Correlation functions of higher-dimensional Luttinger liquids

7 Pages·0.2 MB·English

by  Lorenz Bartosch

Correlation functions of higher-dimensional Luttinger liquids
Lorenz Bartosch and Peter Kopietz
Department of Physics and Astronomy, University of California, Los Angeles, 90095∗
Institut fu¨r Theoretische Physik der Universit¨at G¨ottingen, Bunsenstrasse 9, D-37073 G¨ottingen, Germany
(28 April 1998)
9
wave-vector k . In this case the single-particle Green’s
F
9 Using higher-dimensional bosonization, we study correla- function G(k,ω) satisfies an asymptotic Ward-identity,
9 tion functions of fermions with singular forward scattering.
which can be used to sum the most singular terms in
1
Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)],
the expansion of G(k,ω) to all orders in the interaction
n weconsider density-densityinteractions ind dimensions that
[2]. This Ward identity forms also the basis [3] for the
a diverge in Fourier space for small momentum transfers |q|
J as |q|−η with η = 2(d−1). In this case the single-particle generalization of the bosonization method to arbitrary
dimensions [3–7].
5 Green’s function shows Luttinger liquid behavior. However,
Inthis workwe shallusehigher-dimensionalbosoniza-
in contrast to d = 1, in higher dimensions the singularity of

el] Ftheermmiowmaevnet-uvmectdoirstkrFibuisticohnarnakctfeorrizweadvbe-yveacdtoiffrserkenctloesxeptoontehnet toifohnigthoesrt-uddimyeonnseio-naanldLtuwttoi-npgaerrtilciqleuicdosr.reAlastaiomnifcurnoscctoiopnics
- model system for a NFL in d > 1, we shall consider
than thesingularity of thedensity of statesρ(ω)for frequen-
r
t ciesωclosetozero. Wealsocalculatetheirreduciblepolariza- spinless fermions interacting with a long-range density-
at.s tTioomnoΠn⋆a(gqa,-ωL)utfotirn|gqe|r≈mo2dkFel.ΠW⋆(h±e2rekaFs+inqt,hωe)oenxhe-idbiimtseannsoiomnaa-l dfoernmsitfyqinotfetrhaectiinotnerVa(crti−onr′)is,suchthattheFouriertrans-
m lous scaling, we show that in d> 1 the leading singular cor-
d- rinetcetrioanctsiocnasncinel.d>Th1usle,aedvetno Lthuotutignhgesrinlgiquulaidr bdeehnasivtiyo-rdeonfstihtye fq = qgc2ηe−|q|/qc . (1)
n single-particle Green’s function, gauge invariant correlation | |
o functions such as the polarization or the conductivity show Here g2 is some coupling constant and q k is an
c c c ≪ F
conventionalFermiliquidbehavior. Wediscussconsequences ultraviolet cutoff. Interactions of this type have been
[
fortheeffectof disorderonhigher-dimensional Luttingerliq- studied by Bares and Wen [8], who showed that for

2 uids. η = 2(d 1) the quasi-particle residue vanishes log-
v arithmical−ly at the Fermi surface. The single-particle
5
Green’s function exhibits then anomalous scaling, sim-
0
ilar to the one-dimensional Tomonaga-Luttinger model.
0
In the regime η < 2(d 1) the quasi-particle residue re-
5
−
0 I. INTRODUCTION mainsfinite,andforη >2(d 1)thesingularitiesareeven
−
8 stronger than logarithmic. Note that η d corresponds
9 toaninteractionV(r r′)thatdiverges≥for r r′ .
