Kohn-Luttinger effect in nested Fermion liquids

17 Pages·0.22 MB·

by  Hyok-Jon Kwon

Kohn-Luttinger effect in nested Fermion liquids
Hyok-JonKwon
∗
Department of Physics, Brown University, Providence, RI 02912-1843, USA
(October 2, 1996)
WestudytheKohn-Luttingereffect in atwo-dimensional (2D) nested Fermion liquid with a re-
pulsiveinteractionviatherenormalizationgroupmethodandidentifytheresultingorderparameter
7
symmetry. Using the band structure of the 2D Hubbard model close to half-filling as a prototype,
9
we construct an effective low-energy theory. We use multidimensional bosonization to incorporate
9
thezerosoundchannelandfindmarginalFermiliquidbehaviorintheabsenceofanyinstability. We
1
showananalogoftheLandautheoreminnestedFermionliquids,whichservesasthecriterionofthe

n BCS instability. Including repulsive or antiferromagnetic exchange interactions in the low-energy
a
J theory, we show that the dx2−y2-wave BCS channel is renormalized to be the most attractive. Be-
low half-filling, when thenestingis not perfect, thereiscompetition between thespin-density-wave
9
2 (SDW)andtheBCSchannels;whentheSDWcouplingissmallenough,thereoccursadx2−y2-wave
superconducting instability at sufficiently low temperatures.

]
l PACS numbers: 74.20.Mn, 11.10.Hi, 71.10.Hf, 74.25.-q
e
-
r cond-mat/9610020
t
s
.
t
a I. INTRODUCTION
m
- One question of recent interest concerns the possible breakdown of the Fermi liquid in the presence of singular or
d
evenregularinteractions. OneexampleistheLuttingerliquidinonespatialdimensionwherethequasiparticlepoleis
n
o destroyedby a short-rangeinteractionalthoughin this case the low-energyexcitationremains gapless. The existence
c oftheLuttingerliquidinhigherspatialdimensionshasalsobeenproposedbutisstillcontroversial.1Anotherexample
[ is the Kohn-Luttinger(KL)effect: A Fermiliquidstate is unstableagainstone ormoreBCSchannelsin the presence

of a repulsive interaction.2 It has been shown that a short-range repulsive interaction can induce an attractive BCS
3
v channel at large orbital angular momentum modes. In three-dimensional Fermi gases, the Kohn-Luttinger effect was
0 foundinthep-waveBCSchannel.3 Thisnotionwasextendedtotwo-dimensionalFermiliquidsandaweakerKLeffect
2 was also found in the p-wave channel although the effect is not visible in a second-order perturbative expansion.4
0 However,theKLeffectsinisotropicFermiliquidsareweakbecausetheyarefoundinsub-leadingcorrectionsinthe
0
renormalization group (RG) sense; the resulting T is expected to be low. For example, in three dimensions, the KL
1 c
correction to the β function is found to be O(s 7/4) where s is the dimensionless length scale which is taken to be
6 −
a large number in the low-energy limit.5 But in the case of nested Fermion liquids, the effect is more dramatic since
9
/ the nesting gives rise to a logarithmic flow in the repulsive channel. Baranov et al.3 studied the KL effect in a two-
t
a dimensional Hubbard model at low filling factors where the Fermi surface is almost circular and found an attractive
m channel with d -wave symmetry, but it is also a sub-leading effect which originates from the directional property of
xy
- the square lattice. In this paper we study the KL effect in two dimensions in a nested square Fermi surface.
d The subject of this paper may be of interest in the context of the cuprate high-T superconducting materials.
c
n
The KL effect implies that an initially repulsive interaction between electrons can induce an effective attractive BCS
o
interactionamongquasiparticleslyinginthethinmomentumshellaroundtheFermisurface. Hencethisisonepossible
c
: candidateforanon-phononicmechanismofcupratesuperconductivity.6 Indeed,wefindthattherepulsiveinteraction,
v
Xi iwnhaicchconradtaunrcaellwyictohnetxapinesctaantiaonntsi.f7e,6rromagnetic (AF) exchange channel, favors a dx2−y2-wave BCS instability. This is
The low-energytheory of the two-dimensionalHubbard model is complicatedby severalfeatures of the anisotropic
r
a nested Fermi surface. Especially near half-filling, there are van Hove singularities in addition to non-uniformity of
the Fermi velocities along the Fermi surface. Moreover, in the low-energy limit, the system is not a regular Fermi
liquid due to nesting. For instance, the quasiparticles are expected to have a short lifetime, 1/τ T, near the Fermi
surface, and the existence of a well-defined Fermi surface may be questioned.8 To incorporate t∼hese characteristics,
we introduce a low-energy theory which models the band structure of the two-dimensional Hubbard model near
half-filling. We make use of multidimensional bosonization9–12 to describe the low-energy properties of the model.
Although Fermi liquid theory may break down near the Fermi surface, marginal deviations from a Fermi liquid are
within the scope of bosonization.
