Renormalization group and Fermi liquid theory for many-nucleon systems

0.48 MB·English

by  B. Friman

Renormalization group and Fermi liquid theory
for many-nucleon systems
2 B. Friman1, K. Hebeler2,3 and A. Schwenk2,4,5
1
0 1GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH,
2
64291 Darmstadt, Germany

n [email protected]
a
2TRIUMF,4004 Wesbrook Mall, Vancouver,BC, V6T 2A3, Canada
J

2 3Department of Physics, The Ohio StateUniversity,
1 Columbus, OH 43210, USA
[email protected]
]
h 4ExtreMe Matter InstituteEMMI,
t
- GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH,
l 64291 Darmstadt, Germany
c
u 5Institutfu¨r Kernphysik,TechnischeUniversit¨at Darmstadt,
n
64289 Darmstadt, Germany
[
[email protected]

1
v Summary. We discuss renormalization group approaches to strongly interacting
0 Fermi systems, in the context of Landau’s theory of Fermi liquids and functional
1
methods, and their application to neutron matter.
5
2
.
1
1 Introduction
0
2
1 In these lecture notes we discuss developments using renormalization group
: (RG) methods for strongly interacting Fermi systems and their application
v
i to neutron matter. We rely on material from the review of Shankar [1], the
X
lecture notes by Polchinski [2], and work on the functional RG, discussed in
r the lectures of Gies [3] and in the recent review by Metzner et al. [4]. The
a
lecture notes are intended to show the strengths and flexibility of the RG for
nucleonic matter, and to explain the ideas in more detail.
We start these notes with an introduction to Landau’s theory of nor-
mal Fermi liquids [5, 6, 7], which make the concept of a quasiparticle very
clear. Since Landau’s work, this concept has been successfully applied to a
wide range of many-body systems. In the quasiparticle approximation it is
assumed that the relevant part of the excitation spectrum of the one-body
2 B. Friman, K. Hebelerand A.Schwenk
propagator can be incorporated as an effective degree of freedom, a quasi-
particle. In Landau’s theory of normal Fermi liquids this assumption is well
motivated, andthe so-calledbackgroundcontributionsto the one-body prop-
agator are included in the low-energy couplings of the theory. In microscopic
calculations and in applications of the RG to many-body systems, the quasi-
particle approximation is physically motivated and widely used due to the
great reduction of the calculational effort.
2 Fermi liquid theory
2.1 Basic ideas
Much of our understanding of strongly interacting Fermi systems at low en-
ergies and temperatures goes back to the seminal work of Landau in the late
fifties[5,6,7].Landauwasabletoexpressmacroscopicobservablesintermsof
microscopicpropertiesofthe elementaryexcitations,the so-calledquasiparti-
cles,andtheirresidualinteractions.InordertoillustrateLandau’sarguments
here, we consider a uniform system of non-relativistic spin-1/2 fermions at
zero temperature.
Landau assumed that the low-energy,elementary excitations of the inter-
acting system can be described by effective degrees of freedom, the quasi-
particles. Due to translational invariance, the states of the uniform system
are eigenstates of the momentum operator. The quasiparticles are much like
single-particle states in the sense that for each momentum there is a well-
defined quasiparticle energy. We stress, however,that a quasiparticle state is
not an energy eigenstate, but rather a resonance with a non-zero width. For
quasiparticles close to the Fermi surface, the width is small and the cor-
responding life-time is large; hence the quasiparticle concept is useful for
time scales short compared to the quasiparticle life-time. Landau assumed
that there is a one-to-one correspondence between the quasiparticles and the
single-particle states of a free Fermi gas. For a superfluid system, this one-
to-one correspondence does not exist, and Landau’s theory must be suitably
modified, as discussed by Larkin and Migdal [8] and Leggett [9]. Whether
the quasiparticle concept is useful for a particular system can be determined
by comparison with experiment or by microscopic calculations based on the
underlying theory.
