VOLUME 80, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JANUARY 1998

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Interchain Coherence of Coupled Luttinger Liquids at all Ordersin Perturbation Theory

Enrico ArrigoniInstitut für Theoretische Physik, Universität Würzburg, D-97074 Würzburg, Germany

(Received 18 August 1997)

We analyze the problem of Luttinger liquids coupled via a single-particle hopping parametert'

and introduce a systematic diagrammatic expansion in powers oft'. An analysis of the scalingof the diagrams at each order allows us to determine the power-law behavior versust' of theinterchain hopping and of the Fermi surface warp. In particular, for strong interactions, we fithat the exponents are dominated by higher-order diagrams producing an enhanced coherencefailure of linear-response theory. Our results are valid at any finite order int' for the self-energy.[S0031-9007(97)05115-6]

PACS numbers: 71.10.Pm, 05.30.Fk, 71.10.Hf, 71.15.–m

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The problem of crossover from one to higher dimensions has recently received particular interest [1–9]. Tpuzzling question is whether and how an infinite row oone-dimensional chains of Luttinger liquids (LL) developinterchain coherence and possibly goes over into a Ferliquid state for an arbitrarily weak single-particle hoppint' ø EF between the chains (EF being the Fermi en-ergy), and how this is affected by the correlation expnent a [10] of the unperturbed LL which parametrizethe intrachain interaction. No clear agreement has beereached so far, and different analyses have provided cflicting answers: (i) The system might go over to a Fermliquid state for arbitrarily smallt'; (ii) there might bea finite t' (depending ona), below which the systemsremains in a LL state and no coherence is developbetween the liquids [6]; or finally (iii) one could have adifferent state, possibly with gaps and long-range ordThis ambiguity is due essentially to the fact that the peturbation introduced byt' is relevant[3–5]. This meansthat, no matter how smallt', perturbative results int'

will always be inaccurate for sufficiently low values osome characteristic energyE & E0 ­ st'yEa

F d1ys12ad.Here,E is the largest energy scale between (i) the extenal frequencyv, (ii) the momentum measured from theFermi surfacejk 2 kF j [10], and (iii) the temperatureT .Indeed, in the interesting regionE , E0, where inter-chain coherence starts to set on [5], all perturbation terare of the same order and no definitive prediction canmade about the nature of the ground state. An alternatapproach to this topic is to consider the limit of strong foward scattering, where one can use the higher-dimensiobosonization method [7] or the Ward identities [4]. In thilimit, one can show that a system always becomes a Feliquid in dimensions greater than one if the interactionnot too singular.

In this Letter, we analyze how, as a function ofa,interchain coherencedevelops whent' is switched on.The striking point of the present work is that the resulare valid atany finite orderin t' for the single-particle

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self-energy, in contrast to previous work [2,5,6,8]. Thisimportant because it fixes some essentially exact resultwhich approximate theories should be compared, althouit is clear that an answer to the more interesting quest“Fermi liquid or not” to this degree of accuracyisextremely difficult and will not be addressed here.addition, we show that restricting to lowest-order termmay give incorrect results even in the small-t' limitfor a larger than a certain value. In particular, wheconsidering higher-order diagrams, interchain coherenis increased, and linear-response theory does not woSpecifically, we evaluate the exponent of the power-labehavior int' (i) of the interchain hopping of a particleat the Fermi momentum and (ii) of the Fermi surface (Fwarp. To calculate these quantities we extend a methdeveloped in Ref. [12] to the case of coupled LL anrewrite diagrammatically the perturbation expansion int'

introduced in Ref. [5]. We then consider the low-energscaling behavior of the diagrams at all orders int' andnotice that the scaling is no longer homogeneous fora .

a2p (cf. also Refs. [5,9]). Finally, we use this scalinbehavior to calculate the exponents mentioned above.

We consider a system ofN coplanar coupled identicalchains lying parallel to thex axis with Hamiltonian,H ­P

i HLLsid 1P

ijrs t'ij

Rdxcy

r ,ssx; idcr ,ssx; jd, whereHLLsid describes a (uncoupled) Luttinger liquid in chaii, t'ij is a hopping term between chaini and j, andcr ,ssx; id fcy

r ,ssx; idg is the destruction (creation) operatofor a right-moving sr ­ 11d or left-moving sr ­ 21dfermion at sitex in the chaini with spin s (we will alsoconsider the spinless case). The diagrammatic expansin t around the atomic limit of the Hubbard model introduced by Metzner in Ref. [12] can be readily extendeto the present case by considering perturbations int'

about the exactly solvable Hamiltonian for the isolateLuttinger liquids

Pi HLLsid. We consider the diagram-

matic expansion for the Green’s functionG skk, iv; k'dexpressed in the imaginary frequencysivd and momen-tum representation, wherekk and k' label the Fourier