The RG in the bosonic basis is a suitable method to study the low-energy phases of the model since it puts all
interaction channels on an equal footing. Wilson’s RG algorithm, which we shall make use of, consists of integrating
1
out high-energy degrees of freedom to construct a low-energy effective theory. In a degenerate fermion system, this
amounts to successively eliminating inner and outer sides of the momentum shell around the Fermi surface in which
the low-energytheorylies.5,13,14 Inthis paperweconstructalow-energyeffectivetheoryofanestedfermionliquidby
the RG procedure and then investigate the RG flow of the effective interactions in the frameworkof bosonization. In
other words, we successively integrate out higher-energy modes of the bosonic field φ. In most cases, this bosonized
RG procedure yields results which agree with those obtained in the fermion basis. The distinct advantage of the
RG treatment based on bosonization is that the existence of well-defined quasiparticles is not a priori assumed.
Furthermore,those interactionchannels which preservethe infinite-U(1) symmetry9 onthe Fermi surface are exactly
diagonalized at the outset.15
This paper is organized as follows. In Sec. II, we introduce the model low-energy theory of a two-dimensional
nested Fermion liquid. We use multidimensional bosonization to incorporate the zero sound channel and show that
the resulting low-energy fixed point is a marginal Fermi liquid. In Sec. III, we derive the RG equations for the
BCS channel and discuss the condition of the BCS instability. An analog of the Landau theorem is applicable to
the case of anisotropic Fermi surfaces. In Sec. IV, we incorporate marginally relevant repulsive or AF interactions
and evaluate their RG flow which contribute to the BCS channel. The dx2 y2-wave channel is the most important
one, and we argue that below half-filling competition arises between the spi−n-density-wave (SDW) and dx2 y2-wave
superconductivity. A brief summary and conclusions are given in Sec. V. −
II. MODEL LOW-ENERGY THEORY
The Hubbard model is a convenient descriptionof strongly correlatedfermion systems. For a reliable investigation
into the RG of this model, it is essential to accommodate a properly realistic band structure near the Fermi surface.
Here we present a model low-energy theory which simulates the nested fermion system.
First, consider the band structure of the Hubbard model in two dimensions near half-filling. The Hamiltonian is
H =−t c†iαcjα+U ni↑ ni↓ , (1)
i,j i
X X
where i,j are between nearest neighbors on a square lattice and U is the on-site interaction strength. Here c†iα and
c are fermion creation and annihilation operators. Throughout this paper we assume that U/t 1 holds so that
iα
| |≪
the perturbative expansions are justified. For now we turn off the on-site interaction term and discuss the character
of the Fermi surface that results from the hopping term. The spectrum of the fermions is
E(k)= 2t(cosk a+cosk a), (2)
x y
−
where a is the lattice spacing. At half-filling, the Fermi surface is described by the points in k space which satisfy
cosk a+cosk a=0 . (3)
x y
The Fermi velocity is a function of k given by
v(k)=2ta(xˆ sin(k a)+yˆ sin(k a)) (4)
x y
on the Fermi surface. Singularities in the density of states occur at four corners of the Fermi surface, k= ( π/a,0)
±
or (0, π/a), where the Fermi velocity vanishes.
Tur±ning on the on-site interaction, the RG may be used to integrate out the high-energy degrees of freedom.5 The
resulting low-energy effective theory is expressed in terms of quasiparticles living in a narrow shell around the Fermi
surface of energy cutoff ε . Quasiparticles may then be related to the bare fermion operators by
c
ψk =Zk−1/2(εc) ck , (5)
wherethewave-functionrenormalizationfactorZ (ε )rescalesthediscontinuityinthequasiparticleoccupancyatthe
k c
Fermi surface back to 1. The final quasiparticle weight Z can be obtained by further integrating out the remaining
k
low-energy degrees of freedom, lim Z (ε )= Z . In non-Fermi liquids, Z is zero. Here we stop the RG process
εc 0 k c k k
atthe energyscaleε andinvestigate→the low-energytheorythus obtained. Assuming that the weak nearest-neighbor
c
interaction term does not alter the shape of the Fermi surface or the band structure in Eq. (2), we may linearize the
spectrum of the quasiparticles:
2
E(k)=v sin[k (k) a] q (k) , (6)
F x
| | k
whichissimilartoEq. (2)closetotheFermisurface. Hereq (k) (k k (k)) nˆ(k),wherek (k)isthewave-vector
F F
on the Fermi surface closest to k, k (k) is the x componkent o≡f k (−k), and nˆ·(k) is the unit vector normal to the
F x F
Fermi surface at k (k).
F
ThedifficultywithbosonizingthisFermisurfaceisthatatthefourcornersoftheFermisurfacetheFermivelocities
are zero and the Fermi surface normal vectors are not well-defined in the vicinity of the van Hove points. Below
half-filling, however,the van Hove singularityis not encountered. So we introduce a model spectrum which simulates
the Hubbard model near half-filling but with a smoothed out van Hove singularity. We modify the spectrum in Eq.