The one-to-one correspondence starts from a free Fermi gas consisting
of N particles, where the ground state is given by a filled Fermi sphere in
momentum space, see Fig. 1. The particle number density n and the ground-
state energy E are given by (with ~=c=1)
0
1 k3 p2 3 k2
n= n0 = F and E = n0 = F N, (1)
V pσ 3π2 0 2m pσ 52m
Xpσ Xpσ
Renormalization group and Fermi liquid theory for many-nucleon systems 3
n0 pσ
pσ
✻
1 kF
✲ p
0 kF | |
Fig. 1. Zero-temperature distribution function of a free Fermi gas in the ground
state (left) and with oneadded particle (right).
wherek denotestheFermimomentum,V thevolume,andn0 =θ(k p)
F pσ F−| |
istheFermi-Diracdistributionfunctionatzerotemperatureforparticleswith
momentump,spinprojectionσ,andmassm.Byaddingparticlesorholes,the
distribution function is changed by δn = n n0 , and the total energy
pσ pσ − pσ
of the system by
p2
δE =E E = δn . (2)
0 pσ
− 2m
Xpσ
When a particle is added in the state pσ, one has δn = 1 and when a
pσ
particle is removed (a hole is added) δn = 1.
pσ
−
In the interacting system the corresponding state is one with a quasipar-
ticle added or removed, and the change in energy is given by
δE = ε δn , (3)
pσ pσ
Xpσ
where ε = δE/δn denotes the quasiparticle energy. When two or more
pσ pσ
quasiparticlesare addedto the system, anadditionalterm takesinto account
the interaction between the quasiparticles:
1
δE = ε0 δn + f δn δn . (4)
pσ pσ 2V p1σ1p2σ2 p1σ1 p2σ2
Xpσ p1σX1,p2σ2
Here ε0 is the quasiparticle energy in the ground state. In the next section,
pσ
we will show that the expansion in δn is general and does not require weak
interactions. The small expansion parameter in Fermi liquid theory is the
density of quasiparticles, or equivalently the excitation energy, and not the
strength of the interaction. This allows a systematic treatment of strongly
interacting systems at low temperatures.
The second term in Eq. (4), the quasiparticle interaction f , has
p1σ1p2σ2
no correspondence in the non-interacting Fermi gas. In an excited state with
morethanonequasiparticle,thequasiparticleenergyismodifiedaccordingto
δE 1
ε = =ε0 + f δn , (5)
pσ δn pσ V pσp2σ2 p2σ2
pσ pX2σ2
4 B. Friman, K. Hebelerand A.Schwenk
where the changes are effectively proportional to the quasiparticle density.
The quasiparticle interaction can be understood microscopically from the
second variation of the energy with respect to the quasiparticle distribution,
δ2E δε
f =V =V p1σ1 . (6)
p1σ1p2σ2 δn δn δn
p1σ1 p2σ2 p2σ2
As discussed in detail in Section 2.5, this variation diagrammatically corre-
sponds to cutting one of the fermion lines in a given energy diagram and
labeling the incoming and outgoing fermion by p σ , followed by a second
1 1
variationleadingtop σ .Fortheuniformsystem,theresultingcontributions
2 2
to f are quasiparticle reducible in the particle-particle and in the
p1σ1p2σ2
exchange particle-hole (induced interaction) channels, but irreducible in the
direct particle-hole (zero sound) channel. The zero-sound reducible diagrams
are generated by the particle-hole scattering equation.
In normal Fermi systems, the quasiparticle concept makes sense only for
states close to the Fermi surface, where the quasiparticle life-time τ is long.
p
The leading term is quadratic in the momentum difference from the Fermi
surface [10], 1/τ (p k )2, while the dependence of the quasiparticle
p F
∼ −
energy is linear, ε µ (p k ). Thus, the condition
p F
− ∼ −
1
ε µ , (7)
p
| − |≫ τ
p
which is needed for the quasiparticle to be well defined, is satisfied by states
closeenoughtotheFermisurface.Generally,quasiparticlesareusefulfortime
scales τ τ and thus for high frequencies ω τ 1. In particular, states
p p
≪ | | ≫
deep in the Fermi sea, which are occupied in the ground-state distribution,
do not correspond to well-defined quasiparticles. Accordingly, we refer to the
interacting ground state that corresponds to a filled Fermi sea in the non-
interactingsystemasastate with noquasiparticles.Inaweaklyexcitedstate
the quasiparticle distribution δn is generally non-zero only for states close
pσ
to the Fermi surface.
