PHYSICAL REVIEW B 15 MAY 2000-IVOLUME 61, NUMBER 19

Influence of Coulomb interaction and impurity scattering on lattice dimerizationin a one-dimensional system

Yu-Liang Liu and H. Q. Lin*

Department of Physics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, People’s Republic of China~Received 15 September 1999!

We study the influence of the Coulomb interaction and impurity scattering on the dimerization of a one-dimensional system. An analytic expression for the dimerization parameter on dimensionless interaction pa-rameter and phase shift induced by impurity scattering is obtained. For the impurity scattering free case, theweak Coulomb interaction is in favor of the dimerization, whereas the strong interaction suppresses thedimerization. When the impurity scattering is weak, the dimerization is suppressed in the weak-interactionregion and it decreases slowly as the interaction is increased toward the strong-interaction region. Finally, astrong impurity scattering will completely destroy the dimerization.

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Since the discovery of the metallic phase of doped poacetylene, this material has been extensively studied ovepast years.1 Besides its bright future in industry applicationthe material is also of great theoretical interest. Due toquasi-one-dimensional character, detailed numerical simtions can be done on big lattices so to obtain accuratetrapolation at thermodynamic limit. Many analytical aproaches could also be applied to it with success, such aBethe Ansatz and bosonization method. In the early theoical studies, there was a controversy as to the influence oon-site Coulomb interaction between electrons with oppospins on the dimerization of conducting polymers,2–4 and itis clear now that the weak Coulomb interaction is in favorthe lattice dimerization, while the strong Coulomb interation suppresses it, as demonstrated by extensive numecalculations.5–13 The conclusion was also drawanalytically.14 However, the inclusion of the Coulomb inteaction makes the system very complicated and it is difficto obtain an analytic expression for the dependence ofdimerization parameter on the Coulomb interaction. Thany analytic description of the influence of the Coulombteraction on the dimerization is desirable.

In this paper, using the path integral and bosonizatmethods, we study the influence of the Coulomb interactand the impurity scattering on the dimerization. For ondimensional systems, the spectrum of fermions can be linized at its two Fermi surface points because the physproperty of the system is mainly determined by the loenergy exciting states nearby. Due to this prominent chater of one-dimensional systems, we can easily calculateGreen function of the fermions by a factorization Ansatz15–17

through auxiliary fields. Therefore, we can obtain more reable results for the ground state of the system. Here wea concrete analytic expression of the dimerization paramvarying with the interaction strength and the phase shiftduced by the impurity scattering. We show analytically ththe dimerization is increased by the weak Coulomb intertion, suppressed by the strong Coulomb interaction,could be completely destroyed by a strong-impurity scating.

We consider a one-dimensional interacting spinlessmion system whose Hamiltonian is defined as

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H52 i\vFE dx@cR†~x!]xcR~x!2cL

†~x!]xcL~x!#

1VE dxrR~x!rL~x!1lE dx@cR†~x!cL~x!

1cL†~x!cR~x!#1HK , ~1!

wherecR(x)@cR1(x)# is the field operator of right propaga

ing fermions with wave vectors;1kF andcL(x)@cL1(x)# is

the field operator of left propagating fermions with wavectors ;2kF ; rR(L)(x)5cR(L)

1 (x)cR(L)(x) are fermiondensity operators; the spectrum of the fermions is linearinear the Fermi surface withvF being the Fermi velocity;Vdescribes density-density interaction with momentum traferring much smaller thankF ; HK5 1

2 a0Ll2 describes theelastic potential induced by lattice distortionl with L beingthe length of the system anda0 the spring constant. In theboson representation,18–20 it can be easily shown that thHamiltonian ~1! becomes the usual sine-Gordon modHamiltonian. Therefore, some results obtained for the Hamtonian ~1! are also valid for other systems described bysine-Gordon equation.21

The action of the system can be written as~omitting theelastic potential termHK)

S5E0

b

dtE dx$cR†~x,t!DRcR~x,t!1cL

†~x,t!DLcL~x,t!

