PHYSICAL REVIEW B 15 SEPTEMBER 1997-IIVOLUME 56, NUMBER 12

Renormalization-group approach to the metal-insulator transitions in „DCNQI …2M„DCNQI is N,N8-dicyanoquinonediimine andM 5Ag, Cu…

K. YonemitsuInstitute for Molecular Science, Okazaki 444, Japan

~Received 17 March 1997; revised manuscript received 30 May 1997!

Metal-insulator transitions and different ground-state phases in the quasi-one-dimensional materials,(R1R2-DCNQI)2M (R15R25CH3, I andM5Ag, Cu!, are studied with a renormalization-group method. Weuse one-dimensional continuum models with backward scatterings, umklapp processes, and couplings with 2kF

and 4kF phonons~not static lattice distortion!. We take a quarter-filled band forM5Ag and a sixth-filled bandcoupled with a third-filled band forM5Cu. Depending on electron-electron and electron-phonon couplingstrengths, the ground-state phase becomes a Tomonaga-Luttinger liquid or a state with a gap~s!. For M5Ag, there appears a spin-gap state with a dominant 2kF charge-density-wave correlation, a Mott insulatorwith a dominant 4kF charge-density-wave correlation, or a spin-Peierls state with different magnitudes of spinand charge gaps. Three dimensionality is taken into account by cutting off the logarithmic singularity in eitherthe particle-particle channel or the particle-hole channel. The difference between the ground-state phase of theR15R25CH3 salt ~spin-Peierls state! and that of theR15R25I salt ~antiferromagnetic state! is qualitativelyexplained by a difference in the cutoff energy in the particle-particle channel. ForM5Cu, there appears a Mottinsulator with a charge-density wave of period 3 and a Peierls insulator with a charge-ensity wave of period 6.The conditions for the experimentally observed, Mott insulator phase are strong correlation in the sixth-filledband, moderate electron-phonon couplings, and finite electron-4kF phonon coupling. Resistance is calculatedas a function of temperature with a memory-function approximation in both cases above. It qualitativelyreproduces the differences among theM5Ag and M5Cu cases as well as theR15R25CH3 and R15R2

5I cases.@S0163-1829~97!06035-9#

-f a

b

ate

-liora

el

hgitatie

is

etol-ini-

e,

le

Inanyofss.

hebyionhat

entsngbedt

gerex-re-

ge-

ng-ingle-

I. INTRODUCTION

Quasi-one-dimensional materials, (R1R2-DCNQI)2M~M5Ag, Cu!, show various phases. ForM5Cu, the hybrid-ization of DCNQI p orbitals and Cud orbitals causes in-triguing transport1 and magnetic2 properties. The metalinsulator transition is accompanied by the formation ocharge-density wave~CDW! of period 3.2,3 The low-temperature phase is a Mott insulator, which is causedstrong correlation of d electrons in cooperation withelectron-lattice coupling and cannot be described by a bpicture. In fact, the spin susceptibility in the insulating stais enhanced over the temperature-independent Pauliparamagnetism seen in the metallic state. It has called ftheoretical studies, most of which have used mefields4—the Hartree-Fock approximation and the mean fiapproximation for slave bosons. The 1/N correction to thelatter explains the overall phase diagram rather well. Tcooperative effect has been discussed on phenomenologrounds.5 Here we take a different approach, which qualitively reproduces exactly known, ground-state propernear the metal-insulator transition in one dimension.

For M5Ag, the Ag d level is far away from the Fermenergy so that the DCNQIp bands are quarter filled. In thisense, they are similar to (TMTTF)2X and (TMTSF)2Xsalts, which are also quarter-filled and show various phas6

In the latter, the 4kF anion potential has been consideredproduce the umklapp process7 and then to cause the metainsulator transition.8 Such an extrinsic potential is absentthe (R1R2-DCNQI)2Ag salts, but the metal-insulator trans

560163-1829/97/56~12!/7262~15!/$10.00

y

nd

kethnd

ecal-s

s.

tion occurs, accompanied by the formation of a 4kF CDWfor both of theR15R25CH3 ~denoted by DMe hereafter!andR15R25I ~denoted by DI hereafter! cases.9 At low tem-peratures,~DMe-DCNQI!2Ag becomes a spin-Peierls statwhile ~DI-DCNQI!2Ag becomes an antiferromagnet.10 Thuselectron correlation in thep band also plays an essential roto determine the ground-state phases.

Electron correlation is essential in low dimensions.pure one dimension, the Fermi liquid is unstable againstperturbation. When the perturbation is irrelevant in termsa renormalization group, the excitation spectrum is gapleThen the low-energy limit of the property is described by tTomonaga-Luttinger liquid theory and characterizedpower laws in the density of states and various correlatfunctions.11,12 Otherwise, the spectrum has a gap so tsome of the correlation functions decay exponentially.13

For the metallic~DMe-DCNQI!2Cu, the single-particlespectrum has been observed in photoemission experimand described by a power law of the electron bindienergy.14 This suggests that the metallic phase is describy the Tomonaga-Luttinger liquid theory for low but noextremely low temperatures. The exponent is much larthan the calculated one for the Hubbard model or thetended Hubbard model with on-site and nearest-neighborpulsions only. However, it can be explained by long-raninteractions.15,16 The power law holds up to the binding energy as large as 0.3 eV. This fact would also indicate a lorange interaction.14 Recently, suppression of the interchacoherent hopping by such a large exponent in the sinparticle spectrum is numerically investigated.16

7262 © 1997 The American Physical Society

yole

exde

ones

-ththtinann

rsb

-

serimcera

nehes.rth

thapn

m

anice

ththxtapf

e

esh

arnc

um

lda-een

the

heardpli-The

ns

eor

ixth-e a

yized

theory-

be-

ensi-ter-ue

pe

esumand

gcehow

Vance

ec-re-

ct-de

56 7263RENORMALIZATION-GROUP APPROACH TO THE . . .

In the insulator phase on the other hand, the Cud elec-trons are almost localized so that good one-dimensionalitexpected. Since the nesting is perfect and the tendency tinsulator phase is strong in one dimension when coupwith phonons, the metal-insulator transition must beplained at least in a purely one-dimensional model in orfor it to occur in a quasi-one-dimensional real system. Wthen employ a purely one-dimensional model to study cditions for the observed Mott insulator phase with a gaplspin mode.

Meanwhile,~DCNQI!2Ag salts are insulators for low temperatures. One-dimensionality is expected to be better~DCNQI!2Cu salts because the metal ion does not helpelectron propagation perpendicular to the most conducdirection as in the Cu salt. In fact, the one-dimensional bstructure has been evidenced by the polarized reflectaspectra on single crystals.17 Thus the Tomonaga-Luttingeliquid theory would be a good starting point. Qualitative apects of the experimental findings are in fact explainedconsidering the effects of electron-2kF phonon coupling,electron-4kF phonon coupling, and slight threedimensionality on a one-dimensional continuum model.

In one dimension, a renormalization-group method baon the scaling law is very useful to study the effects of vaous perturbations. The Mott transition is caused by the uklapp process. The temperature and frequency dependenthe conductivity for commensurate and nearly commensufillings has been studied in detail18 with the renormalization-group method combined with a memory-functioapproximation.19 The Mott transition which occurs when thcorrelation strength is changed and that which occurs wthe filling is changed have different physical propertiewhich are clarified for even and odd commensurabilities20

Meanwhile, the electron-phonon interaction produces thetarded attraction. Its effect has also been studied withrenormalization-group method. When it contributes toforward scattering, the backward scattering, and the umklprocess, it enhances the pairing correlation, the formatioa spin gap,21 and the formation of a charge gap,22,23 respec-tively.

Here, we apply the method to more complex systewhich have couplings with 2kF and 4kF phonons with one ortwo bands at even or odd commensurability. We derivenumerically solve the lowest-order equations to learn whphase the Tomonaga-Luttinger liquid approaches as theergy scale is lowered. Although a precise description ofelectronic states at strong-coupling fixed points is beyondscope of the present method, it indicates whether each etation spectrum has a gap or not so that the ground-sphases can be classified accordingly. Such procedure reduces qualitative tendency to the strong-coupling phasesnearly half-filled electron-phonon systems22,23and for a two-coupled-chain system.24 The present paper also follows thprocedure.

To compare with experimental results, it would be necsary to take weak three-dimensionality into account. Tscaling law and the consequent power-law behavior of vous quantities in one dimension result from the interfereof the logarithmic singularity in the 2kF particle-hole chan-nel with that in the particle-particle channel. The contribtions from the corresponding lowest-order bubble diagra

isand-r

e-s

anegdce

-y

d--of

te

n,

e-e

ep

of

s,

dhn-ee

ci-tero-or

-ei-e

-s

are different in signs only. Quasi-one-dimensionality woube simulated by different cutoff energies for the two logrithmic singularities and the consequent imbalance betwthe two channels.7 In the M5Ag case, we find that the dif-ferences between theR15R25CH3 salt of the spin-Peierlsground state and theR15R25I salt of the antiferromagneticground state are qualitatively explained by a difference incutoff energy in the particle-particle channel.

In the M5Cu case, the two-band feature is essential. Tinterference between electron-phonon couplings, backwscatterings, and umklapp processes is much more comcated in the two-band case than in the single-band case.phonons with momenta near 2p/3 are responsible for the4kF scattering in the sixth-filled band and for the 2kF and4kF scatterings in the third-filled band, while the phonowith momenta nearp/3 are responsible for the 2kF scatter-ing in the sixth-filled band. Different combinations of thabove scatterings lead to different backward scatteringumklapp processes, opening charge gaps in both of the sfilled and third-filled bands. The insulator phase can havspin gap only in the third-filled band~Mott insulator! or spingaps in both bands~Peierls insulator!. The former phase isrealized when the sixth-filled band~i.e., p-d hybrid band25!has strong correlation and the electron-2kF phonon couplingis moderate. The electron-4kF phonon coupling is necessarhere, though it may not be strong. The latter phase is realwhen the electron-2kF phonon coupling is strong.