© 1998 The American Physical Society

VOLUME 80, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JANUARY 1998

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transforms of thex and i components, respectivelyUnless otherwise specified,kk will also implicitly containthe other in-chain quantum numbersr and s. Theexpansion ofGskk, iv; k'd in t' is obtained by drawingall connected diagrams composed of two external linand of an arbitrary number of vertices connected wdirected lines, each vertex having a number of enterilines equal to the number of leaving lines (see Fig. 1One then labels each internal linesld by a momentumsksld

k , ksld' d and a frequency variablesivsldd, and the

external lines bykk, k', and iv. Each internal line

sld yields a factor t'fksld' g (the Fourier transform of

t'ij). Each vertex with2n legs s2n $ 2d acquires a

contribution, C0nsks1d

k ivs1d, . . . , ksndk ivsnd j k

s1d0

k ivs1d0

, . . . ,

ksnd0

k ivsnd0 d, where C0n is the n-particle cumulant of the

isolated chain (for example,C02 ­ G0, the LL Green’s

function), andksldk ivsld, k

sld0

k ivsld0

are the frequency andmomenta (which are conserved at the vertex) of tentering and leaving lines, respectively (cf. [5,12]). Thintroduction of higher-order cumulants and verticesnecessary, since the correlation functions of thet' ­ 0Hamiltonian do not satisfy Wick’s theorem. Eventually, one has to multiply each diagram by the ususymmetry and fermion factors, and integrate (or sumover all internal frequencies and momenta. This isdiagrammatic representation of the expansion introducby Boies and co-workers in Ref. [5]. Similar to conventional diagrammatic theory (cf. also Ref. [12]), oncan introduce a “self-energy”Gskk, iv; k'd21. The“inverse self-energy” Gskk, iv; k'd is then obtainedby summing all one-particle irreducible contributionto Gskk, iv; k'd [see Fig. 1(c)]. One then obtainsan analogue of the Dyson equation [cf. Fig. 1(b)G skk, iv; k'd ­ sGskk, iv; k'd21 2 t'fk'gd21.

At this point, it is clear that the approximation used iRefs. [2,5,6,8] (and others) consists of considering onthe contribution toG from the first diagram (the dot,a)in Fig. 1(c) [“single-dot” approximation (SDA)] and thususingG ­ G0. Furthermore, one can obtain the limit o

FIG. 1. Diagrammatic expansion int' of the single-particleGreen’s functionG (gray box). Directed lines give a contri-bution t'fk'g, and a dot withn entering andn leaving linescontributes a factorC0

n (n-particle cumulant). (a) Exampleof single-particle irreducible and reducible contributions toG .(b) Dyson’s equation forG in terms of the inverse self-energyG (gray disk). (c) Example of diagrams contributing toG(some of them are discussed in the text).

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infinite coordinationD ! ` by summing an infinite seriesof “bow” diagrams with “full” self-consistent Green’sfunctions as internal lines (cf. Ref. [12]). Neverthelesswe briefly review the results obtained by others [2,5,8within the SDA. Here, the system (both spinless anspinful) behaves as a Fermi Liquidfor any values of thecorrelation exponenta , 1 [10] in the sense that there isa warped (i.e.,k'-dependent) FSkkFfk'g on which theGreen’s function has a (real) pole with finite weight av ­ 0, and there is afinite region in momentum spacearoundkkFfk'g where the pole survives, remains real, ancontinuously shifts to higher binding energies [13]. Thequasiparticles are thus well defined everywhere on the F(except at thespecialpoints for whicht'fk'g ­ 0). Otherpoles [6] do not describe true quasiparticles, since theimaginary part is always of the same order as their energ

However, the trouble starts to arise when one triesconsider higher-order diagrams. A simple power-countinargument indeed shows that a diagram of ordertn21

' forG diverges such asE2na

F E nsa21d in the low-E limit.If one tries to keep theeffective expansion parame-ter t'E2a