(6) as
E(k)=v sin[k (k) a] q (k) , ǫ/a< k (k) <(π ǫ)/a ,
F x F x
| | k | | −
=vsin(ǫa) q (k) (otherwise) (7)
k
and we smooth the four corners of the Fermi surface as four quarter-circles with a radius √2ǫ/a (Fig. 1). So we
define the Fermi velocity function v (k) as the coefficient of q (k) in Eq. (7). This choice is reasonable because the
F
number of states near the van Hove points here is of order ǫ/ksin(ǫ) O(1), which is finite even if the limit ǫ 0
∼ →
is taken. This artificial smoothing simulates the van Hove singularity as the van Hove points accommodate a fixed
fraction of the total number of states. From now on we set 1/a = 1 and v = 1 for convenience. The factor 1/a may
be recoveredwhenever a comparison with other momentum scales is needed. With this convention, for example, the
nesting vectors are expressed as Q (π,π) or (π, π).
≡± ± −
NowthattheFermivelocitiesarewell-definedoverthewholeFermisurface,themodelmaybebosonized. Wecoarse-
grain the Fermi surface into small squat boxes of a width Λ along the Fermi surface and a height λ(S) ε /v (S)
c F
≡
normalto the Fermi surface,as shownin Fig. 1. Here v (S) is the Fermi velocity function at the squatbox S. Then
F
we introduce a coarse-grainedcharge current which lives in a squat box labeled as S:
Jα(kS;q)= θ(S;k−q) θ(S;k) {ψk† q,α ψk,α−δq,0 nk,α} . (8)
−
k
X
Here nk,α = hψk†,αψk,αi. α is the spin index and the label S identifies a patch on the Fermi surface with a Fermi
momentum k and θ(S;k) = 1 if k lies inside a squat box centered on k and θ(S;k) = 0 otherwise. The two
S S
momentum scales Λ,λ(S) are taken to be small so that 1/a Λ λ(S) at each Fermi surface patch. The latter
≫ ≫
inequality is needed to minimize scattering of fermions outside of the squat box so that the number of fermions in
each patch on the Fermi surface is conserved. Now the limit of ǫ 0 is not permitted since it violates the condition
→
of ǫ/a Λ at the four van Hove points. Instead, we take ǫ as a small non-zero number.
The≫U(1) current algebra reads9–11
[J (k ;q),J (k ;p)]=δ δ δ2 Ω q nˆ , (9)
α S β T αβ S,T q+p,0 · S
which holds when 1/a Λ λ(S). Here Ω Λ(L/2π)2 is the number of states in the squat box divided by λ(S)
≫ ≫ ≡
and nˆ is a unit vector normal to the Fermi surface at patch S. The algebra, Eq. (9), makes it possible to bosonize
S
the currents. The bosonic representation of the currents is
J (k ;x)=√4π nˆ φ (k ;x) , (10)
α S S α S
·∇
wherethe bosonicfieldsφ satisfy thecanonicalcommutationrelations. Thenthe fermionquasiparticlefields ateach
α
patch can be expressed in terms of boson fields as10
ψ (k ;x,t) eikSx θ(S;p)ei(p kS)xψ (t)
α S · − · αp
≡
p
X
Λλ(S)/2 √4π
= eikSxexp i φ (k ;x,t) Oˆ(S) . (11)
s (2π)2 · Ω α S
n o
Here Oˆ(S) is an ordering operator introduced to maintain Fermi statistics along the Fermi surface.
The bosonized effective action of the low-energy theory is
dω
S = [ω v (S)q nˆ ] a α(k ;q) a (k ;q)+S . (12)
F S ∗ S α S I
2π − ·
XS q,λ(S)X>q·nˆS>0Z
3
Here S is the fermion interaction terms which we discuss later. The charge currents are related to the canonical
I
boson operators a and a by
†
Jα(kS;q)= Ω|q·nˆS| [aα(kS;q) θ(q·nˆS)+a†α(kS;−q) θ(−q·nˆS)] , (13)
p
where
[aα(kS;q), a†β(kT;p)]=δαβδS,Tδp2,q . (14)
Hereθ(x)=1ifx>0andθ(x)=0otherwise. FromEq. (11)thefermionGreen’sfunctioncanbeexpressedinterms
of the Fourier-transformedboson correlationfunction and it is given by15
GαF(kS;x,t)≡ihψ†α(kS;x,t) ψα(kS;0,0)i
λ(S)Λ/2 d2q dω
= eikSxexp [ei(qx ωt) 1]
· · −
(2π)2 (2π)2 2π −
hZ Z
(2π)2
a (k ;q) a (k ;q) . (15)
×Λq nˆ h α S †α S i
S
· i
Nextconsidertherenormalizationofthefermioninteractions. AsinthecaseofacircularFermisurface,themarginal
interactions take the form of four-fermion (two-body) interactions, as can be verified by counting the engineering
dimension. Unlike the case of a circular Fermi surface, however, many large-momentum scattering channels are now
marginal due to the geometry of the nesting. Some of the channels give rise to spin-density-wave (SDW) or charge-
density-wave (CDW) instabilities. These logarithmically renormalized channels are discussed in Sec. IV. For now,
the zero-sound (ZS) channel may be incorporated. This channel yields a Fermi liquid solution for a circular Fermi
surface in two dimensions and a Luttinger liquid in one dimension. It is important to include this channel in the
free action before the the large-momentum scattering channels are studied perturbatively. This permits a check on
whether the Fermi surface breaks down in the presence of the ZS channel.