To study how the behavior of resistance depends oncommensurability and model parameters, we use a memfunction approximation. ForM5Ag, the renormalization-group method qualitatively reproduces the differencetween the DMe and DI cases. ForM5Cu, the behavior ofthe resistance above the transition temperature is less stive to changes in correlation strengths than in the quarfilled case. Experimentally, the transition is of first order dto the third-order commensurability energy,26 and it needsthree-dimensionality in phonons, which is beyond the scoof the present study.

This paper is organized as follows: Section II introducbosonized models based on one-dimensional continumodels with backward scatterings, umklapp processes,couplings with 2kF and 4kF phonons. Section III outlines thederivation of renormalization-group equations, clarifyinhow different electron-phonon couplings effectively produdifferent backward scattering or umklapp processes andthey open gaps in different channels. Sections IV andshow phase diagrams for different parameters and resistas a function of temperature forM5Ag andM5Cu, respec-tively, to compare them with the experimental results. Stion VI summarizes the present work. Part of the results psented in this paper were reported briefly elsewhere.27

II. MODELS

We consider a continuum model in which the noninteraing part is a Tomonaga-Luttinger liquid extended to incluthe spin ~denoted by subscripts! and charge~denoted bysubscriptr! degrees of freedom,

d

r-

ec

riormi

x-

of

7264 56K. YONEMITSU

H05 (n5s,r

E dx

2p FunKn@]xun~x!#21un

Kn@]xfn~x!#2G ,

~2.1!

where fieldsfn(x) and (1/p)]xun(x) are conjugate,un andKn are the velocity and the correlation exponent of thenchannel, respectively, which are standard notations andscribed in detail in the previous paper.22 For M5Ag, weconsider a quarter-filled band. ForM5Cu, we take twobands and distinguish them by subscripts:A for the sixth-filled band;B for the third-filled band. Then, the noninteacting part is the sum ofHA0 and HB0 , in which the sum-mation is performed overn5As,Ar and n5Bs,Br,respectively. It is noted thatH (C)0 (C5A,B) ~i.e., H0 ,HA0 , or HB0! contains the~one-electron! kinetic part and theforward scattering. ForM5Cu, we consider, for simplicity,the regime where the Fermi velocity in theA band and thatin theB band do not differ so much. Since we do not exp

e-

t

that the difference in the Fermi velocities affects the scenato the metal-insulator transition, we use the averaged Fevelocity, vF . For later convenience, we defineX(C)s

52(12K (C)s21 ).g(C)1 /(pvF) and X(C)r52(12K (C)r

21 ).(g(C)122g(C)2)/(pvF) for C5A,B, where g(C)1 andg(C)2 are the backward and forward scattering strengths~inthe C band!, respectively. In terms of the single-band etended Hubbard model with on-site (U) and nearest-neighbor (V) repulsions, Xs5U/(pvF), Xr52(U14V)/(pvF) for quarter filling, XAs5(U1V)/(pvF), XAr52(U13V)/(pvF) for sixth filling, andXBs5(U2V)/(pvF), XBr52(U15V)/(pvF) for thirdfilling.

A. Electron-electron interactions

A backward scattering between antiparallel spinsstrengthY(C)s5g(C)1 /(pvF),

re to be

tors, it

el,wnof

nd spin

H ~C!s5Y~C!spvF(sE dxc~C!1,s

† ~x!c~C!2,2s† ~x!c~C!1,2s~x!c~C!2,s~x!, ~2.2!

wherec (C)1,s andc (C)2,s are right- and left-going electrons with spins ~in the C band!, is written with the phase field as

H ~C!s5Y~C!svF

2pa2 E dx cos@2&f~C!s~x!#, ~2.3!

wherea is a cutoff parameter of the order of the inverse of the Fermi wave number.To study metal-insulator transition, we need to include high-order electron-electron scattering processes, which a

produced in the renormalization process when electron-phonon interactions are included. In the 1/m-filled band, the Fermiwave number isp/m so that the umklapp process is written as

HU,m@fr~ ,fs!;Yr#}YrvFam22E dxF(s

c1,s† ~x!c2,s~x!Gm

1H.c., ~2.4!

whereYr is a coupling strength such that the prefactor appears simple in the phase-field representation,a is of the order of thelattice spacing and set to bea, and the power should be understood as point-split hereafter. With the phase-field operais rewritten as

HU,m@fr~ ,fs!;Yr#5H ~YrvF!/~2pa2!E dx cos@m&fr~x!# for even m,

~YrvF!/~&pa2!E dx cos@m&fr~x!#cos@&fs~x!# for odd m,~2.5!

where and hereafter less relevant terms are neglected. Thus, we haveHr5HU,4@fr ;Yr# for the quarter-filled band,HAr5HU,6@fAr ;YAr# for the sixth-filled band,HBrs5HU,3@fBr ,fBs ;YBrs# andHBr5HU,6@fBr ;YBr# ~which is includedthough less relevant thanHBrs! for the third-filled band. In terms of the extended Hubbard modYr53U3/(16p3t2vF)53Ys

3/8, etc. Asm increases, the effect ofHU,m becomes weak on opening a gap, as explicitly shoin the renormalization-group equations later. Note that, for oddm, the umklapp process involves the spin degreesfreedom:20 when the umklapp process flows to a strong-coupling fixed point, there are gaps in both the charge aexcitations. In such a case, soliton excitations carry both charge and spin in contrast to the case with evenm.

In addition, there are interband scattering processes forM5Cu. The most relevant backward scattering is written as

HABrs5~YABrsp2vFa/& !E dxF(s

cA1,s† ~x!cA2,s~x!G2F(

scB2,s

† ~x!cB1,s~x!G1H.c., ~2.6!

whereYABrs is the coupling strength. It is rewritten as

HABrs5YABrsvF

&pa2 E dx cos@2&fAr~x!2&fBr~x!#cos@&fBs~x!#, ~2.7!

heeesring. Mean-

t

ten as

56 7265RENORMALIZATION-GROUP APPROACH TO THE . . .

which involves the charge degrees of freedom in theA band and both the charge and the spin degrees of freedom in tBband. The scaling dimension is the lowest~in the noninteracting limit! among the perturbations involving the charge degrof freedom. So, this is the most likely candidate which causes a metal-insulator transition. When this backward scatte~notumklapp process! flows to a strong-coupling fixed point, there are at least three gaps in the corresponding channelswhile, the most relevant umklapp process is written as

HABr5YABrsp3vFa2E dxF(s

cA1,s† ~x!cA2,s~x!G2F(

scB1,s

† ~x!cB2,s~x!G2

1H.c., ~2.8!

whereYABr is the coupling strength. It is rewritten as

HABr5YABrvF

2pa2 E dx cos@2&fAr~x!12&fBr~x!#, ~2.9!

which involves the charge degrees of freedom in both bands but does not involve the spin degrees of freedom.

B. Electron-phonon interactions

As to phonons, we need to consider those with momenta near 2kF and those with momenta near 4kF . Because we are nointerested in superconductivity here, we do not consider phonons with momenta near 0kF , though the extension isstraightforward.22,27 The wave numbersm52kF , 4kF are m5p/2, p for m54, m5p/3, 2p/3 for m56, andm52p/3,4p/3(522p/3) for m53. The corresponding phonon fields and their conjugate momenta are denoted byfm(x) andPm(x),respectively. The fieldsfp(x) andPp(x) are real, and the others are complex. The phonon parts of the models are writ

Hm5E dx@Pm~x!P2m~x!1vm2 fm~x!f2m~x!# ~2.10!

for the complex fields (mÞp) and

Hp51

2 E dx@Pp2 ~x!1vp

2 fp2 ~x!# ~2.11!

for the real fields, wherevm andvp are the corresponding phonon frequencies.Electron-2kF phonon coupling is generally written as

H1,~C,!m5ApvFvmy~C!1(sE dxc~C!2s

† ~x!c~C!1s~x!fm~x!1H.c., ~2.12!

wherey(C)1 is the strength of the 2kF scattering~in the C band! by a phonon. It is rewritten as

H1,~C,!m5AvFvmy~C!1

ApaE dx cos@&f~C!s~x!#e2 i&f~C!r~x!fm~x!1H.c. ~2.13!

For the quarter-filled band, for example, the coupling strength and the phonon frequencies are given byy15b/ApvFK andvp/25vp5AK/M for the Holstein coupling,28 ( i@bqini1(K/2)qi

21(1/2M )pi2#, with coupling strengthb, spring constantK,

ionic massM , electron densityni at sitei , lattice displacementqi and its conjugate momentumpi , and byy152iaS /ApvFKand &vp/25vp52AK/M for the SSH coupling,29 ( i@aSqi ,i 11(s(ci ,s

† ci 11,s1H.c.)1(K/2)qi ,i 112

1(1/2M )pi2#, with coupling strengthaS , qi ,i 115qi 112qi , ci ,s annihilating an electron with spins at site i , and other

parameters as defined above.Electron-4kF phonon coupling is generally written as

H3,~C,!m52A2p3vFvmy~C!3(s,s8

aE dxc~C!2s† ~x!c~C!2s8

†~x!c~C!1s8~x!c~C!1s~x!fm~x!1H.c., ~2.14!

ng

-r

teeheg

ue

un

erogomo

leth

e

d-

in

ird.seus

by

er.