F E a21 small, then thet'fk'g term in the de-nominator ofG skk, iv; k'd becomes much smaller thanGskk, iv; k'd21 and the system behaves as a sum of uncoupled LL. On the other hand, as soon as one tries to aproach the pole, the approximation fails and higher-orddiagrams become of the same order of magnitude. Fthis reason, nothing reliable can be said about possibFermi-liquid behavior at this order of perturbation, and iis mandatory to consider higher-order diagrams. In addition, the above power-counting argument does not hofor all values ofa. For a greater than a certaina2p,some diagrams turn out to diverge stronger at low frequecies (cf. also Refs. [3,5,9]). For example, the contributioto t'Gsx1 2 x2; k'd [14] from the diagramg [Fig. 1(c)],with internal lines 3, 4, 5 taking ther indices1, 2, 2, re-spectively, is proportional to [15]µ

t'

EaF

∂4

cosk'

Z 5Yi­3

d2xisjx1 2 x3j jx4 2 x5jd212a

3

∑µjx1 2 x5j jx3 2 x4j

jx1 2 x4j jx3 2 x5j

∂2B

2 1

∏3 e2ifargsx12x3d2args x42x5d 3 sterm withx1 ! x2d , (1)

where jxj ;p

x2 1 sjgj 1 ad2, argx ; argsx 1 itd,and the exponentB ­ s1yKr 2 Krdys2Sd. A naivedimensional analysis yields a term~st'yEa

F d4 3

jx1 2 x2j224a , giving a contribution propor-

tional to ft'ysEaF E 12adg4 to the Fourier transform

t'Gskk, iv; k'd. This is correct forE ø EF if theintegral does not diverge at small distances fora ! 0,i.e., as long asB , 1. For B . 1 [10], one picks up ana-dependent contributiona222B , E2B22

F , which must bebalanced byE 222B to give the correct energy dimensionsThis produces a contribution tot'Gskk, iv; k'd propor-tional to ft'ysEa

F E 12adg4sEFyE d2B22, i.e., a strongerdivergence. A similar analysis shows that the 7-leg

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VOLUME 80, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JANUARY 1998

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diagramd in Fig. 1(c) and its generalizations with2n 1 1internal legs (n odd integer) produce a contribution tot'G proportional toft'ysEa

F E 12adg2n12sEFyE d2nsB21d.In fact, this is the strongest low-energy divergence oncan attain for a contribution tot'G at a given ordert2n12

' .The regimeB . 1 corresponds toa . a2p with a2p ­sp

S2 1 1 2 1dyS, i.e., where two-particle processesbecome more relevant than one-particle processes [5,9

We now use the above results to calculate the powelaw behavior for smallt' of the quantity [14]

DnkF ~Z `

2`dvfG skF , iv; pd 2 G skF , iv; 0dg . (2)

DnkF is also equal to the expectation value of the singleparticle hopping operator atkF in the two-chain caseand is thus a measure of thecoherence[6] of single-particle hopping. Fork ­ kF and T ­ 0, the onlyremaining energy scale that can be associated withE

is v. For a , a2p all diagrams scale in the sameway as discussed above. Therefore,at any order in thet' expansionand for v ø EF the difference betweenthe Green’s function of the two bandsGskF , iv; pd 2

G skF , iv; 0d can be written (say, for Imiv . 0) asE2a

F sivda21gft'E2aF sivda21g, where gfxg is a scaling

function. The change of variablesv ­ xt1ys12ad' gives

DnkF~ st'yEFdays12ad times a dimensionless integral,

which converges at largex for a ,12 . To assure

convergence at low frequencies it is essential to expathe self-energysG21d in t' and not directly the Green’sfunction, in order to avoid anunphysicallow-frequency(or low-T) diverence. Fora .

12 the integral diverges for

largex, and one needs to introduce a large-energy cutoyielding a dominant linear behaviorDnkF

~ t'yEF . Thesame exponents would have been obtained by simpusing the SDA forG. Notice also that this result isconsistent with the one in Ref. [4].

In the a . a2p regime, one cannot straightforwardlyextend the above discussion since the diagrams give nhomogeneous contributions. If one stops the expansiin t' for the self-energyat an arbitrary finite ordert

m011' ,

one obtains (restricting to dominant terms at each order

G skF , iv; pd21

­ t'

m0Xm­21

amfsivda21E2aF t'gm

µivEF

∂s12Bdfmg4

, (3)

where fmg4 is defined as4 intfsm 2 2dy4g 1 2, “int”being the (lower) integer part, and theam are constantssa0 ­ 1d possibly dependent on the spin and chargvelocities. GskF , iv; 0d21 is obtained from the sameexpression by replacingt' ! 2t'. By inserting (3)into the integral (2), one can show that the integrais dominated by the constant and by thet

m̄011' terms

in (3). Carrying out the integral yieldsDnkF ~ tRfm̄0g' ,

where the exponentRfm̄0g approaches quite rapidly itsm̄0 ! ` limit R ­ aysB 2 ad. For R . 1 this termwill again be shaded by the linear term as in the ca

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a , a2p. In Fig. 2, we show thedominantand somesubdominant exponents for the spinless and spinful caas a function ofa. As one can see, the occurrence othe two-particle regimesa . a2pd reduces the dominantexponent in the spinless case, and, in particular, it shupwards (from a .