Forsimplicity,wesuppressallmomentumdependence inthe couplingfunctionanddropthe spinindices. Inclusion
of the spin exchange channel does not change the result qualitatively. The ZS channel in the fermion basis is
S = d2x dt F(S,T) ψ (k ;x,t) ψ(k ;x,t) ψ (k ;x,t) ψ(k ;x,t) . (16)
ZS † S S † T T
−
Z
Note that a U(1) phase rotation in each squat box, ψ(k ;x,t) eiΓ(S)ψ(k ;x,t) and ψ (k ;x,t)
S S † S
e iΓ(S)ψ (k ;x,t), leaves the action invariant. Physically, this infinite U→(1) symmetry means that the number→of
− † S
fermions at each patch on the Fermi surface is conserved. This channel can be expressed in terms of U(1) currents
since Eq. (8) is U(1) invariant at each patch. The ZS channel in the bosonized form is succinctly written as
1 dω
S = F(S,T) J(k ;q) J(k ; q) , (17)
ZS S T
−V 2π −
S,T q Z
XX
whichis marginalsince it is bilinear inbosons. Hereq denotes (q,ω). The solutionis a Gaussianbosonparametrized
by the shape of the Fermi surface and the coupling parameter F(S,T). Those which do not have infinite U(1)
symmetry are not expressed as bilinear in currents but they can be studied perturbatively. Here for simplicity the
case F(S,T)=F is studied.
0
Turningourattentiontothe fermionsonthenestedpartofthe Fermisurface,sincethatis wherethemostsingular
behavior may arise, we make use of the generating functional method,16,17 and obtain the boson correlationfunction
ha(kS;q)a†(kS;q)i= ω v (iS)q nˆ +i(2Λπ)2 (ω vq(·Snˆ)Sq nˆ )2 K0(q) −1 , (18)
F S F S
− · − ·
(cid:2) (cid:3)
and the fermion Green’s function is given by Eq. (15). Here,
1 2Λ θ(q nˆ ) q nˆ
S S
K (q)= · ·
0 F − (2π)2 ω v (S)q nˆ
0 F S
S − ·
X
1 √2
= + (2π)2 χ (ω,sinǫ q)+χ (ω,q) , (19)
F (2π)2 0 N
0
h i
4
where χ is the contribution from the nested part of the Fermi surface:
N
1 ω2/q2+cosǫ
4 −
χ (q)= ln k
N q
1 ω2/q2 (cid:20) 1 ω2/q2 cosǫ(cid:21)
− − −
k k
q 4 q 1 ω2/q2 +cosǫ
+ ln − . (20)
⊥
1 ω2/q2 1 ω2/q2 cosǫ
− (cid:20)p − − (cid:21)
⊥ ⊥
Again q is the component of q parallel wpith the Fermi surpface normal and q is the perpendicular component. The
contribuktion from the curved part of the Fermi surface (χ ) in Eq. (19) gives⊥a self-energy correction Im Σ ω2lnω
0
as in a Fermi liquid system.15 We perturbatively estimate the imaginary part of the quasiparticle self-ener∼gy which
originates from Eq. (20). The imaginary part of χ (q) arises when ω < q sinǫ or ω < q sinǫ. To second
N
order in F , the boson correlation function is expanded as | | | k| | | | ⊥|
0
i
a(kS;q)a†(kS;q)
h i≈ ω v (S)q nˆ
F S
− ·
Λ q nˆ √2
+i · S F F2 χ (q) , (21)
(2π)2 (ω v (S)q nˆ )2 0− 0 (2π)2 N
F S
− · (cid:16) (cid:17)
where the χ term is dropped. The first order term in F renormalizes the Fermi velocity at the quasiparticle pole
0 0
just as in the Fermi liquid case, but the velocity correction is δv =F Λ/k (2π)2, which is infinitesimal. Spin-charge
0 F
velocity separation does not occur, as can be easily shown when the spin indices and the exchange channel are
included in the effective action. Inserting Eq. (21) into Eq. (15), expanding the exponential to second order in F ,
0
andperformingaFouriertransforminto(k,ω)spaceyieldstheimaginarycorrectiontothequasiparticleself-energy.15
Then Im Σ(k,ω) ω F2Λ/k , which is of a marginal Fermi liquid form.18 Similarly, corrections of higher orders
∝ 0 F
in F can be estimated and we find that they also yield a marginal Fermi liquid form. This agrees with previous
0
expectationsbyotherswhoobtainedtheinverselifetimeofquasiparticles,1/τ T,asafunctionofthetemperature.8
∼
Since the quasiparticle weight of a marginal Fermi liquid vanishes at the Fermi surface as the inverse of a logarithm,
theFermiliquidfixedpointbreaksdownbutthebreakdownoftheFermisurfaceissmoothenoughforthebosonization
and the subsequent RG procedure to be applicable.17 Therefore we can confidently apply the RG method to analyze
large-momentum channels which do not preserve infinite U(1)-symmetry.