1l@cR†~x,t!cL~x,t!1cL

†~x,t!cR~x,t!#

1VrR~x,t!rL~x,t!2 ifR~x,t!@rR~x,t!

2cR†~x,t!cR~x,t!#2 ifL~x,t!@rL~x,t!

2cL†~x,t!cL~x,t!#%, ~2!

whereDR(L)5]t7 i\vF]x , b51/kBT, T is the temperatureof the system, andfR(L)(x,t) are the auxiliary fieldsaccount for the constraint conditionsrR(L)(x,t)5cR(L)

† (x,t)cR(L)(x,t).After integrating out the fermion fieldscR(L)(x,t), we

obtain the effective action

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Se f f@r,f#5E0

b

dtE dx$W~x,t!1VrR~x,t!rL~x,t!

2 ifR~x,t!rR~x,t!2 ifL~x,t!rL~x,t!%2Tr~M0!, ~3!

where

M05S DR , l

l, DLD ,

W~x,t!5i

2@fR~x,t!GRR~x,t;x8,t8!

1fL~x,t!GLL~x,t;x8,t8!#ux8→xt8→t

,

and the Green functionsGi j (x,t;x8,t8)( i , j 5R,L), satisfythe linear differential equation

S DR1 ifR , l

l, DL1 ifLD S GRR, GRL

GLR , GLLD

52S d~x2x8!d~t2t8!, 0

0, d~x2x8!d~t2t8!D . ~4!

By using the factorization Ansatz15–17(x2x8!1) for Greenfunctions, we have

GRR~x,t;x8,t8!5GR0~x,t;x8,t8!eQ1(x,t)2Q1(x8,t8)

3el(x2x8)1 f (x,t)2 f (x8,t8)

GLR~x,t;x8,t8!5GR0~x,t;x8,t8!eQ2(x,t)2Q2(x8,t8)

3eip/21l(x2x8)1 f (x,t)2 f (x8,t8)

GLL~x,t;x8,t8!5GL0~x,t;x8,t8!eQ2(x,t)2Q2(x8,t8)

3el(x2x8)1 f (x,t)2 f (x8,t8)

GRL~x,t;x8,t8!5GL0~x,t;x8,t8!eQ1(x,t)2Q1(x8,t8)

3eip/21l(x2x8)1 f (x,t)2 f (x8,t8), ~5!

where

DRQ1~x,t!52 ifR~x,t!, DLQ2~x,t!52 ifL~x,t!,

D RGR0~x,t;x8,t8!52d~x2x8!d~t2t8!,

and

D LGL0~x,t;x8,t8!52d~x2x8!d~t2t8!.

Defining Q1252Q215Q1(x,t)2Q2(x,t), we obtain thenonlinear equation

~]t21]x

212l]x! f 52 ilDLQ212DL fDRf 1DLQ21DRf

~6!~]t

21]x212l]x! f 5 ilDRQ122DRfDL f 1DRQ12DL f .

In Eq. ~6! and the following, we use the unit such that\5vF51.

In order to calculate the potentialW(x,t), we need toknow the Green functionsGRR(LL)(x,t;x601,t). As a first-order approximation, we only retain the linear term in t

functions f (x,t), f (x,t), and Q12(21)(x,t), and neglecthigher order terms in Eq.~6!. Under this approximation, thepotential functionW(x,t) contains only quadratic terms othe auxiliary functionsfR(L)(x,t). The resulting linear equations that can be easily solved are

~]t21]x

212l]x! f 52 ilDLQ21

~7!

~]t21]x

212l]x! f 5 ilDRQ12.

With the use of the Fourier transform, the potential funtion can be written as

W~p,v!5fT~2p,2v!Mf~p,v!2bLl

4p@fR~p,v!

2fL~p,v!#dp,0dv,0 ,~8!

M5p

4p

1

p21v22 i2lpS p1 iv2 il, il

il, p2 iv2 il D ,

wherefT(p,v)5@fR(p,v),fL(p,v)#.By integrating out the auxiliary fieldsfR(L)(p,v), we ob-

tain the effective action

Se f f@r#51

bL (p,v

$rT~2p,2v!Mr~p,v!

