15 April 1999

Ž .Physics Letters B 452 1999 98–102

Bosonic description of a Tomonaga-Luttinger modelwith impurities

Victoria Fernandez a, Kang Li a,b, Carlos Naon a,1´ ´a Depto. de Fısica. UniÕersidad Nacional de La Plata. CC 67, 1900 La Plata, Argentina´

b Department of Physics, Zhejiang UniÕersity, Hangzhou 310028, PR China

Received 13 January 1999Editor: R. Gatto

Abstract

We extend a recently proposed non-local version of Coleman’s equivalence between the Thirring and sine-Gordonmodels to the case in which the original fermion fields interact with fixed impurities. We explain how our results can be

Ž .used in the context of one-dimensional strongly correlated systems the so called Tomonaga-Luttinger model to study thedependence of the charge-density oscillations on the range of the fermionic interactions. q 1999 Published by ElsevierScience B.V. All rights reserved.

PACS: 05.30.Fk; 11.10.Lm; 71.10.Pm

w xIn a recent paper 1 a non-local generalization ofColeman’s equivalence between the massive Thirring

w xand sine-Gordon models 2 was established. Specifi-cally it was proved the identity of the vacuum tovacuum functionals corresponding to the followingŽ .Euclidean Lagrangian densities:

1 2 2LL s iCEuCq g d yJ x V x , y J yŽ . Ž . Ž .HT m Ž m . m2

ymCC 1Ž .

1 E-Mail: naon@venus.fisica.unlp.edu.ar

and

21LL s E f xŽ .Ž .SG m2

1 2q d yE f x d x , y E f yŽ . Ž . Ž .H m Ž m . m2

a0y cosbf , 2Ž .2b

Ž . Ž .where J x is the usual fermion current, V x, ym Ž m .Ž . Ž .and d x, y ms0,1 are arbitrary functions ofŽ m .

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00269-5

( )V. Fernandez et al.rPhysics Letters B 452 1999 98–102´ 99

ˆ< < Ž Ž .x y y whose Fourier transforms V p andŽ m .ˆ Ž ..d p must be related byŽ m .

12g

2 2 2ˆ ˆp V qp V qpž /0 Ž1. 1 Ž0.p

b 2

s 3Ž .2 2 2ˆ ˆ4p p qd p qd pž /Ž0. 0 Ž1. 1

For the sake of clarity, we have omitted the p-depen-Ž . 2Ž .dence of the potentials. When V x, y sd xyyŽ m .

Ž .and d x, y s0 one reobtains the usual localŽ m .Žequivalence let us also mention that with the present

convention attractive potentials correspond to posi-.tive values of V . Of course, the usual identityŽ m .

a0ms 4Ž .2b

also holds in this non-local context.The purpose of this letter is to extend the above

result in two directions. First of all we shall considera coordinate-dependent mass in our non-local ver-sion of the Thirring model. At the same time we willalso add a local interaction between the fermioncurrent and a classical background field C . In otherm

Ž .words, we shall modify Eq. 1 above by settingms0 and adding

LL sC CuCym x CC 5Ž . Ž .imp

where the name LL refers to the fact that thisimp

density can be used to study the interaction betweenw x Ž .electrons and impurities 3 . Indeed, if one uses 1

Ž .with ms0 in order to describe the forward scatter-w xing of one-dimensional spinless electrons 4,5 , then

Ž . Ž .the first second term in the rhs of 5 models theŽ .forward backward scattering between electrons and

impurities. In particular, if we choose

C x sVd x yd sm x 6Ž . Ž . Ž . Ž .0 1

where V is a constant, together with

C x s0, 7Ž . Ž .1

our model coincides with the one recently consideredw xin Ref. 6 to study Friedel charge-density oscilla-

w xtions in a 1d Tomonaga-Luttinger liquid 7 with animpurity located at x sd. Moreover, the bosonic1

vacuum to vacuum functional we shall derive here

can also be used to compute the 2-point densityfunction and the electron propagator as functionalsof the electron-electron potential. The former was

w x Žcomputed in Ref. 8 for a Coulomb potential and.without impurities where a Wigner crystal structure

was revealed. Apart from academic interest, theseconnections are the main practical motivation for thepresent computation. Before making contact withthis solid-state application, we now start by consider-