-

-

der.f-

m

i-ns-hein

lingte-mtionhinults

7266 56K. YONEMITSU

wherey(C)3 is the strength of the 4kF scattering~in the Cband! by a phonon. It is rewritten as

H3,~C,!m52AvFvmy~C!3

A2paE dxe2 i2&f~C!r~x!fm~x!1H.c.

~2.15!

For the quarter-filled band, for example, the couplistrength is given byy35b8/A8p3vFK for the electron-electron-phonon coupling,( i(U2b8qi)ni↑ni↓ , with on-siterepulsionU, coefficientb8 of its modulation by lattice dis-placementqi , electron densitynis with spin s at sitei , andby y35 ia8/A2p3vFK for the electron-electron-phonon coupling, ( i ,s,s8(V2a8qi ,i 11)nisni 11s8 , with nearest-neighborepulsionV, coefficienta8 of its modulation by lattice dis-placementqi ,i 11 .

The phonon fields are bilinear, so that they can be ingrated out completely to produce effective retardinteractions,22 which are lengthy and not shown here. In tvm→` limit, the electron-phonon interactions do nothinbut shift the parameters, X(C)s→X(C)s2uy(C)1u2,Y(C)s→Y(C)s2uy(C)1u2, X(C)r→X(C)r2uy(C)1u224uy(C)3u2,Yr→Yr2(y3

21y3*2)/2, YBrs→YBrs1(yB1yB31yB1* yB3* ),

YABrs→YABrs1(yB1yA3* 1yB1* yA3), and YABr→YABr

2(yA3yB31yA3* yB3* ). Below we take the Hamiltonian

H5H01Hs1Hr1Hp/21Hp1H1,p/21H3,p ~2.16!

for M5Ag with a quarter-filled band, and

H5 (C5A,B

~HC01HCs1HCr!1HBrs1HABrs1HABr

1Hp/31H2p/31H1,A,p/31H1,B,2p/31H3,A,2p/3

1H3,B,22p/3 ~2.17!

for M5Cu with coupled sixth- and third-filled bands.

III. RENORMALIZATION EQUATIONS

We have derived the equations following the previostudy.22 We will give an outline for discussions later. Thpresent one-dimensional quantum-mechanical systemmapped to a two-dimensional classical system where Bgers vectors interact with one another. The correlation fution

^Ttei&f~C!n~x,t!e2 i&f~C!n~0,0!& ~C5A,B, n5s,r!

~3.1!

is perturbationally developed and successively integratedchanging the length scale little by little. In this process, thare two possibilities for the fate of Burgers vectors: a pairneutral Burgers vectors are annihilated in the larger lenscale; and a pair of non-neutral Burgers vectors are cbined to produce another Burgers vector. The latter is imptant since it causes various interference effects.22,23 By com-paring the correlation functions in successive length scawe find relations between the effective parameters inlarger energy~i.e., smaller length! scale and those in th

-d

s

isr-c-

byefth

-r-

s,e

smaller energy~i.e., larger length! scale, which are describeby differential equations~called renormalization-group equations! shown below.

A. Quarter-filled band

The combinations for the non-neutral Burgers vectorsthe renormalization process are (y1 ,y1* ,Ys), (y3 ,y3 ,Yr),and (y3* ,y3* ,Yr), where (a,b,c) means that any two of themare combined to produce the complex conjugate of the thIn order to make a comparison with the half-filled caeasier, we use the notation similar to that in the previostudy,22 Y15uy1u2, Y35uy3u2, and a form factorf 5(y3

21y3*2)/(2Y3) which satisfies21, f ,1. Below we

mark the contributions from the particle-particle channelJh( l ) and those from the 2kF particle-hole channel byJk( l ),which are useful to study a three-dimensionality effect latThen, we finally have

dXs~ l !/dl52Jh~ l !@Xs~ l !2Xr~ l !#Xs~ l !/22Jk~ l !

3@Xs~ l !1Xr~ l !#Xs~ l !/22Y1~ l !Dp/2~ l !,

~3.2!

dXr~ l !/dl5Jh~ l !@3Xs2~ l !1Xr

2~ l !#/42Jk~ l !

3$@3Xs2~ l !1Xr

2~ l !#/414Yr2~ l !%2Y1~ l !Dp/2~ l !

24Y3~ l !Dp~ l !, ~3.3!

dYr~ l !/dl5Jk~ l !@228Kr~ l !#Yr~ l !2 f Y3~ l !Dp~ l !,~3.4!

dY1~ l !/dl5Jk~ l !@22Ks~ l !2Kr~ l !2Xs~ l !#Y1~ l !,~3.5!

dY3~ l !/dl5Jk~ l !@224Kr~ l !2 f Yr~ l !#Y3~ l !, ~3.6!

where l 5 ln@EF /E#, E( l ) is a cutoff energy@E(0)5EF#,Kn( l )5(12Xn( l )/2)21, Dm( l ) is the phonon propagator defined by Dm( l )5@vm /E( l )#exp@2vm /E(l)#. We have usedthe relationXs( l )5Ys( l ) due to the spin-rotational symmetry and omitted the equations forun( l ) which does notcouple with the above equations in the present, lowest orInitial conditions atl 50 are determined by treating the efective retarded interactions carefully.22 For Xs(0), Xr(0),and Yr(0), only the phonon propagator is integrated fro2` to 0 with the fixed prefactor:Xs(0)5Xs2Y1(12e2vp/2 /EF), for example. The reason whyXs does notrenormalizeXs(0) or whyY1 is not renormalized atl ,0 inthis example has been discussed already.22 This correctlyreproduces the antiadiabatic limit shown below Eq.~2.15!.

In real materials, slight three-dimensionality would manfest itself at very low temperatures where the electron traport in the perpendicular direction becomes coherent. Trenormalization-group method would not be justified deepthe anisotropic three-dimensional regime, where the scalaw no longer holds. However, the scaling law would deriorate little by little as that regime is approached froabove. Then we can see at least how the renormalizaflow is deflected and which phase it tends to approach witthe present method. The scaling law in one dimension res

dcoin

intreia-a

ivarea

.ua

d.ut

thw

cle

ith

-

-

hteancaononrly

p

n

tun

n-

in

56 7267RENORMALIZATION-GROUP APPROACH TO THE . . .

from the interference of the 2kF particle-hole channel withthe particle-particle channel. The corresponding lowest-orbubble diagrams are logarithmically divergent and haveefficients of equal magnitudes and different signs. Suchterference disappears in higher dimensions becauseFermi surface does not consist of a finite number of poany more. Though the distortion of the Fermi surfacemoves the logarithmic singularity in the particle-hole dgram unless the nesting is perfect, the particle-particle chnel generally becomes less important for repulsinteractions when only the particle-particle diagramssummed infinitely. It is natural to regard quasi-ondimensionality as causing imbalance between the two chnels as the most important effect among all.7 It is realized bydifferent cutoff energies for the two logarithmic diagrams

It should be noted that the renormalization-group eqtions derived from the bosonized model and the mappinga two-dimensional classical system and those derivedrectly from the original fermion model are equivalent11

Renormalization of the velocities and detailed forms of coff functions if any~e.g., one due to a misfit parameter! aregenerally different, but they are not essential. Derivingrenormalization equations again in the second method,can distinguish the contributions from the particle-partichannel and the contributions from the 2kF particle-holechannel, as already shown above. Then we can cut off ethe particle-particle channel by the functionJh( l ) ~deter-mined below! or the 2kF particle-hole channel by the function Jk( l ). Assuming that the logarithm ln(EF /E) is replacedby ln@EF

2/(E21g2EF2)#/2 (g5h,k) in the perturbation expan

sions, we obtain

Jg~ l !5~d/dl !ln@EF2/~E2~ l !1g2EF

2 !#/25~11g2e2l !21,~3.7!

which satisfiesJ0( l )51, JgÞ0( l )!1 for l @2 ln g ~i.e.,E!gEF), andJgÞ0( l )→0 asl→`.

The temperature dependence of the resistance showsthe system approaches the insulating-metallic ground stais a useful property to characterize the metal-insulator trsition, and also a good indicator for the quality of theoretiapproaches. Since we perform perturbative calculatiabove, we perform the perturbative expansion for the cductivity following the study for commensurate and neacommensurate fillings18 with a memory-functionformalism.19 The conductivity is given by s(v)5( i2urKr /p)@v1M (v)#21, where the memory functionM (v) is defined byM (v)5v^ j ; j &v /(^ j ; j &02^ j ; j &v) and^ j ; j &v is the retarded correlation function of the current oerator, j (x)5&urKr]xur(x)/p. The lowest-order term inthe perturbative expansion for M (v) is M (v).(^F;F&v

0 2^F;F&00)/(2v^ j ; j &0), where F is defined by

F5@ j ,H# and ^F;F&v0 stands for the retarded correlatio

function of the operatorF at frequencyv computed in theabsence of perturbations. Later we consider a temperarange below the phonon frequencies, where electron-phointeractions are integrated out. The calculation of^F;F&v

0 isstraightforward. Taking thev→0 limit, we finally have, forthe resistanceR(T)51/s(v50,T) at finite temperatureT,

er--

thes-

n-ee-n-

-toi-

-

ee

er

ow. It-ls-

-

reon

R~T!}Yr2~T!TB2@4Kr~T!,128Kr~T!#cos2@4pKr~T!#,

~3.8!

where B(x) is the beta function andYr(T) denotesYr@ l 5 ln(EF /T)#, etc. If we neglect the temperature depedence ofKr in the equation forYr ~and for k50!, whichwould be valid only in the weak-coupling limit,R(T)}Yr

2T16Kr23.18

B. Coupled sixth- and third-filled bands

The combinations for the non-neutral Burgers vectorsthe renormalization process are now (yA3 ,yB1* ,YABrs),(yA3 ,yB3 ,YABr), (yB1 ,yB3 ,YBrs), (YABrs ,YABr ,YBrs),(YABrs ,YABrs ,YBs), (YBrs ,YBrs ,YBs), (YBrs ,YBrs ,YBr), (yA1 ,yA1* ,YAs), and (yB1 ,yB1* ,YBs). When (a,b,c)is possible, (a* ,b* ,c* ) is also possible but it is not listedabove because it is trivial. Note thaty’s are complex param-eters, whileY’s are real parameters. Finally, we have

dXAs~ l !/dl52XAs2 ~ l !2uyA1~ l !u2Dp/3~ l !, ~3.9!

dXAr~ l !/dl52YABrs2 ~ l !2YABr

2 ~ l !29YAr2 ~ l !