For low-lying excitations, the quasiparticle energy ε and interaction
pσ
f is needed only for momenta close to the Fermi momentum k . It is
p1σ1p2σ2 F
then sufficient to retain the leading term in the expansion of ε µ around
pσ
−
theFermisurface,andtotakethemagnitudeofthe quasiparticlemomentain
f equal to the Fermi momentum. In an isotropic and spin-saturated
p1σ1p2σ2
system (N = N ), and if the interaction between free particles is invariant
↑ ↓
under SU(2) spin symmetry (so that there are no non-central contributions,
such as σ p to the energy), we have
∼ ·
ε µ=ε µ v (p k )+... , (8)
pσ p F F
− − ≈ −
where v = k /m∗ denotes the Fermi velocity and m∗ is the effective mass.
F F
In addition, the quasiparticle interaction can be decomposed as
Renormalization group and Fermi liquid theory for many-nucleon systems 5
f =fs +fa σ σ , (9)
p1σ1p2σ2 p1p2 p1p2 1· 2
where
1 1
fs = f +f and fa = f f . (10)
p1p2 2 p1↑p2↑ p1↑p2↓ p1p2 2 p1↑p2↑− p1↑p2↓
(cid:0) (cid:1) (cid:0) (cid:1)
In nuclear physics the notation f =fs and g = fa is generally
p1p2 p1p2 p1p2 p1p2
used, and the quasiparticle interaction includes additional terms that take
intoaccounttheisospindependence andnon-centraltensorcontributions[11,
12, 13]. However, for our discussion here, the spin and isospin dependence is
not important.
For the uniform system, Eq. (6) yields the quasiparticle interaction only
for forward scattering (low momentum transfers). In the particle-hole chan-
nel, this corresponds to the long-wavelength limit. This restriction, which is
consistent with considering low excitation energies, constrains the momenta
p and p to be close to the Fermi surface, p = p = k . The quasi-
1 2 1 2 F
| | | |
particle interaction then depends only on the angle between p and p . It is
1 2
convenient to expand this dependence on Legendre polynomials
fs/a =fs/a(cosθ )= fs/aP (cosθ ), (11)
p1p2 p1p2 l l p1p2
Xl
and to define the dimensionless Landau Parameters Fs/a by
l
s/a s/a
F =N(0)f , (12)
l l
whereN(0)= 1 δ(ε µ)=m∗k /π2denotesthequasiparticledensity
V pσ pσ− F
of states at the FPermi surface.
The Landau parameterscan be directly relatedto macroscopic properties
of the system. Fs determines the effective mass and the specific heat c ,
1 V
m∗ Fs
=1+ 1 , (13)
m 3
m∗k
c = F k2T , (14)
V 3 B
while the compressibility K and incompressibility κ are given by Fs,
0
1 ∂V 1 ∂n 1 N(0)
K = = = , (15)
−V ∂P n2∂µ n21+Fs
0
9 9V ∂P ∂P 3k2
κ= = =9 = F (1+Fs). (16)
nK − n ∂V ∂n m∗ 0
Moreover,the spin susceptibility χ is related to Fa,
m 0
∂m N(0)
χ = =β2 , (17)
m ∂H 1+Fa
0
6 B. Friman, K. Hebelerand A.Schwenk
for spin-1/2 fermions with magnetic moment β =ge/(4m) and gyromagnetic
ratio g. Finally, a stability analysis of the Fermi surface against small ampli-
tude deformations leads to the Pomeranchuk criteria [14]
Fs/a > (2l+1). (18)
l −
For instance Fs/a < 1 implies an instability against spontaneous growth of
0 −
density/spin fluctuations.
Landau’s theory of normal Fermi liquids is an effective low-energy theory
in the modern sense [1, 2]. The effective theory incorporates the symmetries
of the system and the low-energy couplings can be fixed by experiment or
calculated microscopically based on the underlying theory. Fermi Liquid the-
ory has been very successful in describing low-temperature Fermi liquids, in
particular liquid 3He. Applications to the normal phase are reviewed, for ex-
ample, in Baym and Pethick [10] and Pines and Nozi`eres [15], while we refer
to W¨olfle and Vollhardt [16] for a description of the superfluid phases. The
first applications to nuclear systems were pioneered by Migdal [11] and first
microscopic calculations for nuclei and nuclear matter by Brown et al. [12].