Ž Ž . Ž . .ing the general case C x and m x arbitrary .m

Using a convenient representation of the functionalŽdelta and introducing the vector field A Please seem

w x .Ref. 1 for details , the partition function of themodel under consideration can be written as

Zs DA det iEuqgAuqCuym x eyS w A x , 8Ž . Ž .Ž .H m

with

1 2 2 y1S A s d xd yV x , y A x A y , 9Ž . Ž . Ž . Ž . Ž .H Ž m . m m2

where Vy1 is such thatŽ m .

d2 z Vy1 z , x V y , z sd 2 xyy . 10Ž . Ž . Ž . Ž .H Ž m . Ž m .

As it is known, the massive-like determinant inŽ .Eq. 8 cannot be exactly solved, even in the local

case. However, we were able to write the vacuum tovacuum functional in such a way that non-localterms are not present in the determinant. Then, fol-

w xlowing 1 , we can decouple A and C fromm m

fermions by performing chiral and gauge transforma-tions in the fermionic path-integral measure, with

Ž . Ž .parameters F x and h x respectively:

C x sexp yg g F x q ih x x x ,Ž . Ž . Ž . Ž .Ž .5

11Ž .

Ž Ž ..and a similar expression for C x and writing

1A x se E F x yE h x y C x . 12Ž . Ž . Ž . Ž . Ž .m mn n m mg

Taking into account the non-trivial Fujikawa Jaco-w xbian 9 associated to the fermionic transformation,

Ž .which gives a local kinetic term for F x , we obtain

yS effZs Dx Dx DF Dhe 13Ž .H

( )V. Fernandez et al.rPhysics Letters B 452 1999 98–102´100

where

2 y2 gg F5S sS q d x x iEuxym x x e x ,Ž .Ž .Heff 0 B

14Ž .

g 222S s d x E FŽ .H0 B m2p

1 2 2 y1q d xd ye e V y , xŽ .H ml ms Ž m .2

=E F x E F yŽ . Ž .l s

1 2 2 y1q d xd yV y , x E h x E h yŽ . Ž . Ž .H Ž m . m m2

2 2 y1y d xd y V y , x E h x E F yŽ . Ž . Ž .H Ž0. 0 1

y1yV y , x E h x E F yŽ . Ž . Ž .Ž1. 1 0

12 2 y1y d xd yV x , yŽ .H Ž m .g

=C y e E F x yE h xŽ . Ž . Ž .Ž .m ms s m

12 2 y1q d xd yV x , y C x C y .Ž . Ž . Ž .H Ž m . m m22 g

15Ž .

Note that the last term of S is field independent0 BŽ Ž . .remember that C x is a classical function , thusm

its contribution can be absorbed in a path-integralw xnormalization constant N C which is relevant ifm

one is interested in functional derivatives of Z withŽ .respect to C x . Since here we are mainly con-m

cerned with vacuum to vacuum functionals we shalldisregard this factor. Its contribution will be recov-ered at the end of this work, where the evaluation ofthe fermion current v.e.v. will be sketched. Let usalso mention that one should be more careful withthis normalization when considering the finite tem-perature version of the present calculation, whichcould be easily done by following the lines of Ref.w x Ž .10 . On the other hand, the first term in 15 comesfrom the contribution of the previously mentioned

w xfermionic Jacobian. Exactly as it was done in 1 , thepartition function for the model can be formally

Ž .expanded in powers of m x . In fact, the x depen-dence of this perturbative parameter, together with

Ž .the appearance of C x in the bosonic action arem

the new features of the present computation. As faras these functions are well-behaved one can assumethe existence of every term in the following series:

` 1Zs Ý

n!ns0

=n

2 y2 gg F Ž x .5 j² :d x m x x x e x xŽ . Ž . Ž .ŁH 0j j j jjs1

16Ž .