One might wonder whether the marginal-Fermi-liquid-like behavior of the nested fermion system has relevance
to the linear-T in-plane resistivity of cuprate materials in the normal state.19 However, we only have considered a
simplified single-band Hubbard model with the ZS channel alone whereas a three-band Hubbard model may be a
better description of the cuprate materials. For example, we have not incorporated in the model the strong van
Hove singularity which is present in the cuprate materials.20 Furthermore, the n-doped N Ce CuO (NCCO)
2 x x 4 δ
material exhibits T2 resistivity even though photoemission studies show a nested Fermi surfa−ce.21 This su−ggests that
nestingwiththe ZSaloneisnotsufficientto accountforthe behaviorofthe resistivityinthe normalstatesofcuprate
superconductors.
In the following sections, we study large-momentum scattering channels by the perturbative RG method. We do
not explicitly include the ZS channel at the outset but we assume that it contributes sub-leading corrections to RG
equations. Forexample,thoughbosonsindifferentpatchesarecorrelatedviatheZSchannel,the correlationbetween
fermions in different patches is negligible:
1
ψ (k ;x) ψ(k ;0) e φ(kS;x)φ(kT;0) 0 , (22)
† S T h i
h i∝ L →
where L is the system size.15 Also the self-energy corrections are of order Λa 1 times smaller than the leading
≪
order. So the leading RG equations we obtain in the following sections are assumed to be unchanged even if the ZS
channel is considered.
III. RG OF THE BCS CHANNEL
In this section we study the RG flow of the BCS channel for the nested Fermi surface and find an analog of the
Landau theorem. As for the case of an isotropic circular Fermi surface, the BCS channel in a nested square Fermi
surface flows logarithmically in scale length since the Fermi surface is symmetric under reflection at the origin in
5
momentum space. On a circular Fermi surface, the Landau theorem22 states that there occurs a BCS instability if
any one value of V is negative where V is the lth orbital angular momentum component of the BCS vertex function
l l
V(θ ,θ ) = eil(θ1 θ2)V . Here θ ,θ are the angular variables on the Fermi surface in momentum space. This
1 2 l − l 1 2
theorem can be seen in the RG equation for V derived by Shankar,5,15
l
P
dV 1
l = V2 , (23)
d lns − 2π l
the solution of which is
V
l
V(s)= , (24)
l 1+ Vl lns
2π
for each l. Thus, if any V <0, it eventually leads to instability.
l
InthecaseofthenestedFermisurfacethereisnorotationalinvarianceintheBCSvertexfunction,andsoV(k ,k )
S T
must be expanded in a double Fourier series.3 Consequently the Fourier coefficients are coupled in the resulting RG
equations,unlikeinthecaseofacircularFermisurface. Firstwediscussthepossiblesymmetryoftheorderparameter
inasquarelattice. TheBCSvertexfunctionhasD symmetryandtherearefivesetsofbasisfunctionswhichgenerate
4
the two-dimensionalfunction space to which the vertex function belongs. The five sets are representedas A and A
1 2
(s-wave), B and B (d-wave), and E (p-wave).3 The basis functions in each set are following:
1 2
A : cos2nk ,
1 x
A : sgn(k ) sin2nk ,
2 y x
B : cos(2n+1)k ,
1 x
B : sgn(k ) sin(2n+1)k ,
2 y x
E : a sin(n+1/2)k +b sgn(k ) cos(n+1/2)k , (25)
x y x
wherewehaveparametrizedthebasisfunctionswithk andthesignofk onthesquareFermisurfacewhichisdefined
x y
by cosk +cosk = 0. In general the BCS vertex function on a square lattice can be expressed as a double Fourier
x y
expansionin terms of these basis functions. Due to theD symmetry ofthe BCSvertex function, the expansiondoes
4
not containcrossedterms between basis functions of different symmetry sets, and so the five sets are decoupled from
one another in the RG equation at the outset.