Recently, advances using RG methods for nuclear forces [17] have lead to the
developmentof a non-perturbativeRG approachfor nucleonic matter [18], to
a first complete study of the spin structure of induced interactions [13], and
to new calculations of Fermi liquid parameters [19, 20].
2.2 Three-quasiparticle interactions
In Section2.1, weintroduced Fermiliquid theoryas anexpansionin the den-
sity of quasiparticles δn/V. In applications of Fermi liquid theory to date,
even for liquid 3He, which is a very dense and strongly interacting system,
this expansion is truncated after the second-order (δn)2 term, including only
pairwise interactions of quasiparticles (see Eq. (4)). However, for a strongly
interacting system, there is a priori no reason that three-body (or higher-
body) interactions between quasiparticles are small. In this section, we dis-
cuss the convergence of this expansion. Three-quasiparticle interactions arise
from iterated two-body forces, leading to three- and higher-body clusters in
the linked-clusterexpansion,orthroughmany-bodyforces.While three-body
forces play an important role in nuclear physics [21, 22, 23], little is known
about them in other Fermi liquids. Nevertheless, in strongly interacting sys-
tems, the contributions of many-body clusters can in general be significant,
leading to potentially important (δn)3 terms in the Fermi liquid expansion,
also in the absence of three-body forces:
1 1
δE = ε0δn + f(2)δn δn + f(3) δn δn δn . (19)
1 1 2V 1,2 1 2 6V2 1,2,3 1 2 3
X1 X1,2 1X,2,3
Heref(n) denotes the n-quasiparticleinteraction(the Landauinteractionis
1,...,n
f f(2)) and we have introduced the short-hand notation n p σ .
n n
≡ ≡
Renormalization group and Fermi liquid theory for many-nucleon systems 7
In order to better understand the expansion, Eq. (19), around the inter-
actinggroundstatewithN fermions,considerexcitingoraddingN quasipar-
q
ticles with N N. The microscopic contributions from many-body clusters
q
≪
orfrommany-bodyforcescanbe groupedintodiagramscontainingzero,one,
two,three,ormorequasiparticlelines.Thetermswithzeroquasiparticlelines
contribute to the interacting groundstate forδn=0,whereasthe terms with
one,two,andthreequasiparticlelinescontributetoε0,f(2),andf(3) ,respec-
1 1,2 1,2,3
tively (these also depend on the ground-state density due to the N fermion
lines).Thetermswithmorethanthreequasiparticlelineswouldcontributeto
higher-quasiparticle interactions. Because a quasiparticle line replaces a line
summed over N fermions when going from ε0 to f(2), and from f(2) to f(3) ,
1 1,2 1,2 1,2,3
it is intuitively clear that the contributions due to three-quasiparticle inter-
actions are suppressed by N /N compared to two-quasiparticle interactions,
q
and that the Fermi liquid expansion is effectively an expansion in N /N or
q
n /n [15].
q
Fermi liquid theory applies to normal Fermi systems at low energies and
temperatures, or equivalently at low quasiparticle densities. We first consider
excitationsthatconservethenetnumberofquasiparticles,δN = δn =
pσ pσ
0, so that the number of quasiparticlesequals the number of quaPsiholes.This
corresponds to the lowest energy excitations of normal Fermi liquids. We de-
note their energy scale by ∆. Excitations with one valence particle or quasi-
particle added start from energies of order the chemical potential µ. In the
case of δN = 0, the contributions of two-quasiparticle interactions are of the
same orderas the first-orderδn term, but three-quasiparticleinteractions are
suppressedby∆/µ[24].This isthe reasonthatFermiliquidtheorywithonly
two-bodyLandauparametersissosuccessfulindescribingevenstronglyinter-
acting and dense Fermi liquids. This counting is best seen from the variation
of the free energy F =E µN,
−
δF =δ(E µN)
−
1 1
= (ε0 µ)δn + f(2)δn δn + f(3) δn δn δn ,
1− 1 2V 1,2 1 2 6V2 1,2,3 1 2 3
X1 X1,2 1X,2,3
(20)
which for δN = 0 is equivalent to δE of Eq. (19). The quasiparticle distri-
bution is δn 1 within a shell around the Fermi surface ε0 µ ∆.