1xT~2p,2v!r~p,v!%2bLl2

8p

2Tr~M0!1 12 Tr~M !,

~9!

M5p

p S p2 iv2 il, 2 il1V

2pp

2 il1V

2pp, p1 iv2 il

D ,

where rT(p,v)5@rR(p,v),rL(p,v)#, and xT(p,v)5(21,1)ibLl/4dp,0dv,0 . Using this effective action and thboson representation of the fermion fieldscR(L)(x,t), wecan calculate a variety of correlation functions of the syste

Integrating out the density fieldsrR(L)(x,t), we obtain thetotal energy of the system

E~l!/~bL !51

2a0l22

1

bL FTr~M0!21

2Tr~M !2

1

2Tr~M !G

1 ~l2/8p2!/@V/12 ~V/2p!# . ~10!

The lattice dimerization parameterl can be determined selconsistently by minimizing the total energy,]lE(l)50, andthus after taking thermodynamical limit and lettingb→`,we have the following equation,

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4p S 1

g221D Gl5

3l

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4p

3g~g211!sinh21F 2D

~g211!lG ,

~11!

where g25(12V/2p\vF)/(11V/2p\vF), and D52\vF /a0 is the bandwidth of the system. This resultshown in Fig. 1 by the solid line, as function of 1/g. As it isshown there, in the weak Coulomb interaction region,V,1.3D, the dimerization parameterl is an increasing func-tion of V, and as Coulomb interactionV increases further thedimerization decreases. This result agrees with previcalculations.5,7,10,12,14

Next, we consider the influence of impurity scatteringthe lattice dimerization parameterl by including the back-ward scattering term

Him5U2kF@cR

†~0!cL~0!1cL†~0!cR~0!#, ~12!

which can be obtained from usuald-function-like barrierscattering atx50,

FIG. 1. Dependence of the dimerization parameterl on the interactionstrength parameter 1/g and the phase shiftd induced by the impurity scat-tering. The parametersa0 and bandwidthD are taken asa051.31, andD510. ~a! Solid line: k5cos(d)51.0, i.e., for the impurity scattering freecase.~b! Dotted line:k50.99. ~c! Dashed line:k50.88. ~d! Long-dashedline: k50.77.

s

E dxU~x!c†~x!c~x!

5E dxU~x!@rR~x!1rL~x!1e2 i2kFxcR†~x!cL~x!

1ei2kFxcL†~x!cR~x!#

5U0@rR~0!1rL~0!# 1U2kF

3@cR†~0!cL~0!1cL

†~0!cR~0!#,

and the forward scatteringU0 term is absorbed into thechemical potential~omitted for simplicity!. Using the phaseshift representation method as introduced in Refs. 22 andwe can perform a unitary transformation to eliminate tbackward scattering term such that its influence on themions are incorporated into the interaction terms. Theneffective Hamiltonian of the system can be written as

H52 i\vFE dx@cR†~x!]xcR~x!2cL

†~x!]xcL~x!#

11

2VE dx@arR~x!rR~2x!1arL~x!rL~2x!

12brR~x!rL~x!#1lE dx@cR†~x!cL~x!

1cL†~x!cR~x!#1HK , ~13!

where l5l cos(d ), a5 12 @12cos(2d )#, b5 1

2 @11cos(2d )#,and rR(L)(x)5cR(L)

† (x)cR(L)(x). Note that the impurityscattering free case we have studied is just the specase corresponding to d50, where23 d5arctan@(e/D)121/g tan(U2kF

/\vF)#, is the phase shift in-

duced by the backward scattering potential, ande defined theenergy scale. In the high energy region,e;D and d→U2kF

/\vF;0, whereas in the low energy region,e;e0,

the lowest excitation energy,e0 /D→0, andd→p/2. In Eq.~13!, we omitted the term 2 il sin(d)*2`

` dxsgn(x)