2uyA1~ l !u2Dp/3~ l !24uyA3~ l !u2D2p/3~ l !,

~3.10!

dXBs~ l !/dl52XBs2 ~ l !2

1

4YABrs

2 ~ l !21

4YBrs

2 ~ l !

2uyB1~ l !u2D2p/3~ l !, ~3.11!

dXBr~ l !/dl521

4YABrs

2 ~ l !2YABr2 ~ l !2

9

4YBrs

2 ~ l !29YBr2 ~ l !

2@ uyB1~ l !u214uyB3~ l !u2#D2p/3~ l !, ~3.12!

dyA1~ l !/dl5H 121

2@KAs~ l !1KAr~ l !1XAs~ l !#J yA1~ l !,

~3.13!

dyA3~ l !/dl5@122KAr~ l !#yA3~ l !11

2YABrs~ l !yB1~ l !

21

2YABr~ l !yB3* ~ l !, ~3.14!

dyB1~ l !/dl5H 121

2@KBs~ l !1KBr~ l !1XBs~ l !#J yB1~ l !

11

2YABrs~ l !yA3~ l !1

1

2YBrs~ l !yB3* ~ l !,

~3.15!

dyB3~ l !/dl5@122KBr~ l !#yB3~ l !21

2YABr~ l !yA3* ~ l !

11

2YBrs~ l !yB1* ~ l !, ~3.16!

de-.

7268 56K. YONEMITSU

dYABrs~ l !/dl5H 221

2@4KAr~ l !1KBs~ l !

1KBr~ l !1XBs~ l !#J YABrs~ l !

21

2YABr~ l !YBrs~ l !1@yB1~ l !yA3* ~ l !

1yB1* ~ l !yA3~ l !#D2p/3~ l !, ~3.17!

dYABr~ l !/dl5$222@KAr~ l !1KBr~ l !#%YABr~ l !

21

2YABrs~ l !YBrs~ l !2@yA3~ l !yB3~ l !

1yA3* ~ l !yB3* ~ l !#D2p/3~ l !, ~3.18!

dYBrs~ l !/dl5H 221

2@KBs~ l !19KBr~ l !1XBs~ l !

nc

n-

pon

ht i

w

ser

1YBr~ l !#J YBrs~ l !21

2YABrs~ l !YABr~ l !

1@yB1~ l !yB3~ l !1yB1* ~ l !yB3* ~ l !#D2p/3~ l !,

~3.19!

dYAr~ l !/dl5@2218KAr~ l !#YAr~ l !, ~3.20!

dYBr~ l !/dl5@2218KBr~ l !#YBr~ l !21

4YBrs

2 ~ l !,

~3.21!

where the cutoff energy and the phonon propagators arefined as before,XCs( l )5YCs( l ) (C5A,B) has been usedInitial conditions atl 50 are also determined as before.

For the present system, the resistanceR(T) in a tempera-ture range below the phonon frequencies is given by

R~T!}5YABrs2 ~T!T cos2@$4KAr~T!1KBr~T!1KBs~T!%p/4#B2@$4KAr~T!1KBr~T!1KBs~T!%/4,1

2$4KAr~T!1KBr~T!1KBs~T!%/2#18YABr2 ~T!T cos2@$KAr~T!1KBr~T!%p#

3B2@KAr~T!1KBr~T!,122$KAr~T!1KBr~T!%#19YBrs2 ~T!T cos2@$9KBr~T!1KBs~T!%p/4#

3B2@$9KBr~T!1KBs~T!%/4,12$9KBr~T!1KBs~T!%/2#136YAr2 ~T!T cos2@9KAr~T!p#

3B2@9KAr~T!,1218KAr~T!#136YBr2 ~T!T cos2@9KBr~T!p#

3B2@9KBr~T!,1218KBr~T!#. ~3.22!

edspindesctedthatwen.ytateionshelat-

-are

npho-

l-

In the above formula, any perturbation which containsfAr

or fBr contributes to the resistance. Among them,YABrs hasthe lowest scaling dimension and dominates the resistaIn the weak-coupling limit, we have R(T)}YABrs

2 T4KAr1KBs1KBr23. The actual temperature depedence is modified from this simple power law.

IV. RESULTS FOR QUARTER-FILLED BAND

The fixed points of the spin and charge correlation exnents, Ks* and Kr* , determine the asymptotic correlatiofunctions. Assuming a gap in then spectrum forKn* 50, weclassify the ground-state phases into a gapless state in wall perturbations are irrelevant so that the low-energy limiregarded as a Tomonaga-Luttinger liquid~denoted by TL inthe phase diagrams below!, a state with only a spin gapwhich we call a spin-gap state~denoted by SG!, a state withonly a charge gap which we call a Mott insulator~denoted byMI !, and a state with both spin and charge gaps whichcall a spin-Peierls state~denoted by SP!. When we considerthree-dimensionality, the gapless~thus metallic! state doesnot correspond to a Tomonaga-Luttinger liquid any morethat it is simply denoted by M. The Tomonaga-Luttingliquid has a dominant 2kF SDW correlation;exp@2(Kr*11)lnr# or a dominant 4kF CDW correlation

e.

-

ichs

e

o

;exp@24Kr* lnr#. The spin-gap state has a dominant 2kF

CDW correlation;exp@2Kr* lnr#. The Mott insulator has adominant 4kF CDW correlation.

Usually a spin-Peierls state is described by a localizspin system since the charge gap is much larger than thegap. Here we call even a state with comparable magnituof gaps a spin-Peierls state because it is smoothly connewith the so-called spin-Peierls state. We assumephonons are also one-dimensional for simplicity so thatdo not consider the possibility for a static lattice distortioIn any case, a coupling with 2kF phonons is a necessarcondition here for a spin gap and thus for a spin-Peierls sbecause we consider repulsive electron-electron interactonly. With a coupling with three-dimensional phonons, tspin-Peierls state here would be accompanied by a statictice distortion.

Before we use parameters for~DCNQI!2Ag, we first showhow the electron-4kF phonon coupling and the umklapp process interfere with each other and next show how theyaffected by the electron-2kF phonon coupling, large phonofrequency, and nearest-neighbor repulsion. For the barenon dispersion, we takevp/25vp /&. It is noted that thequantity f appears withYr( l ) as a prefactor in the renormaization equations so that we fixf 521 and allowYr to benegative. We do not cut off the logarithmic singularities,k5h50, unless explicitly mentioned.

vepproexs

.

s

n,

atio

f

rit

thor

s

uaicar

rit

e

dt

seEq

en

use

rge

andthee

in

e

nonndsec-non

ard

56 7269RENORMALIZATION-GROUP APPROACH TO THE . . .

A. Coupling with 4 kF phonons and umklapp process

It is noted again that the perturbative expansion giYr53Xs

3/8 for the extended Hubbard model. The umklaprocess is of third order with respect to electron-electinteraction, so that it is generally weak. In this and the nsubsections, we vary the strength of the umklapp processee the interference effect.

To have a charge gap,Kr* must vanish so thatXr( l ) mustapproach2` in the l→` limit. From the renormalizationequation~3.3!, it requiresYr( l ) not to vanish. Then from Eq~3.4!, Kr( l ) must be smaller than 1/4@Xr( l ) must be smallerthan 26# for large l . For the pure Hubbard model, it iknown that it cannot be smaller than 1/2.30 Even for theextended Hubbard model with nearest-neighbor repulsiois at least 3/16.15 Such a smallKr( l ) is achieved generallyfor long-range repulsion.31 The photoemission experiment14

may indeed suggest a long-range repulsion for~DMe-DCNQI!2Cu. In the ~DCNQI!2Ag salts, the 4kF CDW isobserved9 and the lattice would be modulated. So, it is resonable to regard electron-phonon interaction as cooperawith the umklapp process to have such a small valueKr( l ). This is the case when the umklapp process itselvery weak because it occurs as a high-order process.