| pσ| ∼ | pσ − | ∼
The first-order δn term is therefore proportional to ∆ times the number of
quasiparticles δn =N N(∆/µ), and
pσ| pσ| q ∼
P
N∆2
(ε0 µ)δn . (21)
1− 1 ∼ µ
X1
Correspondingly, the contribution of two-quasiparticle interactions yields
1 1 N∆ 2 N∆2
f(2)δn δn f(2) F(2) , (22)
2V 1,2 1 2 ∼ V h i(cid:18) µ (cid:19) ∼h i µ
X1,2
8 B. Friman, K. Hebelerand A.Schwenk
where F(2) =n f(2) /µisanaveragedimensionlesscouplingontheorderof
h i h i
the Landauparameters.Evenin the stronglyinteracting, scale-invariantcase
f(2) 1/k ; hence F(2) 1 and the contribution of two-quasiparticle
F
h i ∼ h i ∼
interactions is of the same order as the first-order term. However, the three-
quasiparticle contribution is of order
1 n2 N∆3 N∆3
f(3) δn δn δn f(3) F(3) . (23)
6V2 1,2,3 1 2 3 ∼ µ h i µ2 ∼h i µ2
1X,2,3
Therefore at low excitation energies this is suppressed by ∆/µ, compared to
two-quasiparticleinteractions,evenifthedimensionlessthree-quasiparticlein-
teraction F(3) =n2 f(3) /µ is strong (of order 1). Similarly, higher n-body
h i h i
interactions are suppressed by (∆/µ)n−2. Normal Fermi systems at low en-
ergies are weakly coupled in this sense. The small parameter is the ratio of
the excitation energy per particle to the chemical potential. These consider-
ations hold for all normal Fermi systems where the underlying interparticle
interactions are finite range.
The Fermi liquid expansion in ∆/µ is equivalent to an expansion in
N /N ∆/µ, the ratio of the number of quasiparticles and quasiholes N
q q
∼
to the number of particles N in the interacting ground state, or an expan-
sion in the density of excited quasiparticles over the ground-state density,
n /n. For the case where N quasiparticles or valence particles are added to
q q
a Fermi-liquid ground state, δN =0 and the first-order term is
6
N∆
ε0δn µN µ N∆, (24)
1 1 ∼ q ∼ µ ∼
X1
while the contribution of two-quasiparticle interactions is suppressed by
N /N ∆/µ and that of three-quasiparticle interactions by (N /N)2.
q q
∼
Therefore, either for δN = 0 or δN = 0, the contributions of three-
6
quasiparticle interactions to normal Fermi systems at low excitation energies
are suppressed by the ratio of the quasiparticle density to the ground-state
density, or equivalently by the ratio of the excitation energy over the chemi-
calpotential. This holds for excitations that conservethe number ofparticles
(excited states of the interacting ground state) as well as for excitations that
add or remove particles. This suppression is general and applies to strongly
interactingsystemsevenwithstrong,butfinite-rangethree-bodyforces.How-
ever,thisdoes notimply thatthe contributionsfromthree-bodyforcestothe
interactingground-stateenergy(the energyofthe corenucleusinthe context
of shell-model calculations), to quasiparticle energies, or to two-quasiparticle
interactions are small. The argument only applies to the effects of residual
three-body interactions at low energies.
2.3 Microscopic foundation of Fermi liquid theory
A centralobject in microscopicapproachesto many-body systems is the one-
body (time-ordered) propagator or Green’s function G defined by
Renormalization group and Fermi liquid theory for many-nucleon systems 9
Imω Imω
✻ ✻
p2 +iδ
2m
✲ ✲
Reω Reω
p2 iδ
2m−
Fig. 2. Analyticstructureofthefreeone-bodyGreen’sfunctionG inthecomplex
0
ω plane with simple poles for p>k (left) and p<k (right).