Now we inspect the RG equation of the BCS obtained in bosonization language,15
dV(k ,k ) Λ 1
S T = V(k ,k ) V(k ,k ) . (26)
d(lns) −(2π)2 v (U) S U U T
F
U
X
Notice that there is a weight function of 1/v (U) in the sum. To analyze the RG equation, we modify the basis
F
functions in Eq. (25) so that they are orthonormal under a summation over the Fermi surface with the weight
function 1/v (U). The modified basis functions are given by
F
v (k) 1/2 v (k) 1/2
a (n,k) F cos2nk , a (0,k) F ,
1 x 1
≡ 2π√2 ≡ 4π√2
(cid:20) (cid:21) (cid:20) (cid:21)
v (k) 1/2
a (n,k) sgn(k ) F sin2nk ,
2 y x
≡ 2π√2
(cid:20) (cid:21)
v (k) 1/2
b (n,k) F cos(2n+1)k ,
1 x
≡ 2π√2
(cid:20) (cid:21)
v (k) 1/2
b (n,k) sgn(k ) F sin(2n+1)k ,
2 y x
≡ 2π√2
(cid:20) (cid:21)
v (k) 1/2
e(n,k) F c sin(n+1/2)k +c sgn(k ) cos(n+1/2)k . (27)
1 x 2 y x
≡ 2π√2
(cid:20) (cid:21)
(cid:2) (cid:3)
The BCS vertex function can be expanded as
V(k ,k )= Cnm a (n,k ) a (m,k )+Cnm a (n,k ) a (m,k )
S T 1 1 S 1 T 2 2 S 2 T
Xn,mn
+Cnm b (n,k ) b (m,k )+Cnm b (n,k ) b (m,k )
3 1 S 1 T 4 2 S 2 T
+Cnm e(n,k ) e(m,k ) . (28)
5 S T
o
6
This expansionis valid as long as min v (k) >0 which holds in the model consideredhere. In fact, min v (k)
F F
| | | |
is zero only at exactly half-filling. (cid:16) (cid:17) (cid:16) (cid:17)
Now insert Eq. (28) into Eq. (26) to obtain the RG equations of the Fourier coefficients:
dCnm 1 ∞
i = Cnl Clm (29)
dlns − (2π)2 i i
l=0
X
where i=1,...,5. Now the analog of the Landau theorem in the nested fermion liquid can be obtained. Namely, the
diagonal Fourier coefficients obey the equations
dCnn 1 ∞
i = Cnl 2 . (30)
dlns − (2π)2 | i |
l=0
X
Herethediagonalcoefficientsdecreaseasthelengthscaleincreases. Thus,ifanyonevalueofCnn startsoutnegative,
i
a BCS instability occurs. This is similar to the Landau theorem in an isotropic circular Fermi surface. This criterion
of instability permits us to identify the possible symmetry of the resulting order parameter,since the gap function is
related to the BCS vertex function by the following:
∆ = V(k,k) ∆k′ , (31)
k ′
− 2E(k)
k′ ′
X
where k,k are within the thin shell around the Fermi surface and E(k) is the spectrum of the low-lying excitation
′ ′
in the superconducting state. In the next section we apply this result to search for a possible BCS instability in the
presence of a repulsive interaction.
IV. RG OF REPULSIVE INTERACTIONS
In this section we discuss the RG flow of repulsive interactions. We only consider those channels which flow
logarithmically in length scale s. To simplify the analysis we take the coupling functions to be independent of
momentum.
There are three classes of interaction channels which are logarithmically renormalized in the presence of nesting.
The first class is the CDW or SDW channel. In case of a repulsive bare Hubbard interaction, it is the SDW channel
that leads to an instability. This channel is shown to have little overlap with the BCS channel and is practically
decoupled from the BCS channel. The second and third classes are Q = 0 and Q = 2k repulsive (or AF exchange)
F
channels which renormalize the BCS channel. In the following two subsections we study the RG equations in the
SDW (CDW) channel and in the other repulsive (or AF) channel which renormalize the BCS and calculate the
Kohn-Luttinger effect.
A. CDW and SDW channels
In this subsection we derive the RG flows of density-wave channels, which have negligible overlap with the BCS
channel. The channel is
F
S = d2x dt DW ψ α(k ;x) ψ (k +Q ;x)
DW 2 † S β S i
Z S,T i
XX
ψ γ(k ;x) ψ (k +Q ;x) [δβδδ δβδδ] . (32)
× † T δ T j α γ − γ α
Here Q ,Q are two of the four nesting vectors (Fig. 2). Due to the nesting, this channel flows logarithmically in
i j
scale length and causes a CDW or a SDW instability, depending on the sign of F . The spin indices in Eq. (32)
DW
are appropriately symmetrized according to Fermi statistics.14 Separating the interaction F into decoupled CDW
DW
and SDW channels,
1 ′
S = d2x dt F S (S;x) S (T;x)+F n (S;x) n (T;x) , (33)
DW SDW Q Q CDW Q Q
2 ·
Z S,T
X(cid:8) (cid:9)
7
where
1
SQ(S;x)= 2 ψ†α(kS;x) σαβ ψβ(kS+Q;x) (34)
and
n (S;x)=ψ α(k ;x) ψ (k +Q;x) (35)
Q † S α S
are the spin density and the charge density, respectively. The primed summation over S,T indicates that the sum is
over the nested parts of the Fermi surface. First, we investigate the RG flow of the CDW term. For simplicity we
drop the spin indices. From Eq. (11) the bosonized form of the CDW action is
1 ′
SCDW = 2 dt d2x FCDW(S,T) ψ†(kS;x) ψ(kS+Qi;x) ψ†(kT;x) ψ(kT+Qj;x)
S,T Z
X
= 1 Λ 2 ′ dt d2x F (S,T) λ(S) λ(T)
2 (2π)2 CDW 4
S,T Z
(cid:0) (cid:1) X
√4π
exp i [φ(k +Q ;x)+φ(k +Q ;x) φ(k ;x) φ(k ;x)] , (36)
× Ω S i T j − S − T
n o
where x denotes (x,t). The scaling of ε , x , t follows
c
t t/s , x x /s , ε sε , (37)
→ ′ k → ′k c → ′c
where s is the scale of the length. Dimensional analysis shows that S is scale invariant and thus marginal.