t/ whIenthsepritceororfelaintetdenesleectthroeonrsetinicatlwoeffdoirmtse,nstihoensqcuaenstieoxn- Forsimplicity,wesha−llthereforerestrictour|se−lves|i→nt∞his
a
work to η < d [9]. At the first sight long-range interac-
m hibit a non-Fermi liquid (NFL) ground state still has no
tions of this type seem to be irrelevant to real physical
generally accepted answer. The solution to this problem
- systems. Note,however,thatstrongcorrelationsbetween
d may be the key to a proper understanding of supercon-
electronscaninsomecasesbedescribedintermsoflong-
n ductivity in the cuprates. There is little doubt that all
o experiments probing the normal state of the optimally range gauge forces. In fact, recently several authors [10]
c have used gauge field theories to explain the experimen-
doped cuprates can be consistently interpreted in terms
: tally observed non-Fermi liquid behavior in the normal
v of NFL behavior [1]. However, from the theoretical side
Xi there existsno agreementonthe microscopicmechanism stateofthehigh-temperaturesuperconductors[1]. Asal-
readypointedoutbyBaresandWen[8],thecarefulanal-
responsible for the experimentally seen NFL behavior.
r ysis of the long-range interactions of the type (1) might
a The difficulty lies in the fact that a possible breakdown
alsoshedsomelightontothe natureofthe Luttinger liq-
ofthe Fermiliquidstate cannotbe describedby treating
uid state due the coupling to gauge fields in d= 2. Our
the electron-electron interactions perturbatively. There-
approach is complementary to the philosophy adopted
fore,thedevelopmentofnon-perturbativemethodsindi-
in Refs. [11,12], where a model Green’s function of the
mensions d>1 is rather important.
Luttingerliquidtype ind>1isassumedwithoutmicro-
Recently, considerable progress has been made for the
scopic derivation. In this paper we shall show that not
special case that the interaction between the electrons
all of the well known properties of one-dimensional Lut-
is dominated by forward scattering. This means that
tinger liquids survive in d > 1. As discussed in Sec.IV,
the typical momentum q transfered by the interaction
this has importantconsequences for the effect of impuri-
has a magnitude that is small compared with the Fermi
ties on higher-dimensional Luttinger liquids.
1
II. THE SINGLE-PARTICLE GREEN’S where for later reference we have introduced a Debye-
FUNCTION Waller factor Qαα′(r,τ) with two patch indices α and
α′,
To begin with, let us briefly outline the calcula-
Qαα′(r,τ)=Sαα′(0,0) Sαα′(r,τ), (7)
tion of the single-particle Green’s function of interacting
−
fermionswithdominantforwardscatteringwithinhigher-
dsuimrfaencesioisnaslubbdosivoindiezdatiionnto[3a–7fi]n.itFeirnsutmobfearllM, thoefFsemrmalil Sαα′(r,τ)= β1 f[RiωPA(q,viωαm)qc][oisω(q·rv−α′ωmqτ]) .
patches with diameter Λ such that qc Λ kF. De- V qXωm m− · m− ·
≪ ≪
noting by kα, α=1,...,M a set of wave-vectorson the (8)
Fermi surface pointing to the centers of the patches, the
non-interacting energy dispersion ǫ is locally expanded Here is the volume of the system, and
k
ǫ µ vα q+q2/2mα, where µ = ǫ . In the V
kα+q kα
− ≈ · f
simplest approximation,the inverse effective mass 1/mα fRPA(q,iω )= q (9)
is set equal to zero. By doing this we neglect the curva- m 1+fqΠ0(q,iωm)
ture of the Fermi surface, so that the patches are com-
is the screened interaction within the random-phase ap-
pletely flat. Following Refs. [3,5–7,13], we represent cor-
proximation (RPA).
relationfunctionsasimaginarytimeGrassmannianfunc-
It should be mentioned that a priori it is not clear
tionalintegralsanddecouplethetwo-bodyinteractionby
whether it is allowed to linearize the energy dispersion
meansofHubbard-StratonovichfieldsVα(q,iω )(where
m (correspondingtotakingthelimit1/mα 0)ornot. As
ωm =2πm/β is a bosonic Matsubara frequency, β is the discussedinRef.[3],indimensionsd>1→thisapproxima-
inverse temperature). Integrating over the fermions, we
tion becomes more questionable than in d = 1, because
finallyarriveatthefollowingexpressionforthereal-space
for momenta q parallel to the Fermi surface the linear
imaginary-time Green’s function
termvα qvanishes,so thatthe quadratictermq2/2mα
·
is the leadingone. InRefs.[3,6]a systematicmethod for
M
G(r r′,τ τ′)= eikα·(r−r′) α(r,r′,τ,τ′) . (2) including into the bosonization procedure the effects as-
− − hG i sociatedwith the finiteness of1/mα hasbeen developed.