F F
G(1,2)= i 0 ψ(1)ψ†(2)0 , (25)
− h |T | i
where 0 denotes the ground state of the system, is the time-ordering
| i T
operator, ψ and ψ† annihilate and create a fermion, respectively, and 1, 2
are short hand for space, time and internal degrees of freedom (such as spin
andisospin).Foratranslationallyinvariantspin-saturatedsystemthatisalso
invariant under rotations in spin space, the Green’s function is diagonal in
spin and can be written in momentum space as
δ
G(ω,p)δ = d(1 2)G(1,2)eiω(t1−t2)−ip·(x1−x2) = σ1σ2 ,
σ1σ2 Z − ω p2 Σ(ω,p)
− 2m −
(26)
where Σ(ω,p) defines the self-energy.For anintroductionto many-body the-
ory and additional details, we refer to the books by Fetter and Walecka [25],
Abrikosov, Gor’kov and Dzyaloshinski [26], Negele and Orland [27], and Alt-
land and Simons [28].
Without interactions the self-energy vanishes and consequently the free
Green’s function G reads
0
1 1 n0 n0
G (ω,p)= = − p + p , (27)
0 ω p2 +iδ ω p2 +iδ ω p2 iδ
− 2m p − 2m − 2m −
where δ = δsign(p k ) and δ is a positive infinitesimal. The free Green’s
p F
−
function has simple poles, as illustrated in Fig. 2, and the imaginary part
takes the form
p2
ImG (ω,p)= π(1 2n0)δ ω . (28)
0 − − p (cid:18) − 2m(cid:19)
The single-particle spectral function ρ(ω,p) is determined by the imaginary
part of the retarded propagator
GR(1,2)= iθ(t t ) 0 ψ(1),ψ†(2) 0 , (29)
1 2
− − h | | i
1 (cid:8) (cid:9)
ρ(ω,p)= ImGR(ω,p), (30)
−π
10 B. Friman, K. Hebelerand A.Schwenk
where , denotes the anticommutator.The retarded propagatoris analytic
{ }
in the upper complex ω plane and fulfills Kramers-Kronig relations, which
relate the real and imaginary parts. Physically this implies that all modes
are propagatingforwardin time andcausalityis fulfilled. Therefore, response
functions are usually expressed in terms of the retarded propagator.
In a non-interacting system the retarded propagatoris given by
1
GR(ω,p)= , (31)
0 ω p2 +iδ
− 2m
p2
ImGR(ω,p)= πδ ω , (32)
0 − (cid:18) − 2m(cid:19)
which implies that the free spectral function is a delta function ρ (ω,p) =
0
δ(ω p2 ). This simple form follows from the fact that single-particle plane-
− 2m
wave states are eigenstates of the non-interacting Hamiltonian.
In the interacting case the situation is more complicated. Here the quasi-
particle energy is given implicitly by the Dyson equation
p2
ε = +Σ(ε ,p). (33)
p p
2m
At the chemical potential ω = µ, the imaginary part of the self-energy van-
ishes,
ImΣ(µ,p)=0, (34)
and the quasiparticle life-time τ for p k .1 For ω = µ, the imagi-
p F
→∞ | | → 6
nary part of the self-energy obeys
ImΣ(ω,p)<0, for ω >µ, (35)
ImΣ(ω,p)>0, for ω <µ. (36)
The retarded self-energy, which enters the retarded Green’s function
1
GR(ω,p)= , (37)
ω p2 ΣR(ω,p)
− 2m −
is related to the time-ordered one through
ReΣR(ω,p)=ReΣ(ω,p), (38)
+ImΣ(ω,p)<0, for ω >µ,
ImΣR(ω,p)= (39)
(cid:26) ImΣ(ω,p)<0, for ω <µ.
−
Using Eq. (30), one finds the general form of the spectral function
1 Atnon-zerotemperature,theimaginarypartoftheself-energynevervanishesand
the quasiparticle life-time is finite. However, for T ≪µ and ω ≈µ, the life-time
is large and the quasiparticle concept is useful.