3@cR†(x)cR(2x)1cL

†(2x)cL(x)# because~i! the overlap

between cR(L)† (x) and cR(L)(2x) becomes exponentially

small asx goes aways from 0;~ii ! the integration is from2` to ` and the sign function sgn(x) cancels most contri-butions. By the same approach as we used above, aftergrating out the fermion fieldscR(L)(x), we obtain the effec-tive action

Se f f@ r#51

bL (p,v

H rT~2p,2v!Xr~p,v!11

2xT~2p,2v!r~p,v!J 2

bLl2

8p2Tr @M0~ l !#1

1

2Tr @M ~ l !#,

X5p

2p S p2 ivsz2 i l1S bV

2p2 i l Dsx,

aV

2pp

aV

2pp, p1 ivsz2 i l1S bV

2p2 i l Dsx

D , ~14!

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PRB 61 12 577BRIEF REPORTS

where rT(p,v)5@rR(p,v),rL(p,v),rR(2p,v),rL

(2p,v)#, xT5(21,1,21,1)ibLl/4dp,0dv,0 , M0(l)5M0(l→l), M (l)5M (l→l), andsx,z are the Pauli ma-trices.

Integrating out the density fieldsrR(L)(x,t), we obtain thetotal energy of the system

E8~l!/~bL !51

2a0l22

1

bL H Tr M0@~ l !#21

2Tr @M ~ l !#

21

2Tr @M ~ l !#J 1

l2

64p S 7

g2~d!28D

1l2

64p

1

g2~d!~15!

whereg2(d)5@12V/2p cos(2d )#/@11V/2p cos(2d )#. Againby minimizing the total energy,]lE8(l)50, we obtain theequation that determines dimerization implicitly

a0l52l cos2~d!

32p F 7

g2~d!1

1

g228G1

3l

2pcos2~d!sinh21

3F D

l cos~d!G2l

8pg~g211!cos2~d!sinh21

3@2D/~g211!l cos~d!#2~l/8p! g~d!@g2~d!11#

3cos2~d!sinh21@2D/~g2~d!11!l cos~d!#, ~16!

which is valid for any phase shiftd. The influence of theimpurity scattering on the lattice dimerization parameterl,are plotted in Fig. 1 with several values of phase shift. Tfollowing characters are observed:~1! in the weak-interaction region, the dimerization parameterl is drasticallysuppressed by the impurity scattering;~2! for the weak-

le

e

impurity scattering @1.cos(d ).cos(d 0);0.6#, the maxi-mum values ofl are shifted to the stronger-interaction sidand in the strong-interaction region the dimerization paraeter l slowly varies with the interaction parameter 1/g; ~3!for the strong impurity scattering@cos(d ),cos(d 0)#, thedimerization parameterl is drastically suppressed by thCoulomb interaction, even though it is very small. Therefowe conclude that the impurity scattering strongly influencthe lattice dimerization, and in general it is against the dimization. According to this result, we conjecture that a dimeized system can be completely destroyed by sufficient disder ~random static impurities, as the case studied here!. Thisconclusion is consistent with previous result that the interation induced by disorder can completely destroy the lattidimerization.24

In summary, we have studied the influence of the Colomb interaction and impurity scattering on the lattice dimeization. An analytic expression for the lattice dimerizatioparameterl as function of the Coulomb interaction~ex-pressed in terms of 1/g) and the phase shiftd, induced by theimpurity scattering, is obtained. For the impurity scatterinfree case, the weak Coulomb interaction is in favor of tlattice dimerization, while the strong Coulomb interactiosuppresses it, a result that is consistent with previous studWhen impurity scattering is introduced, the lattice dimeriztion of the system is drastically influenced. In the weak Colomb interaction region, even for small impurity scatterinthe dimerization is heavily suppressed. However, in thegion of strong Coulomb interaction and weak-impurity scatering, the dimerization parameterl decreases slowly as theinteraction strength is increased. As the impurity scatterbecomes stronger, the lattice dimerization is completelystroyed.

This work was supported in part by the Earmarked Grafor Research from the Research Grants Council~RGC! of theHong Kong Government.

*Author to whom all correspondence should be addressed. Etronic address: hqlin@phy.cuhk.edu.hk

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