In the absence of the electron-phonon coupling, the ccal value for a finite charge gap is large~Fig. 1! and beyondthe scope of the perturbative regime. However, withelectron-4kF phonon coupling, it becomes smaller. FYr.0 corresponding tof 521, e.g., for lattice modulationof the nearest-neighbor repulsion, the umklapp procesconstructively interfered with the electron-4kF phonon cou-pling, as is expected from the renormalization-group eqtions. This is the reason for the negative slope of the critvalue ofY3 at Yr50 in the figure. It is noted that, even foYr,0 corresponding tof 511, e.g., for lattice modulationof the on-site repulsion, the critical value ofY3 does notbecome so large. This is becauseYr( l ) changes the sign afteit decreases due to the destructive interference at the inhigh-energy scale. Thus, the interference is constructivlow-energy scales irrespectively of the sign off . This situa-tion is in contrast to the half-filled and nearly half-fillecases, where the opening of a spin gap is sensitive toform factor of the electron-2kF phonon coupling.22,23

B. Coupling with 2kF phonons

The electron-2kF phonon coupling is expected to decreaXr( l ), enhancing the tendency for a charge gap, from

FIG. 1. Phase diagram forXs50.4, Xr520.8, vp50.1EF ,andY150.

s

ntto

it

-ngf

is

i-

e

is

-l

ialat

he

.

~3.3!. However, its tendency is very weak as clearly sefrom a comparison of Fig. 1 with Fig. 2 and directly fromFig. 3. There is no interference of the electron-2kF phononcoupling with the umklapp process in Eq.~3.4!. This indi-cates again how the interference effect is important to cathe metal-insulator transition.

The role of the electron-2kF phonon coupling is mainly toenhance the tendency for a spin gap. For a sufficiently lacoupling, the system has a finite spin gap~Fig. 3! as expectedfrom Eq. ~3.2!. Then, the 2kF CDW correlation becomesdominant. Note the phase boundary between TL and SGthat between MI and SP depends considerably onelectron-4kF phonon coupling. This is due to the fact that thelectron-2kF phonon coupling is affected by both of the spand charge correlation exponents in Eq.~3.5!. Meanwhile,the electron-4kF phonon coupling is affected by only thcharge correlation exponent in Eq.~3.6!. Even in the weak-coupling limit @Ks( l ).Kr( l ).1#, Y1( l ) is largely affectedby the correlation exponents as shown in Eq.~3.5!, althoughY3( l ) always decreases in Eq.~3.6!. Thus, the opening of aspin gap is rather sensitive to the electron-4kF phonon cou-pling.

C. Phonon frequency and nearest-neighbor repulsion

Now we have a clear idea about how the electron-phocouplings interfere with the electron-electron interaction ahow they determine the ground-state phase. In this subtion, we show how the phase diagram depends on the phofrequency and the relative magnitude ofuXru to Xs , i.e., thenearest-neighbor repulsion in terms of the extended Hubbmodel. Note that2Xr /Xs5114V/U in this model. ForV/U51/4 and 1/2, the relative magnitude ofuXru be-

FIG. 2. Phase diagram forXs50.4, Xr520.8, vp50.1EF ,andY150.2.

FIG. 3. Phase diagram forXs50.4, Xr520.8, Yr50.375Xs3 ,

andvp50.1EF .

baa

onaeF

ecbeondcaes

e

f

r

M

sth

usenote to

ndiveforce

m-apsthsurrelyga-

onr-l-am.

a-orusde-

se,iblethe

in-

e-less

s

s

an-

7270 56K. YONEMITSU

comes twice and three times as large as that for the Hubmodel, respectively, so that the results can be largelyfected by the nearest-neighbor repulsion strength.

The phonon frequency relative to the Fermi energy ctrols the energy scale of the interference. In the antiadiablimit, the electron-phonon coupling simply shifts thstrengths of the electron-electron scattering parameters.finite phonon frequency below the Fermi energy, the efftive interaction is retarded and the interference effectcomes largest at an energy scale comparable to the phfrequency as in the half-filled and nearly half-fillesystems.22,23 As the phonon frequency increases, the criticoupling strength ofY3 for opening a charge gap decreaswhile that ofY1 for opening a spin gap increases@Figs. 4~a!and 4~b!#. Then the Mott insulator phase occupies a widarea in the phase diagram.

As the magnitude ofuXru increases, the initial value oKr decreases. Then the critical value ofY3 for opening acharge gap decreases@Fig. 4~c!#. The slope is steeper fosmaller phonon frequencies. Meanwhile, theY1 coordinateof the cross section of the phase boundaries among TL,SG, and SP does not depend so much onuXru.

D. Phase diagram for„DCNQI …2Ag

According to the local density functional theory,25 thebandwidth for~DMe-DCNQI!2Ag is about 0.9 eV. Phonondue to the dimerization are observed at about 0.08 eV in

FIG. 4. Y1 andY3 coordinates of the cross section of the phaboundaries among TL, MI, SG, and SP,~a! as a function ofvp /EF for Xr520.8, ~b! as a function ofvp /EF for Xr521.2,and~c! as a function ofuXru for vp50.1EF . The other parameterare the same as in Fig. 3.

rdf-

-tic

or--on

l,

r

I,

e

infrared spectrum.32 We use&vp/25vp50.4EF . The con-dition for the Mott insulator phase is easily achieved becathe phonon frequency relative to the Fermi energy issmall and because the nearest-neighbor repulsion relativthe on-site repulsion is not small either.33 We show the data27

again for completeness~Fig. 5!.The author does not know the electron-electron a

electron-phonon interaction strengths, but the qualitataspect of the present results would not changedifferent electron-electron interaction strengths. Sin~DMe-DCNQI!2Ag becomes a spin-Peierls state at low teperatures~in the sense that both of the spin and charge gopen!, we expect that its electron-phonon coupling strengare in the SP phase of the figure. Note that all of the fopossible phases appear in the phase diagram in the puone-dimensional case. This is because the TomonaLuttinger liquid phase is stable for weak electron-phoncouplings. We consider a cutoff in the logarithmic singulaity below, where the Tomonaga-Luttinger liquid is not reaized. Then some phases disappear from the phase diagr

E. Effects of a cutoff in the logarithmic singularity

We consider three-dimensionality by cutting off the logrithmic singularity in either the particle-particle channelthe 2kF particle-hole channel as explained in the previosection. The most important effect appears in the chargegrees of freedom. It is obvious from Eq.~3.3! that a cutoff inthe particle-particle channel,hÞ0, leadsKr* to 0, while acutoff in the 2kF particle-hole channel,kÞ0, leadsKr* toinfinity. Therefore, a charge gap opens in the former cawhile it does not open in the latter case. Then, possphases are a Mott insulator and a spin-Peierls state informer case~Fig. 6!, and a gapless metallic state and a spgap state in the latter case~Fig. 7!.

If the nesting property is maintained in the quasi-ondimensional case, the particle-particle channel becomes

e

FIG. 5. Phase diagram forXs50.4, Xr520.8, Yr50.375Xs3 ,

andvp50.4EF .

FIG. 6. Phase diagram with a cutoff in the particle-particle chnel, h50.1. The other parameters are the same as in Fig. 5.

thg

p

eron

nlo

atio.8awp

utivhedn-

naerthbithstaakffsupee-

eth

ete

s

nceis

nsi-iveex-

ee-

sis-lic at

he

n-

ngto

an-istheiveif-

theny

e

m

56 7271RENORMALIZATION-GROUP APPROACH TO THE . . .

important at low energies. Then, it is reasonable thatnesting causes a finite charge gap or makes the chargelarger. Note that the phase boundary for a finite spin gashifted to the right in Fig. 6, as expected from Eq.~3.2!. Asthe cutoff increases, a transition occurs from the spin-Peistate to the Mott insulator at zero temperature. This is csistent with the experimental results.~DMe-DCNQI!2Ag hasstrong anisotropy so that it is regarded as a good odimensional material and it becomes an insulator beabout 120–150 K. Meanwhile,~DI-DCNQI!2Ag has a con-siderable conductivity in the transverse direction and it isinsulator already at room temperature. From the activaplot, the charge gap is estimated to be 490 K10

~DMe-DCNQI!2Ag becomes a spin-Peierls state at aboutK so that it has a finite spin gap at zero temperature. Mewhile, ~DI-DCNQI!2Ag becomes antiferromagnetic belo5.5 K ~Ref. 10! so that the spin excitation spectrum is galess. The Mott insulator in the present study is expectedbecome an antiferromagnetic@or spin-density-wave~SDW!#state when weak three-dimensionality is taken into accobecause the repulsive interaction would produce an effecantiferromagnetic coupling in the transverse direction. Tabove result is reminiscent of the earlier work for localizspins at half filling which showed the instability of the spiPeierls state in quasi-one dimension against a SDW.34

If the nesting property is lost in the quasi-one-dimensiocase, the possibility for a finite charge gap would be vstrong electron correlation, which is beyond the scope ofperturbative renormalization-group approach. This possiity may not be excluded, but we can say at least whethertemperature dependence of the resistance within the preapproach is qualitatively consistent with the experimendata or not. It will be done in the next subsection. The weened tendency for a spin gap is common with both cutoNote that the tendency for a spin gap is substantially spressed here~Fig. 7, kÞ0! compared with the other cas~Fig. 6, hÞ0!. This is reasonable in that, in the thredimensional case free from nesting (kÞ0), a spin gap openswhen the phonon-mediated effective attraction overcomthe repulsion, while such condition is not necessary forspin gap in a spin-Peierls state.

F. Temperature dependence of the resistance

Our previous work27 has shown results which are not yconverged so that we show the convergent results here. Htemperatures are in the unit ofEF.2400 K. When theparticle-particle channel is cut off, the resistance increaand the metal-insulator transition temperature increases~Fig.