α=1
X Although the expressions derived in these works have so
Here ... denotesaveragingwithrespectto theeffective far not been analyzed in detail, it can be shown [3] that
h i
action of the Hubbard-Stratonovich field, and α is the for the calculation of the momentum distribution in the
G
solution of presence of the singular density-density interactions dis-
cussedhere it is indeed sufficient to work with linearized
2
∂ +ivα + ∇r Vα(r,τ) α(r,r′,τ,τ′)= energy dispersion.
− τ ·∇r 2mα − G If the system is a Fermi liquid, then the quasi-particle
(cid:20) (cid:21)
δ(r r′)δ∗(τ τ′), (3) residueforwave-vectorsclosetokαcanbeidentifiedwith
− −
[3,5]
wδ∗h(eτre τV′)αi(srt,hτe) an=ti-perqiωomdiceii(mq·ra−gωinmaτr)yV-tαim(qe,iδω-fmu)n,ctiaonnd. Zα =eSαα(0,0) . (10)
In th−e limit 1/mα P0 (i.e. for linearized energy disper-
→ On the other hand, non-Fermi liquid behavior in the
sion) Eq.(3) can be solved exactly [16,3,5,13], with the
single-particleGreen’sfunctionmanifests itselfinthe di-
result
vergenceofSαα(0,0)inthelimit andβ . In
α(r,r′,τ,τ′)=Gα(r r′,τ τ′)eΦα(r,τ)−Φα(r′,τ′) , fact, this quantity is closely relatVed→to∞the lead→ing∞non-
G 0 − − trivial vertex correction shown in Fig.1,
(4)
1
where Gα(r,τ) is the solution of Eq.(3) for Vα =0, and Λ1(k,iω˜n;q,iωm)=−β fRPA(q′,iωm′)
0 V qX′ωm′
Φα(r,τ)= ei(q·r−ωmτ) Vα(q,iω ). (5) ×G0(k+q′,iω˜n+m′)G0(k+q′+q,iω˜n+m′+m), (11)
iω vα q m
qXωm m− · where ω˜n = 2π(n+ 21)/β, and G0(k,iω˜n) = [iω˜n−ǫk+
µ]−1 is the non-interacting Matsubara Green’s function.
Furthermore, for 1/mα 0 the closed loop theorem
Noting that for linearized energy dispersion
→
[2,3,14] justifies the Gaussian approximation for the av-
eraging in Eq.(2), so that we obtain 1
G (kα+q,iω˜ )= , (12)
0 n iω˜ vα q
n
M − ·
G(r,τ)= eikα·rGα(r,τ)eQαα(r,τ) , (6) and comparing Eqs.(11) and (8), it is clear that in the
0
limit β we may identify [17]
α=1
X →∞
2
Sαα(0,0)=−Λ1(kα,0;0,0)≡−Λ1(0). (13) Qαα′(r,τ) √d κη2 Ωd−1 πdϑsind−2ϑ
Thus, Eq.(10) amounts to an exponentiation of the first ≈− 2 kFd−1 Ωd Z0
∞
non-trivial vertex correction. dq qd−2−η2e−2qqc 1 e−ωq|τ|cos(q r cosϑ) . (16)
Obviously,in the above derivation we have ignored all ×Z0 h − | | i
scattering processes that transfer momenta between dif-
HereΩ isthesurfaceareaofad-dimensionalunitsphere,
ferentpatchesontheFermisurface(theso-calledaround- d
and we have used ν =(Ω /(2π)d)mkd−2.