12 to a .

23 ), the region where

DnkFbecomes linear int'. It is interesting to compare

this result with the prediction of linear-response theorwhich is expected, for sufficiently smallt', either togive the correct behavior or to diverge, thus signalina sublinear behavior. The linear-response resultDnk ­t'sA 1 Bjkk 2 kF j2a21d indeed shows a divergence akk ­ kF for a ,

12 , i.e., it predicts a sublinear behavio

in this region. However, our calculation shows that,the spinless case,DnkF

is sublinear int' in a largerregion, namely, up toa ­

23 . Unfortunately, this effect

is more difficult to see in the case of electrons witspin, since, in this case,a2p ø 0.62 .

12 and the effect

is “shaded” by the linear behavior occurring fora .12 .

Nevertheless, this effect should be detectable, e.g., infirst derivative of DnkF with respect tok or in the FSwarp, as discussed below. Notice that the exponentRhas been obtained by cutting the series int' for the self-energyG21 at a given order. Fora . a2p, a differentchoice, such as, e.g., cutting the seriesfor G, may lead toa different (although in this case unphysical) result, whichowever, preserves the qualitative effects, in particular tdecrease of the exponentR. It should be mentioned thatrecently Capponi, Poilblanc, and Arrigoni [16] evaluatethe exponent ofDnkF

by exact diagonalization of smallladders supplemented by a careful finite-size extrapolatcarried out by means of an appropriate scaling functioThese numerical results are in very good accordance wthe exponents ofDnkF predicted here and clearly showthe change of behavior between the two regimesa , a2p

anda . a2p.

FIG. 2. Exponents controlling the behavior as a functioof t' of DnkF (thick solid line), and of the FS warpDkF (thick dashed line) for theN-chain system. Thin linesindicate subdominant exponents forDnkF : ays1 2 ad (solidline), aysB 2 ad (dashed line), and unity (dotted line). Theexponents are plotted as a function of the correlation expona for the spinless and spinful cases.

VOLUME 80, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JANUARY 1998

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We now study the behavior of the Fermi surfacas a function oft'. The FS consists of the pointskkFfk'g given by the solution of the equation [10]ReGskkFfk'g, iv ­ 01; k'd21 ­ 0. As already dis-cussed [2,5,8], within the SDA a solution exists angives kkFfk'g 2 kF ~ jt'fk'gj1ys12ad. At higher order,G skk, iv ­ 01; pd21 has the same form as (3), withivreplaced bykk 2 kF and with different coefficientsan,wherebyB is formally replaced by 1 fora , a2p . Theequation for thesk' ­ pd pole can be written in termsof a scaling functionf as fft'yskk 2 kFd12ag ­ 0. Ifffxg ­ 0 has a solution at some pointx ­ xp, then thebehavior of the FS is similar to the one for the SDAi.e., DkF ; kkFfpg 2 kkFf0g ­ st'yxpd1ys12ad [17].This quantity measures the FS warp produced byt'.However, it is not easy to find out whether the imaginarpart of G21 vanishes fast enough at the FS, so that thabove result does not necessarily imply that the systegoes into a Fermi-liquid state. Moreover, considerinhigher-order terms forffxg, the solution of ffxg ­ 0may not exist. In fact, several gaps open in the differemodes of aN-chain system [18], which could possiblyprevent the pole equation from having a solution fosomeN and somek' points of the FS. In this case, as ameasure of the FS warp, one could define a “pseudo”by the pointkk for which jG21j has a minimum. Alsoin this case, the FS warp turns out to behave the saas t

1ys12ad' . Finally, for a . a2p the behavior for small

t' is again dominated by the dot diagramsad and bythe higher-order diagrams of typeg and d, as for thecalculation ofDnkF

. In a similar way, one obtains, for

the FS warp,DkF ~ tRya' (cf. Fig. 2). This means that

the Fermi surface is warped even fora ! 1, in contrastwith the SDA result and with the expectation cominform the fact thatt' is “irrelevant” for a . 1. Notice,however, that particle-hole instabilities producing a gaat the Fermi surface are likely to occur for largea.