FIG. 7. Phase diagram with a cutoff in the particle-hole channk50.1. The other parameters are the same as in Fig. 5.

eap

is

ls-

e-w

nn

0n-

-to

nte

e

lyel-e

entl-.-

se

re,

es

8!. With a slight three-dimensional component, the resistaincreases rapidly below the transition temperature. Thisobtained by renormalizing bothYr and Kr . If only Yr isrenormalized, the resistance is not so steep below the tration temperature. Including this, the overall and qualitatbehavior of the resistance is indeed consistent with theperimental one for ~DMe-DCNQI!2Ag and~DI-DCNQI!2Ag.10 Thus, we consider that a cutoff in thparticle-particle channel imitates the effect of thredimensionality for ~DCNQI!2Ag salts. When the 2kFparticle-hole channel is cut off on the other hand, the retance decreases and the system finally becomes metallow temperatures~Fig. 9!. This is obviously contradictorywith the experimental data, as it is expected from trenormalization-group equations.

The real difference between~DMe-DCNQI!2Ag and~DI-DCNQI!2Ag would not be limited to the difference inthe three-dimensionality. In fact,~DI-DCNQI!2Ag hassmaller bandwidth25 so that the relative strength of electroelectron interaction would be larger than~DMe-DCNQI!2Ag. We calculated the resistance with increasicoupling strengths and found that the behavior is similarthe case with increasing cutoffs in the particle-particle chnel ~not shown!. However, the spin degrees of freedomalmost unaffected. Therefore, the three-dimensionality iskey to understand their differences, although the relatcoupling strength is also different and contributes to the dference in the resistance.

V. RESULTS FOR COUPLED SIXTH-AND THIRD-FILLED BANDS

The ground-state phases can be classified according tochannel~s! whose excitation spectrum has a gap. When a

l, FIG. 8. Logarithm of resistance~in arbitrary unit! as a functionof temperature~in the unit of EF! with different cutoffs in theparticle-particle channel,h50, 0.02, 0.04, 0.06, 0.08, and 0.1 fromthe bottom. The parameters areY150.25, Y350.6, and otherwisethe same as in Fig. 5.

FIG. 9. Logarithm of resistance~in arbitrary unit! as a functionof temperature~in the unit of EF! with different cutoffs in theparticle-hole channel,k50, 0.01, 0.02, 0.03, 0.04, and 0.05 frothe top. The parameters areY150.4, Y350.8, and otherwise thesame as in Fig. 5.

rgid

ba

isrmethedrgn

r aanm

lite

onuhs

nxe

a

ns

or

ef

r

iooa

r

es. In

r

, a

tione

ctsor-

-ses.

e

-ins-a-ua-

her

o

n-lay

l

sen-

r-

do

r-

7272 56K. YONEMITSU

perturbation is irrelevant and vanishes in the low-enelimit, the limit is described as a Tomonaga-Luttinger liquwithout any gap. When all of the fourXn’s diverge and bothbands have spin and charge gaps, the system would becally described as a band insulator~if the spin and chargegaps are comparable in the magnitude!. It would be realizedif a CDW of period 6 is formed and the Brillouin zonefolded at6p/6 so that both bands have a gap at the Fepoints. Then, it may be called a Peierls insulator becauscan occur by the Peierls mechanism only, i.e., withouthelp of electron correlation. The experimentally observinsulator phase have a CDW of period 3. There are chagaps in both bands and a spin gap in the third-filled baonly. In fact, it has a long-ranged antiferromagnetic ordelow temperature. Since it can not be described by a bpicture, we call it a Mott insulator. In the phase diagrabelow, each phase is denoted by the number of gap~s!. Thethree important phases above, the Tomonaga-Luttingeruid, the Mott insulator, and the Peierls insulator, are denoby 0, 3, and 4, respectively.

There are also other phases which do not generallycupy a wide area in the phase diagrams below. They caartifacts of the present, lowest-order renormalization-groapproach if the occupied area is very narrow. The mecnism of each phase is discussed below. Phase 1 has agap either in theB band~if the correlation in theA band iscomparable to or stronger than that in theB band! or in theA band~if the correlation in theB band is much stronger!.The former would correspond to a freely moving~unpinned!CDW of period 3. Phase 2 has spin gaps in both bands~ify1Þ0! or charge gaps in both bands~if y150!. The formerwould correspond to a freely moving, CDW of period 6.

A. Interference between different interactions

Before looking at the numerical results, we should cosider what is suggested by the equations. When the fipoint of the correlation exponentKn* is zero, there is a gap inthen channel. For that,Xn( l ) must diverge to2` as l goesto ` in the corresponding equation among Eqs.~3.9!–~3.12!.Since the phonon propagators exponentially decreasethus the electron-phonon interactions~y’s! are finally inte-grated out, the behavior of the electronic perturbatioYABrs( l ), YABr( l ), YBrs( l ), YAr( l ), andYBr( l ), is of par-ticular interest. If some of them diverge, they make the crespondingXn( l ) diverge to2` in Eqs. ~3.9!–~3.12!. Thebehavior of each perturbationY( l ) largely depends upon thfactor multiplied by Y( l ) on the right-hand side o

dY( l )/dl @for example, 22 12 @4KAr( l )1KBs( l )1KBr( l )

1XBs( l )# for YABrs( l )#. Note that, before the equations foYAs( l ) and YBs( l ) ~not shown! are linearized to retain thespin-rotational symmetry, the factors were 222KAs( l ) forYAs( l ) and 222KBs( l ) for YBs( l ). If we neglect the inter-ference effect, i.e., if we do not consider the renormalizatprocess where a pair of non-neutral Burgers vectors are cbined to produce another Burgers vector, these factorsgiven by ~l dependence is implicit hereafte!

22 12 (4KAr1KBs1KBr) for YABrs , 222(KAr1KBr) for

YABr , 22 12 (KBs19KBr) for YBrs , 2218KAr for YAr ,

2218KBr for YBr , 222KAs for YAs , and 222KBs for

y

si-

iit

e,edtd

s

q-d

c-bepa-pin

-d

nd

,

-

nm-re

YBs . They are 22 ‘‘scaling dimensions’’ in the field-theoretical terminology. As the scaling dimension becomlower, the perturbation generally becomes more relevantthe weak-coupling limit~Xn→0 andKn→1!, they are21for YABrs , 22 for YABr , 23 for YBrs , 216 for YAr andYBr , and 0 forYAs andYBs . The corresponding factors fothe electron-phonon interactions are 0 foryA1 and yB1 and21 for yA3 andyB3 . Namely,YAs , YBs , yA1 , andyB1 aremarginal and the others are irrelevant in this limit. Thencharge gap would not open in either band.

Of course, the relevance depends upon the correlastrengths. AsKn’s become smaller, the perturbations becommore relevant. More importantly, the interference effelargely affect the relevance of each perturbation in the renmalization process. The strengths ofYABrs , YABr , YBrs ,YAr , andYBr are initially very small~set to be zero in thenumerical calculations in the next section! because they correspond to high-order electron-electron scattering procesHowever, the electron-2p/3 phonon interactions,yA3 , yB1 ,and yB3 , interfere with one another to produceYABrs ,YABr , YBrs , and the latter three then interfere with onanother as well as with the electron-2p/3 phonon interac-tions. OnceYABrs becomes relevant and diverges,XAr ,XBs , andXBr diverges to2` and gaps open in the corresponding channels. IfYABr becomes relevant, gaps openthe channelsAr and Br. If YBrs becomes relevant, gapopen in the channelsBs andBr. Since the scaling dimension of YABrs is the lowest among the three, the first sitution would always occur when the second or the third sittion is realized. Without the electron-4kF phononinteractions, yA3 and yB3 , on the other hand,YABrs ,YABr , YBrs , andYBr remain zero~when they are initiallyset to be zero!, so that a charge gap does not open in eitband. In such a case, the situation foryB1 and XBs is thesame as that foryA1 andXAs discussed below. In contrast tYABrs , YABr , YBrs , and YBr , the perturbationYAr doesnot interfere with any other perturbation. Its scaling dimesion is much higher than the others so that it would not pan important role.

It is noted that, ifYABrs or YBrs is finite ~though it maybe very small in the real materials!, it and yB1 would inter-fere and produceyA3 or yB3 , respectively. This is also apossibility for opening charge gaps if an infinitesimal~orvery small! value ofyA3 or yB3 is necessary in the numericaresults with a vanishing~initial value of! YABrs or YBrs inthe next section. In the present two-band system, it is estial that phonons with momenta near 2p/3 contribute toyA1 , yA3 , and yB3 processes. In the single-band quartefilled case, the coupling with 2kF phonons interferes with thebackward scattering, and the coupling with 4kF phonons in-terferes with the umklapp process, but the two couplingsnot interfere with each other~Sec. III A! in contrast to thepresent case.

The electron-p/3 phonon interactionyA1 does not inter-fere with another electron-phonon interaction. It only inteferes withXAs . After yA1 is integrated out, i.e.,uyA1u2Dp/3vanishes at l . l 0 for some l 0 , XAs followsXAs( l )5@ l 2 l 01XAs

21( l 0)#21. The fixed point is eitherXAs* 50@KAs* 51# or XAs* 52`@KAs* 50#, depending uponthe sign ofXAs( l 0). In other words, when the electron-p/3

o-

titi-ilathteth

e

itn

,

A

-

nd

e

esin

nal

se.

eam

rengu-in

in

a--x-

pen

56 7273RENORMALIZATION-GROUP APPROACH TO THE . . .

phonon interactionyA1 is strong enough forXAs( l 0) to benegative, a gap opens in the channelAs. Otherwise, theexcitation spectrum of this channel is gapless. If the electr4kF phonon interactions,yA3 andyB3 , are absent, the behavior of yB1 and XBs is exactly the same as that ofyA1 andXAs mentioned above. Without the electron-2kF phonon in-teractions,yA1 andyB1 , YABrs andYBrs remain zero~whenthey are initially set to be zero!, so that a spin gap does noopen in either band for the repulsive case, where the inconditions forXAs andXBs are positive. In short, electron2kF phonon coupling is necessary for a spin gap, whelectron-4kF phonon coupling is necessary for a charge gunless a sufficiently strong long-range interaction makesscaling dimension of a high-order electron-electron scating process lower than 2. This statement holds also forsingle-band quarter-filled case~Sec. IV!.