the-cornerprocesses). Onlythentheclosedlooptheorem d F
Let us now focus on the case η = 2(d 1). As men-
[2,3,14] is valid, which implies that the effective interac- −
tionedabove,in this case perturbationtheoryis plagued
tion in Eq.(8) is simply given by the RPA. The neglect
by logarithmic singularities. The long-distance behavior
of the around-the-corner processes is one of the main of Qαα′(r,0) can be easily extracted, and we obtain for
approximationsinherentinhigher-dimensionalbosoniza-
r
tion [3,4]. We are not awaveof any quantitative calcula- | |→∞
tion of the single-particle Green’s function which takes eQαα′(r,0) (q r)−γ , (17)
these processes explicitly into account. However, for ∼ c| |
q Λ it is reasonable to expect that these processes
c where the anomalous dimension is given by
≪
can indeed be neglected, provided the globaltopology of
theFermisurfacedoesnotplayacrucialrole. Thereason 1
√d κ d−1 dmg2 2
is thatforqc Λthe number ofaround-the-cornerscat- γ = = c . (18)
teringprocess≪esisalwaysparamatricallysmallerthanthe 2 (cid:18)kF(cid:19) 2d+1πd2Γ(d2)kFd !
number of forward scattering processes [3].
From this we can obtain the momentum distribution
Let us now evaluate the above expressions in the
nα(q) = n(kα +q) near the non-interacting Fermi sur-
case of the singular interaction of the type (1). Pre-
face. For γ <1 we find
viously, Bares and Wen [8] have considered only static
properties, such as the momentum distribution. To the
1 sin(πγ) qα γ
best of our knowledge, a calculation of dynamic proper- nα(q)= 1 sgn(qα)Γ(1 γ) 2 k , (19)
ties, such as the frequency-dependent density of states 2" − k − π2γ (cid:12)qc(cid:12) #
ρ(ω), cannot be found in the literature. Such a calcula- (cid:12) (cid:12)
(cid:12) (cid:12)
tion is straightforward within the framework of higher- whereqα istheprojectionofqontokα. Sim(cid:12) ila(cid:12)rlywecan
k
dimensionalbosonization. Thecalculationsaresimplified compute Qαα′(0,τ) for η =2(d 1). We find for τ
if one uses the fact that the non-Fermiliquid behavior is − →∞
γ
due to the existence of a collective plasmon mode ωq eQαα′(0,τ) κ (κv τ )−γ˜ . (20)
in the density-density correlation function. The long- ∼ q F| |
(cid:18) c(cid:19)
wavelength dispersion of this mode is [3,18]
where for 1 d < 2 the dynamic anomalous dimension
≤
v κ q 1−η/2 γ˜ is
F
ωq ∼ √d |κ| , (14) γ
(cid:18) (cid:19) γ˜ = . (21)
2 d
where κ=(g2ν)1/η, and ν is the non-interacting density −
c
Recall that we have excluded the case d = 2, which for
of states. It turns out that this mode completely deter-
η = 2(d 1) corresponds to η = 2 [9]. The small fre-
minestheinfraredbehaviorofthesingleparticleGreen’s
−
quency behavior of the density of states can now be cal-
function. Because for small wave-vectors ω v q,
q F
≫ | | culated, and we finally obtain
wemayrestrictthefrequencyintegrationinEq.(8)tothe
regime ω v q. Evidently,inthisregimetheterms
m F 1
vα qin|the|d≫enom|in|atorofEq.(8)canbeignored,which ρ(ω)= ImG(r=0,ω+i0+) ω γ˜ . (22)
· −π ∝| |
impliesthattheleadinglong-distanceorlong-timebehav-
ioroftheDebye-WallerfactorQαα′(r,τ)doesnotdepend We conclude that in dimensions 1 < d < 2 the singu-
on the patch indices α and α′. Furthermore,theRPAin- laritiesin the momentum distributionandin the density
teraction fRPA(q,iω ) can be replaced by its limit for of states of our higher-dimensional Luttinger liquid are
m
ω v q, which is given by characterized by different anomalous dimensions. Al-
m F
| |≫ | |
though we have shown this only for a specific class of
1 κ η ω2 models, we believe that this is a general property of
fRPA(q,iω ) m e−|q|/qc . (15)
m ≈ ν q ω2 +ω2 higher-dimensionalLuttinger liquids. The reason is that
(cid:18)| |(cid:19) m q ind>1theFermisurfaceconsistsofad 1-dimensional
−
Aftersomestraightforwardmanipulationswefindthatin continuum, so that loop integrations will always involve
the limit and β the dominant contribution some kind of angular averaging over the Fermi surface.
