗,
N(0)[δσ,σ′2πδ(θ)+N(0)fσσ′(θ)]=χ∗σ−σ′1(θ), obtained when Λ(t)=0, are physical (observable) quan-
tities. The Landau f function is therefore related to χ∗.
χ∗σσ′(θ−θ′)=4π2βδh−σ1(δθ()2δ)hlnσ′Z(θ[h′)](cid:12) . (5.5) t[Nheotperetcheadtinχg∗sceocrtrioesnps.o]nds to what we denoted by χ in
(cid:12)h=0
(cid:12)
(cid:12)
(cid:12)
9
A. one-loop order Eqs. (5.14) have to be supplemented with the one-loop
RG equations for the vertex functions Aq and Bq. As
The integration of high-energy degrees of freedom shown in Ref. 8, the latter can be written as
(|k−k |>Λ )leadingtothelow-energyeffectiveaction
F 0 dAq β dAΩ
(5.1) will in general generate a wave-function renormal- l =− R Aq2+ l ,
ization factor zΛ0 <1 and an external field renormaliza- dt cosh2βR l dt
dtiiosncuzs(she)d|Λi0n. sWecetiiognnoVreBt.heTsehucos,mtphleiciantiitoinasl wcohnidchitiwoinllsboef ddBtlq =−cosβhR2β Blq2+ dBdtlΩ. (5.15)
the RG equations, besides Γ| =ΓΛ0, are R
Λ0
The first terms of the rhs of Eqs. (5.15) are the con-
zσ(hσ)′(θ)|Λ0 =δσ,σ′2πδ(θ), tribution of the ZS graph to the renormalization of Γq.
χσσ′(θ)|Λ0 =0. (5.8) The contribution of the ZS’ and BCS graphs is taken
into account via the second terms on the rhs of (5.15)
The latter equation follows from the condition T ≪ Λ
0 (see Fig. 1 in Ref. 10). Because the thermal factor
and the fact that the external field h creates excitations
β /cosh2β is a strongly peaked function of Λ(t) near
only in the vicinity of the Fermi surface. R R
Λ(t)=0 when T →0, these equations have the approxi-
The externalfieldrenormalizationat one-looporderis mate solutions8–10
given by (see Fig. 2a in Ref. 10)
dzσ(hσ)′(θ−θ′)=Xσ′k′ Z d2′θπ′′zσ(hσ1)′′(θ−θ′′)Γqσ′′σ′,σ′σ′′(θ′′−θ′) BAqql((ττ))== 1+(1BA−ΩllΩ∗∗τ)AΩl ∗,, (5.16)
× F dk′′ [G¯(k˜′′)]2, (5.9) l 1+(1−τ)BΩ∗
2π β l
Z ω′′
X
whereG¯(k˜)=(iω−v (k−k ))−1 istheQPpropagator. for Λ(t) <∼ T/vF. AΩl ∗ = AΩl |Λ(t)=0 and BlΩ∗ =
′ indicates that theFintegraFtion is restricted to the de- BlΩ|Λ(t)=0 are the fixed-point values of AΩl and BlΩ. We
have introduced the parameter τ = tanhβ and used
R
greesof freedom that are in the infinitesimal momentum
tanh(βv Λ /2) ≃ 1 for T ≪ Λ /v . Since Eqs. (5.16)
R F 0 0 F
shell to be integrated out. Using
hold beyond one-loop order, we postpone their detailed
′ derivation to section VB. Eqs. (5.14) show that the RG
1 β
dk′′[G¯(k˜′′)]2 =− R dt, (5.10) flowofz(h)becomessignificantonlyatsmallenergywhen
β Xω′′ Z vF cosh2βR Λ(t)<∼T/vF. Thisallowstoinsert(5.16)into(5.14)and
obtain
we obtain
1
dzσ(dhσt)′(l) =−cNos(h02)ββR zσ(hσ)′′(l)Γqσ′′σ′,σ′σ′′(l). (5.11) z+(h)(l)= 1+(1−τ)AΩl ∗,
R σ′′ 1
X (h)
z (l)= , (5.17)
We have introduced the dimensionless inverse temper- − 1+(1−τ)BΩ∗
l
ature β = v βΛ(t)/2, and expanded in circular har-
R F
monics the quantities appearing in (5.9), zσ(hσ)′(θ) = using the initial conditions z+(h)(l)|Λ0 =z−(h)(l)|Λ0 =1.
z(h)(l)eilθ... Eq. (5.11) is solved by introducing The renormalization of χ is given by (see Fig. 2b in
l σσ′ Ref. 10)
P z(h)(l)=z(h)(l)+z(h)(l),
+ ↑↑ ↑↓ dθ′′
z−(h)(l)=z↑(h↑)(l)−z↑(h↓)(l), (5.12) dχσσ′(θ−θ′)=− σ′′ Z 2π zσ(hσ)′′(θ−θ′′)zσ(h′σ)′′(θ′−θ′′)
X
and the spin symmetric (Aq) and antisymmetric (Bq) k ′
parts of the two-particle vertex defined by × F dk′′[G¯(k˜′′)]2. (5.18)
2πβν
ω′′ Z
2N(0)Γq (l)=Aqδ δ +Bqτ ·τ , (5.13) X
σi l σ1,σ4 σ2,σ3 l σ1σ4 σ2σ3
Expanding(5.18)incircularharmonicsandusing(5.10),
whereτ denotesthePaulimatrices. Eq.(5.11)decouples
we obtain
into two independent equations:
dlnzd+(th)(l) =−cosβhR2β Aql, dχσdστ′(l) =−Nν(0)Xσ′′ zσ(hσ)′′(l)zσ(h′σ)′′(l). (5.19)
R
dlnz(h)(l) β To derive (5.19) from (5.18), we have used the fact that
d−t =−coshR2β Blq. (5.14) zσ(hσ)′(θ) is an even function of θ. Thus, we obtain
R
10