B. Phase diagrams

In the phase diagrams below, the charge gaps in phasand 4 are brought about byYABrs , while the spin gap in theA band in phases 2 and 4 are brought about byyA1 . The spingap in theB band is caused by eitherYABrs or yB1 . Weinitially set yA15yB15y1 andyA35yB35y3 and varyy1 andy3 for several sets of (XAs ,XAr ,XBs ,XBr), whereXAr andXBr are negative. The strengths ofYABrs , YABr , YBrs ,YAr , andYBr are initially ~i.e., at l 52`! set to be zero inthe numerical calculations since they are of high order wrespect to electron-electron scatterings. For the bare phodispersion, we take 2vp/352v2p/3 /)50.4EF as in the pre-vious section unless explicitly mentioned.

When electron correlation is weak in both bands~Fig. 10!,the phase diagram mainly consists of phase 0~Tomonaga-Luttinger liquid! and phase 4~Peierls insulator!, as expectedunless only the electron-4kF phonon couplingy3 is strong.As the electron-2kF phonon couplingy1 increases, a spingap opens in theB band either mainly byyB1 ~phases0→1! or by constructive interference ofyB1 with yA3 andyB3 ~phases 0→3!. Phase 1 may be changed into phase 3.y1 further increases, a spin gap opens also in theA band~phases 1→2, phases 3→4!. It should be noted that the correlation strength (XAs ,XAr), the electron-2kF phonon cou-pling strength (yA1), and the electron-4kF phonon couplingstrength (yA3) in theA band are the same as the correspoing coupling strengths in theB band in this figure, but a spingap opens first in theB band. This is due to the constructivinterference ofyB1 with yA3 and yB3 . In fact, the criticalcoupling strengths for spin gaps are the same fory350 and

FIG. 10. Phase diagram for (XAs ,XAr ,XBs ,XBr)5(0.4,20.4,0.4,20.4).

n-

al

eper-e

s 3

hon

s

-

different for y3Þ0. As y3 increases, the difference becomlarge because the critical coupling strength for a spin gapthe B band becomes small.

When electron correlation is strong in theA band ~Fig.11!, a stronger electron-2kF phonon couplingy1 is necessaryfor a spin gap in theA band, as expected. If the correlatioremains weak in theB band, the relation between the criticcoupling strength for a spin gap in theB band and that forcharge gaps is very similar to the weakly correlated caNamely, asy1 increases, a spin gap opens in theB bandeither mainly byyB1 ~phases 0→1! or by constructive inter-ference ofyB1 with yA3 and yB3 ~phases 0→3!. Then, thephase 1 is changed into phase 3. Finally, wheny1 overcomesthe strong correlation in theA band, a spin gap opens in thA band~phases 3→4!. As a consequence, the phase diagrmainly consists of phases 0~Tomonaga-Luttinger liquid!, 3~Mott insulator with a CDW of period 3!, and 4 ~Peierlsinsulator with a CDW of period 6!. It should be noted againthat the electron-4kF phonon coupling is necessary focharge gaps. On they1 axis (y350), the possible phases ar0, 1, and 2. Therefore, the condition for phase 3 is strocorrelation in theA band, moderate electron-phonon coplings ~not too strong to overcome the electron correlationtheA band!, and finite electron-4kF phonon coupling thoughit may be small.

On the other hand, when electron correlation is strongthe B band ~Fig. 12!, a stronger electron-2kF phonon cou-pling y1 is necessary for a spin gap in theB band, unlessy3 is strong. Meanwhile, the critical coupling strength forspin gap in theA band remains small if the correlation remains weak in theA band. This situation does not correspond to~DCNQI!2Cu salts because the correlation is epected to be stronger in theA band. Ify3 is small, a spin gapin the A band opens first and then the other three gaps o

FIG. 11. Phase diagram for (XAs ,XAr ,XBs ,XBr)5(0.8,20.8,0.4,20.4).

FIG. 12. Phase diagram for (XAs ,XAr ,XBs ,XBr)5(0.4,20.4,0.8,20.8).

eens

nd

t

nspth

hde-

wit

-th

sio

ae

ydy.

ses at

hesitionast,theulti-e

thee-nearis-the

nce

in

on-ro-s

h

ss,

on-edxci-

r atg a

re-

.

or

7274 56K. YONEMITSU

as y1 increases. Ify3 is large on the other hand, the thregaps except the spin gap in theA band open first. Note phas1 here is different from that in the other figures in the sethat theA band has a spin gap here.

When electron correlation is strong in both bands atheir strengths are the same~Fig. 13!, phase 1 disappears anphase 2 exists only ify350 ~spin gaps! or if y150 ~chargegaps!. The critical coupling strength for a spin gap in theBband is the same as that in theA band for y350 and be-comes smaller than the latter oncey3Þ0 as in Fig. 10. Thetwo curves for these critical coupling strengths are shiftedthe right for smally3 , compared with Fig. 10.

So far we tookXAs5uXAru and XBs5uXBru, assumingonly the on-site repulsion in electron-electron interactioWhen we consider nearest-neighbor repulsion, for examsuch a relation no longer holds. Recall that, in terms ofsingle-band extended Hubbard model,XAs5(U1V)/(pvF), XAr52(U13V)/(pvF) for sixth filling, andXBs5(U2V)/(pvF), XBr52(U15V)/(pvF) for thirdfilling. Even for the same coupling strengths in terms of textended Hubbard model, the scattering strengths depenthe filling factor. Neglecting the difference in the Fermi vlocities for simplicity and taking theX values of the single-band extended Hubbard model for sixth and third fillings,study the effect of the nearest-neighbor repulsion wU/(pvF)50.4 andV/(pvF)50.1 ~Fig. 14!. The resultantphase diagram is similar to Fig. 11 becauseXAs.XBs .When compared with the result forU/(pvF)50.4 andV/(pvF)50 ~Fig. 10!, the nearest-neighbor repulsion increases the critical coupling strength for a spin gap inA band and decreases that for a spin gap in theB band, asexpected from the fact that the nearest-neighbor repulincreasesXAs and decreasesXBs .

Experimentally, pressure induces the metal-insulator trsition. The transition is of first order due to the third-ord

FIG. 13. Phase diagram for (XAs ,XAr ,XBs ,XBr)5(0.8,20.8,0.8,20.8).

FIG. 14. Phase diagram for (XAs ,XAr ,XBs ,XBr)5(0.5,20.7,0.3,20.9).

e

d

o

.le,e

eon

eh

e

n

n-r

commensurability energy26 and it needs three-dimensionalitin phonons, which is beyond the scope of the present stuAs temperature decreases, the resistance abruptly increathe transition under pressure for (R1R2-DCNQI)2Cu withR15R25CH3 or I, but above the transition temperature tresistance behaves as at ambient pressure where no tranoccurs. The pressure would change the bandwidth at lebut it hardly affects the behavior of the resistance abovetransition temperature. We decrease the bandwidth by mplying X’s, uyu2’s, andv’s ~all of which scale as the inversof the band width! by a common factor~1.1–1.4! and calcu-lated the temperature dependence of the resistance incase of Fig. 11 withy1 andy3 near the boundary between thphases 0 and 3~Fig. 15!. The overall behavior of the resistance is insensitive to changes in the parameters eventhe metal-insulator transition, which is qualitatively constent with the experimental results. This is in contrast toquarter-filled case appropriate for~DCNQI!2Ag salts, wherethe correlation strength affects the behavior of the resistarather sensitively~Sec. IV!.

VI. SUMMARY

Metal-insulator transitions and electronic phases(DCNQI)2M ~M5Ag, Cu! salts are studied with therenormalization-group method for the one-dimensional ctinuum models with backward scatterings, umklapp pcesses and couplings with 2kF and 4kF phonons. These saltare in contrast to the quarter-filled (TMTTF)2X and(TMTSF)2X salts, where the extrinsic 4kF anion potentialproduces the umklapp process. For the present salts witM5Ag, such a potential is absent but the electron-4kF phononcoupling interferes constructively with the umklapp procethereby causing a metal-insulator transition. The 4kF CDWis therefore a product of the cooperation of the electrelectron and electron-phonon interactions. It can be viewas a Mott insulator in the sense that it has gapless spin etations and it cannot be described by a band picture.

Experimentally, the physical properties of (R1R2-DCNQI)2Ag depend uponR1R2 . For R15R25CH3, one-dimensionality is rather good. It becomes first an insulatoabout 120 K and then a spin-Peierls state at 80 K, openinspin gap. Meanwhile, forR15R25I, electron transfer in theperpendicular direction is not negligible and electron cor

FIG. 15. Logarithm of resistance~in arbitrary unit! as a functionof temperature~in the unit ofEF! for different correlation strengthsThe parameters are (XAs ,XAr ,XBs ,XBr)5(0.8,20.8,0.4,20.4),y150.4, y350.3, andv’s as before for the curve at the bottom. Fthe other curves, the parametersX’s, uyu2’s, and v’s are 1.1, 1.2,1.3, and 1.4 times the above values.

nthercangn-a

gealv

e

eodmtaio

otth

inot

ngdeths

hengb-

e.,th-gsron

de-to

tran-b-ng,of

theas-

e re-rials

i,ingfulforlicc-

56 7275RENORMALIZATION-GROUP APPROACH TO THE . . .

lation is expected to be stronger due to the narrower bawidth. It is already an insulator at room temperature andspin excitation spectrum remains gapless to zero tempture. In fact, it becomes antiferromagnetic below 5.5 K. Sua qualitative difference is explained by the present approif the three-dimensionality is taken into account by cuttioff the logarithmic singularity in the particle-particle chanel. The cutoff suppresses the opening of a spin gapenhances the charge gap. The stronger correlation forR15R25I would contribute to further enhancing the chargap. The temperature dependence of the resistance is clated with a memory-function approximation and its behaior is consistent with the experimentally observed one.