CDW
Nextthefields maybe splitintohigherandlower-energymodesφ(k ;x) φ(k ;x)+h(k ;x)wherehisthe field
S ′ S S
→
of higher-energy modes whose momentum ranges λ(S)/2 < p < λ(S)/2 where λ(S) ε /v (S). The equal-time
′ | | ′ ≡ ′c F
correlation function of h fields is
Ω2 nˆ x+2is/λ(S)
hh(kS;x) h(kT;0)i=δS,T 4π ln nˆS· x+2i/λ(S) ; |nˆS×x|≪1
(cid:18) S· (cid:19)
=0 ; nˆ x 1 . (38)
S
| × |≫
We study how the partition function is renormalizedafter the fast momentum modes are integrated out:
1
e SCDW = 1 S + S2 +...
h − i h − CDW 2 CDW i
=exp −SC′ DW . (39)
Here the average in the brackets is taken only over th(cid:2)e high-en(cid:3)ergy fields h and S is the renormalized CDW
C′ DW
action. The third term on the right hand side (RHS) of Eq. (39) gives the second order in the F correction to
CDW
S . Using the formula eA eB =e AB+(A2+B2)/2 and Eq. (38), we obtain
CDW h i
h i
1 1 Λ 4 ′ λ(S) λ(T) λ2(U)
S2 S 2 = dt d2x F2
2 h CDWi−h CDWi 2 (2π)2 CDW 16s4
h i (cid:20) (cid:21) SX,T,UZ
√4π
exp i [φ(k +Q ;x)+φ(k +Q ;x) φ(k ;x) φ(k ;x)]
× Ω S i T j − S − T
n 4π o
du d2y exp h(k ;y,u) h(k ;0,u) + h(k +Q ;y,u) h(k +Q ;0,u)
× Ω2 h U U i h U l U l i
+sZubleading termns (cid:2) (cid:3)o
1 Λ 4 ′ ′ λ2(U) 2π 4s2/λ(U)2
dt d2x F2 du d(nˆ y)
≈ 2 (2π)2 CDW 4s2 Λ U· [v (U)u]2+(nˆ y)2
(cid:20) (cid:21) S,T Z U Z F U·
X X
λ(S) λ(T) √4π
exp i [φ(k +Q ;x)+φ(k +Q ;x) φ(k ;x) φ(k ;x)]
× 4s2 Ω S i T j − S − T
+subleading termns . o (40)
8
Here v (U) denotes the Fermi velocity at the patch U. Replacing λ(S) with sλ(S) [λ(S) is the new cutoff] and
F ′ ′
performing the integrals over nˆ y and u, we extract the correction to the CDW coupling function
U
·
dFCDW(s) = Λ ′ 1 F2 . (41)
dln(s) − 2(2π)2 v (U) CDW
F
U
X
Therefore, we expect instability in this channel if F is strictly negative. This agrees with the result obtained by
CDW
Shankar.5 F < 0 corresponds to a bare attractive interaction (U < 0) and the CDW instability leads the phase
CDW
transition into a charge-density ordering with period (π,π) and (π, π).
−
Likewise, a similar RG transformation can be performed to obtain the flow equation for the SDW channel:
1 ′
S = d2x dt F S (S;x) S (T;x) (42)
SDW SDW Q Q
− 2 ·
Z S,T
X
yielding
dFSDW(s) = Λ ′ 1 F2 (43)
dln(s) − (2π)2 v (U) SDW
F
U
X
As expected there occurs an instability when F <0, which corresponds to a bare repulsive interaction (U >0).
SDW
The SDW instability leads to an AF spin-density ordering with period (π,π) and (π, π) since the on-site repulsive
−
interaction induces an AF exchange interaction with the nearest neighbors.
Since the RG equation for the BCS channel is of interest, we now discuss the overlap between the BCS and the
SDW (CDW) channels. In Fig. 3 we show a typical scattering channel which belongs to both the BCS and the
density-wave channel. In terms of the BCS, it can be written as
Soverlap = d2x dt V(kS,kT) ψ†α(kS;x) ψ†β( kS;x)
−
S,T Z
X
ψ ( k ;x) ψ (k ;x) δ . (44)
× β − T α T kT,kS+Q
Obviously there is a severe phase space restriction due to the factor of δ in the overlapped channel, which
kT,kS+Q
amounts to a factor of Λa 1. Therefore, the SDW (CDW) is practically decoupled from the BCS channel. The
≪
renormalization of the BCS channel comes exclusively from the other repulsive or AF exchange channels which we
discuss in the next subsection. Here we can evaluate the energy scale of the SDW instability for future reference.