In conclusion, using a diagrammatic representationthe expansion in power oft', we have determined thepower-law behavior as a function oft' of the differ-ence in occupationDnkF

and of the FS warp in aN-chainsystem, our results being exact atany finite orderfor theself-energy. In the single-particle regimesa , a2pd, theexponents are given correctly by the low-order approxmations. In the two-particle regimesa . a2pd higher-order diagrams give dominant contributions and reduthe exponents (thus indicating an increased coherenwith respect to the SDA result. In this regime, linearresponse theory int' is not reliableeven in thesmall-t'

limit. The increase of the exponents witha, as seen inFig. 2, shows that transverse coherence is reduced withcreasing interaction but never completely suppressed sinDnkF and the Fermi surface warp are finite for any finitvalue of t' also for largea. We expect, however, thatat some value ofa, a gap opens due to particle-hole instabilities, possibly producing incoherence effects in th

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interchain transport. For a finite number of chains thegaps seems to open even for smalla [18]. It would beinteresting to establish whether a finite critical value ofa

exists in theN ! ` limit.I am grateful to J. Voit for stimulating my interes

in the topic of quasi-one-dimensional systems. I alacknowledge many interesting and fruitful discussionwith M. G. Zacher, S. Capponi, D. Poilblanc, B. BrendeG. Hildebrand, and W. Hanke. This work was supporteby the EC-TMR program ERBFMBICT950048.

[1] L. P. Gor’kov and I. E. Dzyaloshinskii, Sov. Phys. JETP40, 198 (1974); D. G. Clarke, S. P. Strong, and P. WAnderson, Phys. Rev. Lett.72, 3218 (1994).

[2] X. G. Wen, Phys. Rev. B42, 6623 (1990).[3] H. J. Schulz, Int. J. Mod. Phys.5, 57 (1991).[4] C. Castellani, C. Di Castro, and W. Metzner, Phys. Re

Lett. 69, 1703 (1992);ibid. 72, 316 (1994).[5] D. Boies, C. Bourbonnais, and A.-M. S. Tremblay, Phy

Rev. Lett.74, 968 (1995).[6] D. Clarke and S. Strong, J. Phys. Condens. Matter9, 3853

(1997).[7] A. Houghton and J. B. Marston, Phys. Rev. B48, 7790

(1993); P. Kopietz, V. Meden, and K. SchönhammePhys. Rev. Lett.74, 2997 (1995).

[8] K. Schönhammer, cond-mat/9710343; A. M. Tsvelikcond-mat/9607209.

[9] S. A. Brazovskii and V. M. Yakovenko, Sov. Phys. JETP62, 1340 (1985).

[10] Throughout this paper, we use the conventionsRef. [11] for the correlation exponentsa andKr , and re-strict to a, Kr , 1. Moreover, we set thenoninteractingFermi velocityyF ­ 1 and the chemical potentialm ­ 0.kF indicates the Fermi momentum of thet' ­ 0 system.

[11] J. Voit, Rep. Prog. Phys.58, 977 (1995).[12] W. Metzner, Phys. Rev. B43, 8549 (1991).[13] Notice that, in this case, one has a special Fermi liqu

since the imaginary part of the self-energy isidenticallyzero in a finite region aroundkkFfk'g.

[14] For definiteness, we consider the right-moving brancwith external indexr ­ 1.

[15] Here,G is expressed in real space and imaginary time fthe intrachain coordinatesx ; sx, itd and in momentumspace for the interchain coordinate. For simplicity, wwill set the spin and charge velocities equal toyF s­1d. This does not change the scaling behavior we aconsidering. In the following, the indexS equals 1 (2)in the spinless (spinful) case,a ~ E21

F is a short-distancecutoff, andT ­ 0. Moreover, we takeN even and restricttij to nearest neighbors. Notice that the21 in squarebrackets in Eq. (1) comes from the disconnected pathat has to be subtracted in order to obtain the cumulaC0

4 . This term is important, otherwise the integral wouldiverge in the infinite-volume limit.

[16] S. Capponi, D. Poilblanc, and E. Arrigoni, Phys. Rev.(to be published).

[17] Here, we used the symmetrykkFf0g ­ 2kF 2 kkF fpg.[18] See, e.g., H.-H. Lin, L. Balents, and M. P. Fisher, Phy

Rev. B (to be published).

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