For M5Cu with coupled sixth- and third-filled bands, thMott transition is accompanied by the formation of a CDWIn real materials, three-dimensionality would not be nglected. Reasons why we take the one-dimensional mare: a Tomonaga-Luttinger behavior is observed in thetallic phase in photoemission experiments; and the meinsulator transition should be explained in one dimensalso.

In order for gaps to open in the charge excitations in bbands and in the spin excitation in the third-filled band,high-order backward scatteringYABrs must be relevant. Itsscaling dimension is higher than 2 in the noninteractlimit. So, it may look as an irrelevant perturbation. It is nthe case if the interference of the electron-2kF phonon cou-pling yB1 with the electron-4kF phonon couplings,yA3 andyB3 , is taken into account. These electron-phonon coupliinterfere with one another and produce effective high-orbackward and umklapp scatterings. The interference ofelectron-phonon couplings and the high-order scattering

Y.ys.

ys,

R.

-hy

i,

-

d-ea-hch

nd

cu--

.-el

e-l-n

he

g

sreis

constructive and can makeYABrs a relevant perturbation.In order for a gap not to open in the spin excitation in t

sixth-filled band, the electron correlation must be stroenough. Therefore, the condition for the experimentally oserved, Mott insulator phase with a CDW of period 3, i.with a gapless spin mode, is strong correlation in the sixfilled, p-d hybrid band, moderate electron-phonon couplinwhich are not too strong to overcome the strong electcorrelation in the sixth-filled band, and finite electron-4kFphonon coupling which may be small. The temperaturependence of the resistance is found to be insensitivechanges in the parameters even near the metal-insulatorsition, which is again consistent with the experimentally oserved one. If electron-phonon couplings were too strothe insulator phase would be accompanied by a CDWperiod 6 and have gaps in all channels.

Showing these results, we have demonstrated thatrenormalization-group method reproduces the qualitativepects of the ground-state phases and the behavior of thsistance in these quasi-one-dimensional organic matevery well.

ACKNOWLEDGMENTS

The author would like to thank A. Chainani, K. Hirakand K. Kanoda for enlightening discussions and showhim data prior to publication and H. Fukuyama for usecomments. This work was supported by a Grant-in-AidScientific Research on Priority Areas ‘‘Anomalous MetalState near the Mott Transition’’ and ‘‘Novel ElectroniStates in Molecular Conductors’’ from the Ministry of Education, Science, Sports and Culture.

K.

to,t.

.

,

1H. Kobayashi, A. Miyamoto, R. Kato, F. Sakai, A. Kobayashi,Yamakita, Y. Furukawa, M. Tasumi, and T. Watanabe, PhRev. B47, 3500~1993!; S. Aonuma, H. Sawa, R. Kato, and HKobayashi, Chem. Lett.1993, 513~1993!; H. Sawa, M. Tamura,S. Aonuma, R. Kato, M. Kinoshita, and H. Kobayashi, J. PhSoc. Jpn.62, 2224~1993!; Y. Kashimura, H. Sawa, S. AonumaR. Kato, H. Takahashi, and N. Moˆri, Solid State Commun.93,675 ~1995!.

2T. Takahashi, K. Kanoda, T. Tamura, K. Hiraki, K. Ikeda,Kato, H. Kobayashi, and A. Kobayashi, Synth. Met.55–57,2281 ~1993!; K. Hiraki, Y. Kobayashi, T. Nakamura, T. Takahashi, S. Aonuma, H. Sawa, R. Kato, and H. Kobayashi, J. PSoc. Jpn.64, 2203~1995!.

3A. Kobayashi, R. Kato, H. Kobayashi, T. Mori, and H. InokuchSolid State Commun.64, 45 ~1987!.

4Y. Suzumura and H. Fukuyama, J. Phys. Soc. Jpn.61, 3322~1992!; H. Fukuyama,ibid. 61, 3452 ~1992!; 62, 1436 ~1993!;Prog. Theor. Phys. Suppl.113, 125~1993!; Y. Suzumura and Y.Ono, J. Phys. Soc. Jpn.62, 3244 ~1993!; T. Ogawa, and Y.Suzumura,ibid. 63, 1494~1994!; 63, 2066~1994!; Phys. Rev. B51, 10 293 ~1995!; 53, 7085 ~1996!; Y. Suzumura and T.Ogawa, Synth. Met.70, 1083~1995!; S. Suzuki, T. Ogawa, andY. Suzumura, J. Phys. Soc. Jpn.65, 2577~1996!.

5H. Fukuyama, inCorrelation Effects in Low-Dimensional Electron Systems, edited by A. Okiji and N. Kawakami~Springer-

.

.

s.

Verlag, Berlin, 1994!, p. 128; Synth. Met.71, 1861~1995!.6T. Ishiguro and K. Yamaji,Organic Superconductors~Springer-

Verlag, Berlin, 1990!.7V. J. Emery, R. Bruinsma, and S. Barisˇic, Phys. Rev. Lett.48,

1039 ~1982!.8V. J. Emery, Phys. Rev. Lett.37, 107 ~1976!.9Y. Nogami, R. Moret, J.-P. Pouget, S. Hayashi, K. Oshima,

Hiraki, and K. Kanoda~unpublished!.10K. Hiraki and K. Kanoda, Mol. Cryst. Liq. Cryst.285, 157

~1996!; Phys. Rev. B54, 17 276~1996!.11J. Solyom, Adv. Phys.28, 209 ~1979!.12V. J. Emery, inHighly Conducting One-Dimensional Solids, ed-

ited by J. T. Devreeseet al. ~Plenum, New York, 1979!.13A. Luther and V. J. Emery, Phys. Rev. Lett.33, 589 ~1974!.14A. Sekiyama, A. Fujimori, S. Aonuma, H. Sawa, and R. Ka

Phys. Rev. B51, 13 899 ~1995!; J. Electron Spectrosc. RelaPhenom.78, 473 ~1996!.

15N. Kawakami and S.-K. Yang, Phys. Rev. Lett.67, 2493~1991!;H. J. Schulz,ibid. 71, 1864 ~1993!; F. Mila and X. Zotos, Eu-rophys. Lett.24, 133 ~1993!; S. Yunoki and S. Maekawa, JPhys. Chem. Solids56, 1779~1995!; P. Kopietz, V. Meden, andK. Schonhammer, Phys. Rev. Lett.74, 2997~1995!.

16S. Capponi, D. Poilblanc, and F. Mila, Phys. Rev. B54, 17 547~1996!.

17K. Yakushi, A. Ugawa, G. Ojima, T. Iida, H. Tajima, H. Kuroda

q.

e

,

s.

et.

7276 56K. YONEMITSU

A. Kobayashi, R. Kato, and H. Kobayashi, Mol. Cryst. LiCryst. 181, 217 ~1990!.

18T. Giamarchi, Phys. Rev. B44, 2905~1991!; T. Giamarchi and A.J. Millis, ibid. 46, 9325~1992!; T. Giamarchi, Physica B230–232, 975 ~1997!.

19W. Gotze and P. Wo¨lfle, Phys. Rev. B6, 1226~1972!.20H. J. Schulz, inStrongly Correlated Electronic Materials: Th

Los Alamos Symposium 1993, edited by K. S. Bedellet al.~Addison-Wesley, Reading, Massachusetts, 1994!, p. 187.

21J. Voit and H. J. Schulz, Phys. Rev. B37, 10 068~1988!; J. Voit,Phys. Rev. Lett.64, 323 ~1990!.

22K. Yonemitsu and M. Imada, Phys. Rev. B54, 2410~1996!.23K. Yonemitsu and M. Imada, Physica C263, 82 ~1996!; Mol.

Cryst. Liq. Cryst.285, 223 ~1996!.24M. Fabrizio, Phys. Rev. B48, 15 838 ~1993!; D. V. Khvesh-

chenko and T. M. Rice,ibid. 50, 252~1994!; 50, 380~1994!; H.

J. Schulz,ibid. 53, 2959~1996!; L. Balents and M. P. A. Fisheribid. 53, 12 133~1996!.

25T. Miyazaki, K. Terakura, Y. Morikawa, and T. Yamasaki, PhyRev. Lett.74, 5104~1995!.

26Y. Suzumura and H. Fukuyama, J. Phys. Soc. Jpn.49, 2081~1980!.

27K. Yonemitsu, Synth. Met.85, 1635~1997!.28T. Holstein, Ann. Phys.~N.Y.! 8, 325 ~1959!; 8, 343 ~1959!.29W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett.42,

1698 ~1979!; Phys. Rev. B22, 2099~1980!.30H. J. Schulz, Phys. Rev. Lett.64, 2831~1990!.31H. J. Schulz, Phys. Rev. Lett.71, 1864~1993!.32M. Meneghetti, G. Lunardi, R. Bozio, and C. Pecile, Synth. M

41–43, 1775~1991!.33K. Hiraki and K. Kanoda~private communication!.34S. Inagaki and H. Fukuyama, J. Phys. Soc. Jpn.52, 3620~1983!.