It is not difficult to convince oneself that the samew xmanipulations used in 1 , in order to compute

fermionic and bosonic v.e.v’s at every order of per-turbation theory, also work in the present case. Thisallows to write

` k12 2Zs d x d y m x m yŽ . Ž .Ý ŁH i i i i2k! is1Ž .ks0

=

2d p 2pexp yH 2 2½ p2pŽ .

2 y1 2 y1 2ˆ ˆ2p g V p qV pŽ .Ž . Ž .0 0 1 1y

2 y1 2 y1 2 2 y1 y1 4ˆ ˆ ˆ ˆg V p qV p p qp V V pŽ .Ž . Ž . Ž . Ž .0 0 1 1 0 1

=D p , x , y D yp , x , yŽ . Ž .i i i i 5=

2 y1ˆ ˆd p 2 iV yp C ypŽ . Ž .Ž .m mexp yH 2½ D pŽ .2pŽ .

= C p p y2e p B p D yp , x , yŽ . Ž . Ž .Ž .m mn n i i 517Ž .

where, for simplicity, we have gone to momentumspace and defined

2g12 y1 2 y1 2ˆ ˆA p s p q V p p qV p p ,Ž . Ž . Ž .Ž0. 1 Ž1. 022p

18Ž .1 y1 2 y1 2ˆ ˆB p s V p p qV p p , 19Ž . Ž . Ž . Ž .Ž0. 0 Ž1. 12

y1 y1ˆ ˆC p s V p yV p p p , 20Ž . Ž . Ž . Ž .Ž0. Ž1. 0 1

DsC 2 p y4 A p B p , 21Ž . Ž . Ž . Ž .

( )V. Fernandez et al.rPhysics Letters B 452 1999 98–102´ 101

and

D p , x , y s ei p x i yei p y i . 22Ž . Ž . Ž .Ýi ii

ˆ ˆIn these expressions F ,h and V are the Fourierˆ Ž m .Ž .transforms of F ,h and V respectively. Eq. 17 isŽ m .

our first non-trivial result. It is the extension of thew x Ž .result presented in 1 to the case m™m x and

Ž .C x /0.m

Let us now briefly make our second relevantobservation which concerns the way of constructinga purely bosonic model that gives the same expan-

Ž .sion as in 17 . It is straightforward to show thatŽ .such model can be obtained from Eq. 2 by setting

a s0 and adding0

a xŽ .0XLL sF x E f x y cosbf x 23Ž . Ž . Ž . Ž .imp m m 2b

Ž .where F x represents a couple of classical func-m

Ž .tions to be related to the C ’s and a x is them 0

x-dependent version of a , to be related, of course,0Ž .to m x . Again, going to momentum space and

employing standard procedures to evaluate eachv.e.v., the partition function ZX corresponding to thisgeneralized sine-Gordon model can be expressed as

2` k1 a x a yŽ . Ž .i iX 2 2Z s d x d yÝ ŁH i i 2 2k! b bis1ks0

=

° 2 2b d p~exp y H 24 2p¢ Ž .

=

¶D p , x , y D yp , x , yŽ . Ž .i i i i •

1 1 ß2 2 2ˆ ˆp q d p p qd p pŽ . Ž .Ž .Ž . Ž .0 0 1 12 2

=

2 ˆd p F yp pŽ .m mexp bH 2 2 2 2½ ˆ ˆp qp d p qp d pŽ . Ž .2pŽ . 0 0 1 1

=D yp , x , y 24Ž . Ž .i i 5

Ž . Ž .By comparing Eq. 24 with Eq. 17 , we find thatŽ .both series are identical if not only Eq. 3 holds, but

one also has:

a xŽ .0m x s 25Ž . Ž .2b

and

p C p y2e p B pŽ . Ž .m mn ny1ˆ ˆ2 iV yp C ypŽ . Ž .Ž m . mD pŽ .

F̂ yp pŽ .m msb 26Ž .