V →∞ →∞
to the Debye-Waller-factor can be written as Thisdestroysthesymmetrybetweenspaceandtimevari-
ables, which is only preserved in d=1.
3
III. THE POLARIZATION and the Debye-Waller factor is given by
Q2kα(r,τ)
Letusnowstudyhowsingularinteractionsofthetype
(1) affect the low-frequency behavior of the irreducible =Qαα(r,τ)+Qα¯α¯(r,τ) Qα¯α(r,τ) Qαα¯(r,τ)
− −
polarization Π (q,ω). First of all, let us recall that for 1 4(vα q)2
⋆ = fRPA(q,iω ) ·
small momenta q and small frequencies ω the leading β m [ω2 +(vα q)2]2
singular corrections to Π⋆(q,ω) cancel, so that V qXωm m ·
[1 cos(q r ω τ)]. (28)
m
× − · −
Π (q,ω) Π (q,ω) , q k , ω E , (23)
⋆ 0 F F
≈ | |≪ | |≪ Note that this Debye-Waller factor is different from
where Π is the non-interacting polarization. Eq.(23) is the Debye-Waller factor Qαα(r,τ) of the single-particle
0
aconsequenceofacompletecancellationofthemostsin- Green’s function. It should also be mentioned that for
gular self-energy and vertex corrections, which happens any finite number of flat patches the non-interacting po-
not only in d = 1 [14] but also in higher dimensions larization Π2kα(r,τ) exhibits in d > 1 logarithmic sin-
0
[2,3,5]. For q =O(k ) this cancellationdoes in general gularities due to the artificial nesting symmetry, which
F
| |
not occur. In fact, in the one-dimensional Tomonaga- is generated by replacing the curved Fermi surface by
Luttinger model it is well known that for momentum a finite number of flat patches. However, we have iso-
transfers q 2k the irreducible polarization (as well lated possible non-trivial effects due to interactions in
F
as various| o|t≈her two-particle correlation functions) ex- the Debye-Waller factor Q2kα(r,τ).
hibits anomalous scaling behavior, very much like the ItisnotdifficulttoshowthatinonedimensionEq.(28)
single particle Green’s function, although with different correctly reproduces the irreducible polarization of the
anomalous exponents [15]. We now show that this prop- Tomonaga-Luttinger model for momenta close to 2kF
erty of one-dimensionalLuttinger liquids does in general [15]. Note that in this case the large-distance and long-
not survive in higher dimensions. We would like to em- time decay of Π⋆(r,τ) is characterized by non-universal
phasize that the interaction of the type (1) has no mo- powerlaws,analogoustothesingle-particleGreen’sfunc-
mentum transfers of order O(k ), so that bosonization tion. The anomalous scaling manifests itself via a log-
F
can be used to sum the dominant singularities. It is the arithmic growth of the Debye-Waller factor (28) in the
external momentum transfer of the irreducible polariza- large-distanceorlong-timelimit. Theimportantobserva-
tion which is approximately 2k . tionisnowthatinthehigher-dimensionalnon-Fermiliq-
F
Within the same approximations as described in uid state due to the singulardensity-densityinteractions
Sec.II, the real-space, imaginary-time Fourier transform (1) the irreducible polarization Π2kα(r,τ) does not ex-
⋆
of Π (q,iω ) can be written as hibit any anomalous scaling. The case of the Tomonaga-
⋆ m
Luttinger model (corresponding to d = 1 and η = 0)
Π⋆(r r′,τ τ′)= M ei(kα−kα′)·(r−r′) imsosdpeeciisall,inbeaecraiunseqin, stohitshcaatsietthhaesctohlelecsatimvee polradsemroonf
− − −αα′=1 magnitude as the s|in|gle-particle excitation energy of the
X
α(r,r′,τ,τ′) α′(r′,r,τ′,τ) . (24) bare fermions. The fact that in d > 1 the leading sin-
×hG G i gularities of Qαα′(r,τ) cancel in Q2kα(r,τ) follows triv-
Performing the averaging within the Gaussian approxi- ially from the observation that the infrared behavior of
mation, we obtain the Debye-Waller factor Qαα′(r,τ) is independent of the
patch indices, so that the four terms in the second line
M
Π (r,τ)= ei(kα−kα′)·rGα(r,τ)Gα′( r, τ) of Eq.(28) cancel. As already mentioned in Sec.II, this
⋆ − 0 0 − − is due to the fact that for η > 0 the plasmon energy at
αXα′=1 long wavelengths is larger than vα q, so that the inte-
exp Qαα(r,τ)+Qα′α′(r,τ) grand in Eq.(28) contains an addit·ional small factor of
×
(cid:2)Qαα′(r,τ) Qα′α(r,τ) . (25) (vα ·q/ωq)2 as compared with the integrand in Eq.(8).
