We introduce a path-integral approach that allows to compute charge density
oscillations in a Luttinger liquid with impurities. We obtain an explicit
expression for the envelope of Friedel oscillations in the presence of
arbitrary electron-electron potentials. As examples, in order to illustrate the
procedure, we show how to use our formula for contact and Coulomb potentials.Comment: 11 pages, no figures, latex. Revised version to appear in PR

Baik, Sanghum

Castanier, Matthew P.

Pierre, Christophe

English

arXiv

We introduce a path-integral approach that allows us to compute charge-density oscillations in a Luttinger liquid with impurities. We obtain an explicit expression for the envelope of Friedel oscillations in the presence of arbitrary electron-electron potentials. As examples, in order to illustrate the procedure, we show how to use our formula for contact and Coulomb potentials. © 2001 The American Physical Society.Fil: Fernandez, Victoria Ines. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; ArgentinaFil: Naon, Carlos Maria. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Física; Argentin

Fernandez, Victoria Ines

Naon, Carlos Maria

CONICET Digital

PHYSICAL REVIEW B, VOLUME 64, 033402Friedel oscillations in a Luttinger liquid with long-range interactionsVictoria I. Ferna´ndez1,2,* and Carlos M. Nao´n1,2,†1Instituto de Fı´sica La Plata, Departamento de Fı´sica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CC 67,1900 La Plata, Argentina2Consejo Nacional de Investigaciones Cientificas y Te´cnicas, 1900 La Plata, Buenos Aires, Argentina~Received 5 October 2000; revised manuscript received 5 March 2001; published 8 June 2001!We introduce a path-integral approach that allows us to compute charge-density oscillations in a Luttingerliquid with impurities. We obtain an explicit expression for the envelope of Friedel oscillations in the presenceof arbitrary electron-electron potentials. As examples, in order to illustrate the procedure, we show how to useour formula for contact and Coulomb potentials.DOI: 10.1103/PhysRevB.64.033402 PACS number~s!: 05.30.Fk, 71.10.PmIn the last few years there has been much activity ad-dressed to the study of condensed matter and statistical me-chanics problems through field-theoretical methods.1 In par-ticular, the physics of one-dimensional ~1D! systems ofstrongly correlated particles has become a very interestingsubject since one can take advantage of the simplicity of themodels at hand and, at the same time, expect to make contactwith experiments. For instance, the recently built quantumwire2 is a good realization of a 1D electron gas. From thetheoretical side, the simplest formulation of a 1D electronicsystem is given by the Tomonaga-Luttinger ~TL! model,3which has been successful in describing some qualitative fea-tures of a Luttinger liquid such as spin-charge separation andnonuniversal exponents in the decay law of correlationfunctions.4 There are, however, two crucial issues that arenot considered in the original versions of this model: thepresence of a nontrivial interaction between electrons andimpurities5 and the effect of long-range ~LR! electron-electron interactions.6,7 As is well known, the former leads tothe occurrence of Friedel oscillations in the charge-densityprofile, at least for Fermi liquids.8 On the other hand, as thedimensionality of a system decreases, charge screening ef-fects become less important and the LR interaction betweenelectrons is expected to play a central role in determining theproperties of the system. In fact, from a theoretical point ofview, the effects of LR interactions have been recently dis-cussed in connection to several problems such as the Fermi-edge singularity,9 the insulator-metal transition,10 and therole of the lattice through umklapp scattering and size-dependent effects.11 Thus, it is quite interesting to study theinterplay between impurities and LR interactions by consid-ering Friedel oscillations in a 1D system. Some time ago,Egger and