2 2 2ˆ ˆp qd p qd pŽ0. 0 Ž1. 1

Therefore, we have obtained an equivalence be-tween the partition functions Z and ZX correspondingto the non-local Thirring and sine-Gordon models

Ž .with extra interactions defined above, in Eqs. 5 andŽ . Ž .23 respectively. Note that, apart from Eq. 3 which

w x Ž .was already obtained in 1 and Eq. 25 which is aŽ . Ž .trivial generalization of Eq. 4 , Eq. 26 should be

considered the specific original contribution of thepresent work. As stated in the introductory para-graph, this path-integral identification allows to makecontact with recent descriptions of 1d electronic

Ž .systems the so called Tomonaga-Luttinger modelŽ .in the presence of fixed not randomly distributed

w ximpurities 6 . We shall devote the remainder of thisletter to briefly discuss this possibility. To be spe-cific we shall consider the mean value of thefermionic current defined by:

1 dZ² :j sy 27Ž .m Z d Cm

This is an interesting object because j is the charge0

density whereas j is the electric current. Please1

recall that the usual version of the Tomonaga-Lut-tinger model, where only density-density fluctuations

w xare taken into account, corresponds to the choice 1

V̂ s0 28Ž .Ž1.

Concerning the impurity, it is evident that afterŽ .performing the functional derivative in Eq. 27 one

Ž . Ž .has to use Eqs. 6 and 7 for C and C . At this0 1

point the way of exploiting the above depicted pro-

( )V. Fernandez et al.rPhysics Letters B 452 1999 98–102´102

cedure, i.e. the equivalence between Z and ZX, as abosonizing scheme, becomes apparent. Indeed, one

Ž . Ž . Ž .has just to enforce condition 28 in 3 and 26 andŽ . Xthen use Eq. 27 with Z instead of Z. For the most

relevant case of j one has to keep in mind that there0

will be an extra term coming from the disregardedŽC -dependent factor in Z Please remember the com-0

w x Ž ..ments on N C that followed Eq. 15 . Howeverm

this is not a big trouble, since the correspondingcontribution can be obtained by inspection yielding

1 dZX 1y1² :j x sy q VV x ydŽ . Ž .X0 Ž0. 12Z d C x gŽ .0

29Ž .

We want to stress that this expression gives thecharge density of a Tomonaga-Luttinger electronicliquid, with a non-magnetic impurity located at x s1

d, as functional of the electron-electron potential.Thus our present proposal can be considered as analternative path-integral approach to explore the jointeffect of impurities and potentials on the chargedensity behavior. The explicit calculation of the firstterm would allow to check the result recently re-

w xported in Ref. 6 , where a strong dependence of thedensity oscillations on the range of the interactionwas obtained. We hope to consider this issue in a

w xforthcoming article 11 .

Acknowledgements

K.L. thanks Prof. R. Gamboa Saravı for hospital-´Ž .ity in Universidad Nacional de La Plata UNLP ,

Argentina. V.F. and C.N. are partially supported byŽ .Universidad Nacional de La Plata UNLP and Con-

sejo Nacional de Investigaciones Cientıficas y´Ž .Tecnicas CONICET , Argentina. The authors also´

recognize the support of the Third World AcademyŽ .of Sciences TWAS .

References

w x Ž .1 K. Li, C. Naon, J. Phys. A 31 1998 7929.´w x Ž .2 S. Coleman, Phys. Rev. D 11 1975 2088.w x Ž .3 C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 1992 15, 233.w x Ž .4 J. Voit, Rep. on Progr. in Phys. 58 1995 977.w x5 C. Naon, M.C.von. Reichenbach, M.L. Trobo, Nucl. Phys. B´

w x Ž .435 FS 1995 567.w x6 Q. Yuan, H. Chen, Y. Zhang, Y. Chen, Phys. Rev. B 57

Ž .1998 1, 24.w x Ž .7 S. Tomonaga, Prog. Theor. Phys. 5 1950 544: J. Luttinger,

Ž .J. Math. Phys. 4 1963 1154; E. Lieb, D. Mattis, J. Math.Ž .Phys. 6 1965 304.

w x Ž .8 H.J. Schulz, Phys. Rev. Lett. 71 1993 1864.w x Ž .9 K. Fujikawa, Phys. Rev. D 21 1980 2848.

w x Ž .10 M. Manıas, C. Naon, M.L. Trobo, Phys. Lett. B 416 1998´ ´157.

w x11 V. Fernandez, K. Li, C. Naon, in preparation.´ ´