− − This factor cancels the leading singularity.
For q k theFouriertransformΠ (q,i(cid:3)ω )ofEq.(25) It is instructive to relate the constant part of the
F ⋆ m
is do|m|i≪nated by the diagonal terms kα = kα′, and we Debye-Waller factor (28) to vertex corrections, similar
recoverEq.(23). Similarly,for q 2kα k ,Eq.(25)is to Eq.(13). Consider therefore the leading interaction
F
dominated by kα = kα′, so t|ha−t the|F≪ourier transform correctionstotheirreduciblepolarizationformomentum
of Π (2kα+q,iω )−is for q k given by transfers close to 2kα shown in Fig.2. Calculating these
⋆ m | |≪ F correctionsinrealspaceandimaginarytime,wefindthat
Π2⋆kα(r,τ)=Π20kα(r,τ)eQ2kα(r,τ) , (26) tQhαeαy′(cra,nτb)edeexfipnreedssiendEinqst.e(r7m)saonfdth(8e)Daesbfyoel-loWwasl,lerfactor
where (using the notation kα¯ =−kα) Πα1,αself(r,τ)=2Πα0α(r,τ)Qαα(r,τ), (29)
Π2kα(r,τ)= ei2kα·rGα(r,τ)Gα¯( r, τ), (27) Παα′ (r,τ)= 2Παα′(r,τ)Qαα′(r,τ). (30)
0 − 0 0 − − 1,vertex − 0
4
From these expressions it is obvious that an expan- a very important role, so that even for a δ-function cor-
sion of Eq.(28) to first order in the RPA-interaction ex- related random potential the low-frequency behavior of
actly reproduces perturbation theory. Not surprisingly, τ (ω) and τ (ω) can be completely different.
1 tr
bosonization amounts to an exponentiation of the lead- To substantiate this point, let us use the above non-
ing perturbative corrections in real space and imaginary perturbative results of Sec.II and III to study the effect
time. Just like in the case of the single-particle Green’s of disorder on the clean system perturbatively. As usual
function (see Eq.(13)), we can identify the constant part we model the disorder by coupling the charge density to
S2kα(0,0)ofthe Debye-Waller factor(28) with a certain aGaussianrandompotentialU(r)withzeroaverageand
combination of vertex corrections. Assuming again that δ-function correlations
S2kα(0,0) is finite, we see that in the limit of large dis-
tancesorlongtimesinteractionssimplyleadtoanoverall U(r)U(r′)=γ0δ(r r′), (33)
−
multiplicative renormalizationofthe non-interactingpo-
larization by a factor of eS2kα(0,0). Note that S2kα(0,0) wheretheoverlinedenotesaveragingovertheprobability
distribution of U(r) and the parameter γ is a measure
is always greater than zero and is finite for interactions 0
of the strength of the disorder. Within the lowest order
of the type (1) in dimensions d > 1. From Eq.(11) it is
Bornapproximationthe disordergivesriseto the follow-
easy to see that
ing self-energy correction to the single-particle Green’s
Sαα′(0,0)= Λ (kα′,0;kα kα′,0) function,
1
− −
Λ1(kα kα′ ). (31) Σ (ω)= γ0 G(k,ω)=γ G(r=0,ω), (34)
≡− | − | 1 0
It follows that S2kα(0,0) can also be written as V Xk
whereGis the Green’s functionofthe interactingmany-
S2kα(0,0)=2[Λ (2k ) Λ (0)], (32)
1 F − 1 body system without disorder. The single-particle scat-
tering time τ (ω) is defined by
which should be compared with Eq.(13). From Eq.(32) 1
we can obtain an alternative explanation why for η
2(d 1)thenon-Fermiliquidnatureofthesingle-partic≥le 1/τ1(ω)=−2ImΣ1(ω+i0+). (35)
−
Green’s function does not affect the polarization: al-
Using Eq.(22), we see that in our model for η =2(d 1)
though in this case both vertex corrections Λ (2k ) and −
1 F the single-particle scattering time diverges within lowest
Λ1(0) are divergent, their difference Λ1(2kF) − Λ1(0) order Born approximation as τ1(ω) ω −γ˜, where the