From Eq. (43), we obtain the critical energy at which the SDW instability occurs:
1
E E exp . (45)
SDW ∼ F − 8√2 ln2 F
(cid:20) (2π)2 ǫ | SDW|(cid:21)
B. Kohn-Luttinger effect
The two classes of repulsive channels which renormalize the BCS are shown in Fig. 4 and Fig. 5 and are given by
1
S = d2x dt F ψ α(k ;x) ψ (k +∆p+Q ;x)
R 2 R † S β S i
S,T ∆pZ
XX
×ψ†γ(kT;x) ψδ(kT−∆p−Qi;x) [δαβδγδ −δγβδαδ] , (46)
which is the zero momentum process (Fig. 4) and
1
S = d2x dt F ψ α(k ;x) ψ (k +∆p+Q ;x)
U 2 U † S β S i
S,T ∆pZ
XX
ψ γ(k ;x) ψ (k ∆p+Q ;x) [δβδδ δβδδ] , (47)
× † T δ T− i α γ − γ α
9
which is the 2k =Q process allowedbecause 2Q is a commensurate wave-vector(Fig. 5). In both channels we take
F
the coupling functions to be momentum-independent and the spin indices are anti-symmetrized.14
Now we expand the partition function to second order in S + S , integrate out the high-energy modes, and
R U
study the resulting effective action generated by this procedure. In the course of this RG procedure, S and S are
R U
renormalized as well as the BCS channel, and so the resulting flow equation will be a complicated set of coupled
differential equations. We do not attempt to perform a complete analysis of the flow equations but we are only
interested in the trend of initial growth of the BCS channel; the higher-order coupled equations are important only
in the very-long-length scale. Thus we only study how S +S renormalizes the BCS channel in the initial stages
R U
of growth near s = 1. The second-order correction to the BCS channel is calculated in the same manner as in the
previous subsection:
1
2h(SR+SU)2i≈lns (FR2 +4 FU2) d2x dt ψ†α(kS;x) ψ†β(−kS;x)
S ∆pZ
XX
ψ (k +∆p+Q;x) ψ ( k ∆p Q;x)
β S α S
× − − −
√2 1 1+cos(ǫ+ ∆p/2)
ln | |
×2(2π)2 cos∆p/2 1 cos(ǫ+ ∆p/2)
(cid:16) − | | (cid:17)
lns √2 1+cosǫ 1
+ F2 ln +
2 R (2π)2 1 cosǫ 2π
XS,T h (cid:16) − (cid:17) i
d2x dt ψ α(k ;x) ψ β( k ;x) ψ ( k ;x) ψ (k ;x)
† S † S β T α T
× − −
Z
+ non BCS terms . (48)
−
The second term in Eq. (48) contributes to the s-wave BCS channel. Since it is repulsive even in the first order to
begin with, the s-waveBCS channel is not a candidate for instability. The first term in Eq. (48) gives RG flow to all
BCS symmetry channels since it has a strong wave-vectordependence. More succinctly, the RG equationof the BCS
channel is given by
dV (k ,k ) √2 1 1+cos(ǫ+ ∆p(S,T)/2)
BCS S T =(F2 +4 F2) ln | |
dlns R U 2(2π)2 cos∆p(S,T)/2 1 cos(ǫ+ ∆p(S,T)/2)
(cid:16) − | | (cid:17)
Λ 1
V (k ,k ) V (k ,k ) , (49)
− (2π)2 v (U) BCS S U BCS U T
F
U
X
if S and T are in the same quadrant on the Fermi surface. When S and T are in different quadrants,
dV (k ,k ) Λ 1
BCS S T = V (k ,k ) V (k ,k ) . (50)
dlns − (2π)2 v (U) BCS S U BCS U T
F
U
X
∆p(S,T)isavectoralongthenestedFermisurfacewhichisshowninFig. 4. Forinstance,∆p(S,T)= (k k ) xˆ
S T
| − · −
π ifk andk arebothinthefirstquadrantonthe(k ,k )plane. Theexpressionof∆p(S,T)intheotherquadrants
S T x y
|
can be obtained by inverting the signs of k or k appropriately.
S T
NowwearereadytocheckwhichsymmetrychannelsareattractivebyevaluatingthegeneralizedFouriercoefficients
of the double expansion shown in Sec. III. Among the basis functions we introduced in Eq. (27), we especially focus
on the fundamental modes whose Fourier coefficients are presumably the most significant. After performing the
generalized Fourier transform of the RHS of Eq. (49) we find that there are two diagonal Fourier coefficients which
flow to negative numbers among the fundamental modes. Their rate of change is evaluated as follows:
dC11 1 dV (k ,k )
2 =Λ2 BCS S T a (1,k ) a (1,k )
dlns v (k ) v (k ) dlns 2 S 2 T
F S F T
S,T
X
0.2 (F2 +4 F2) C11 2/(2π)2 , (51)
≈− R U −| 2 |
dC00 1 dV (k ,k )
3 =Λ2 BCS S T b (0,k ) b (0,k )
dlns v (k ) v (k ) dlns 1 S 1 T
F S F T
S,T
X
0.7 (F2 +4 F2) C00 2/(2π)2 , (52)
≈− R U −| 3 |
10