remains finite. In this respect our higher-dimensional ∝ | |
dynamic anomalous dimension γ˜ is given in Eq.(21). As
non-Fermi liquid behaves fundamentally different than
shown in Ref. [12], this result turns out to be only self-
the one-dimensional Tomonaga-Luttinger model, where
consistent for γ˜ > 1/2. For smaller γ˜ this divergence is
Λ (2k ), Λ (0), and the difference Λ (2k ) Λ (0) are
1 F 1 1 F 1 removedif one usesinsteadofEq.(34)theself-consistent
−
all singular.
Born approximation, which amounts to the replacement
G(k,ω) [G−1(k,ω) Σ (ω)]−1 (36)
1
IV. DISORDERED LUTTINGER LIQUIDS IN → −
HIGHER DIMENSIONS on the right-hand side of Eq.(34). However, even if
τ (ω) diverges, it would be incorrect to conclude that
1
The conventionalstrategyto study interactionsindis- the conductivity and the diffusion coefficient diverge as
ordered Fermi systems is to treat first the effect of dis- well. Such a conclusion is only valid as long as vertex
order as accurate as possible, and only after that incor- corrections are unimportant, which is certainly not the
porate the effects of interactions perturbatively. It has caseforhigher-dimensionalLuttingerliquids. Toseethis
recently beenpointed outby severalauthors [12,19]that more clearly, one should keep in mind that the Drude
forstronglyinteractingFermisystemsthisapproachmay formula σ = ne2τ /m∗ for the conductivity (n is the
tr
not correctlycapture the interplaybetweendisorder and density of the electrons and m∗ is their effective mass)
interactions in low dimensions. In fact, in some cases involves the transport scattering time τ , which for in-
tr
it may be better to solve first the interaction problem teracting fermions is in general not identical with the
without disorder,and then study the additional effect of single-particle scattering time τ defined in Eq.(35). Mi-
1
disorder. Here we would like to point out that in this croscopically,the transportscatteringtime τ (ω)canbe
tr
approach it is crucial to distinguish between the single- calculated from the imaginary part of the memory func-
particle scattering time τ (ω) (which is defined in terms tion M(ω+i0+) [20,21],
1
ofthesingle-particleGreen’sfunction)andthetransport
scattering time τtr(ω) (which is related to the conduc- 1/τtr(ω)=ImM(ω+i0+). (37)
tivity). This difference is important even if the impurity
potential is dominated by s-wave scattering. The rea- Forarbitrarycomplexfrequenciesz thememoryfunction
son is that in non-Fermi liquids vertex corrections play is defined in terms of the conductivity via
5
ne2 i momentumtransferscloseto2k doesnotshowanynon-
σ(z)= . (38) F
m∗ z+M(z) Fermi liquid behavior. Although we have obtained these
results for a rather special class of models, we believe
Expanding M(z) to first order in the impurity concen- that they are quite generaland reflectgeneric properties
tration, one obtains in d dimensions [20] of higher-dimensional Luttinger liquids. In the language
ofmany-bodyperturbationtheory,the qualitativediffer-
γ q2 Π(q,z) Π(q,0)
0 ence between the low-frequency behavior of the single-
M(z)= − . (39)
dn m∗ z particle Green’s function and the polarization is due to
V q
X theimportanceofvertexcorrections,whichind>1tend
Thus, to lowestorderin γ the transportscatteringtime tocancelingaugeinvariantcorrelationfunctions suchas
0
is determined by the (reducible) polarization Π(q,z) of the polarization, even for large momentum transfers.
the interacting many-body system. Of course, for non- We would like to point out that our results have been
interacting electrons and a δ-function correlatedrandom obtained for singular density-density interactions. Re-
potential (i.e. s-wave impurity scattering) both defini- centlymanyauthors[10,23]havestudiedanotherclassof
tions (35) and (37) lead to identical results for the scat- higher-dimensionalnon-Fermiliquids,wherethecurrent-
tering times in the limit ω 0. However, for the class current interaction mediated by transverse gauge fields
→
of higher-dimensional Luttinger liquids studied in this is responsible for the non-Fermi liquid behavior. Our
paper the low-frequency behavior of τ (ω) and τ (ω) is results are not applicable to these more complicated in-
1 tr
completely different. This is obvious from the fact that teractions. Incontrasttothedensity-densityinteractions
in these systems the polarization does not exhibit any discussed here, for current-current interactions it is not
anomalous behavior, so that τ (0) is simply a constant. allowedto ignore the curvature of the Fermi surface in a
tr
Thus, to lowest order in the impurity concentration the bosonization approach. Although we know [3,6] how to
static conductivity σ(0) is finite. Note that this implies includecurvatureeffectsintothebosonizationcalculation
also a finite diffusion coefficient = σ(0)/(e2∂n/∂µ), of the single-particle Greens function, it is not straight-
D
where ∂n/∂µ = lim Π (q,0) is the compressibility forward to generalize this method for the polarization.
q→0 ⋆
[22]. Hence,withinaperturbativeapproachtolowestor- Finally, in Sec.IV we have shown that in the pertur-
derintheimpurityconcentrationthereisnosignforadi- bative treatment of disorder in higher-dimensional Lut-
vergentconductivityinourhigher-dimensionalLuttinger tingerliquids itis veryimportantto distinguishbetween
liquids,evenintheregimewherethe single-particlescat- the single-particle scattering time τ1(ω) and the trans-
teringtime τ1(ω)divergesforω 0. Ofcourse,the true portscatteringtimeτtr(ω),even if the impurity potential
low-frequency behavior of the c→onductivity most likely is dominated by s-wave scattering. In particular, a pos-
requires a non-perturbative treatment of disorder, and sible divergence of τ1(ω) does not necessarily indicate a
cannot be calculated by simply expanding the memory similardivergenceoftheconductivityorthediffusionco-
function to first order in the impurity concentration. efficient.
V. SUMMARY ACKNOWLEDGEMENTS
Inthisworkwehaveusedhigher-dimensionalbosoniza- We would like to thank Sudip Chakravarty for discus-
tion to study the correlation functions of a particular sionsandtheDeutscheForschungsgemeinschaftforfinan-
class of higher-dimensional non-Fermi liquids, which are cial support.
characterizedby singulardensity-density interactionsdi-
verging in momentum space as q−η for small momen-
tum transfers q. We have paid particular attention to
the case η = 2(d 1) in dimensions 1 d < 2, which
− ≤
can be considered as a possible generalization of the
Tomonaga-Luttinger model to higher dimensions. The
∗
main result of this work is the observation that in d>1 Address from October 1997 - April 1998.
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the Coulomb interaction in a plane that is embedded in
three dimensional space, while η =2 corresponds to the tex. Here, k is short for (k,iω˜n) and q is short for (q,iωm).
The double wavy line is the RPA screened interaction, the
Coulomb potential in a strictly two-dimensional world.
In the latter case V(r−r′) ∝ ln(|r−r′|). The collec- wavy line is the external field, and the solid arrows denote
non-interactingsingle-particle Green’s functions.
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7