Grabert12 analyzed this phenomenon. By combin-ing the techniques of standard bosonization13 with the self-consistent harmonic approximation14 and quantum MonteCarlo simulations,15 they were able to get explicit results forboth weak and strong impurity scattering regimes. Later on,the authors of Ref. 16 used bosonization and a scatteringdescription to get some exact results for the short-range caseand for a special value of the coupling constant, equivalent tothe so called ‘‘Toulouse point’’ in the anisotropic Kondoproblem. More recently, the authors of Ref. 17 again usedstandard bosonization to address the same problem empha-0163-1829/2001/64~3!/033402~4!/$20.00 64 0334sizing the equivalence between the TL model in the presenceof a single nonmagnetic impurity and a boundary Sine-Gordon model.In this report, we present an alternative approach to thisproblem based on a path-integral bosonization technique pre-viously developed in the context of nonlocal quantum fieldtheories.18 This method seems to be specially adequate toconsider LR interactions. Indeed, it has recently provided astraightforward derivation of the electronic Green’s functionin the presence of noncontact potentials.19 Then, our mainpurpose here is to show how to extend this formulation to thecomputation of Friedel oscillations.We start from a modified nonlocal Thirring model20 de-scribed by the following ~Euclidean! Lagrangian density:L5iC¯ ~]1g0kF!C1E d2yJm~x !U (m)~x ,y !Jm~y !1C¯ C C2M ~x !C¯ C , ~1!where x5(tx ,x)5(x0 ,x1), and Jm5C¯ gmC . The functionsU (m)(x ,y) are forward2scattering potentials. Setting U (0)5U (1)52d2(x2y), one gets the covariant and local ver-sion of the Thirring model usually studied in the context of(111) quantum field theories ~QFT8s!.On the other hand, the choice U (0)(x ,y)5U(ux2yu) d(tx2ty) and U (1)(x ,y)50, yields the sim-plest version of the Tomonaga-Luttinger ~TL! model with aninstantaneous distance-dependent potential and no current-current fluctuations. The last two terms in Eq. ~1! corre-spond to forward and backward electron-impurity scattering,respectively.The main purpose of the present paper is to evaluate thevacuum expectation value ~VEV! of the charge density:^r~x !&5^C†C1e22ikFxCR† CL1e2ikFxCL†CR&, ~2!for an arbitrary electron2electron potential U (m)(x ,y). Also,we will be especially interested in case the impurity termsare C0(x)5Vd(x2d)5M (x) and C1(x)50, where V is aconstant proportional to the impurity tunneling barrier situ-ated at x5d . Using a suitable representation of the functionaldelta and introducing an auxiliary vector field Am ~see Ref.18 for details!, the partition function of the model underconsideration reads©2001 The American Physical Society02-1BRIEF REPORTS PHYSICAL REVIEW B 64 033402Z5NE DAm e2S[A] det~ i]1A2 A 1C 1s1ig0kF2M ~x !1smemne2ikFxGn!, ~3!where S@A# is the free quadratic action for Am , G05I andG15g5. Equation ~2! can be obtained by functional deriva-tion of Eq. ~3! with respect to the source s0.As it is known, the massivelike determinant in Eq. ~3!cannot be exactly solved, even in the local case. However,we can take advantage of the fact that the vacuum to vacuumfunctional can be written in such a way that nonlocal termsare not present in the determinant. Therefore, the termsA2 A 1C 1s1ig0kF can be decoupled from fermions byperforming chiral and gauge transformations in the fermionicpath2integral measure. Indeed, decomposing Am(x) in lon-gitudinal and transverse piecesAm~x !5emn]nS F~x !2 ikFxA2 D 1]mh~x !21A2@Cm~x !1sm~x !# , ~4!where F and h are boson fields ~to be associated to thenormal modes of the system! and applying, as anticipated,functional bosonization techniques18 to express the fermionicdeterminant in terms of F and h , one finally obtainsZ5NE Dx¯DxDFDh exp~2Sbos!exp~2S f er!3exp~2S@M ,sm#!, ~5!where S f er corresponds to free massless fermions (x and x¯ )andS@M ,sm#5E d2x x¯ @sm~x ! emn e2ikFxGn2M ~x !#e22gg5Fx .~6!Concerning Sbos , it can be more briefly described in mo-mentum space:Sbos5E d2p~2p!2@Fˆ ~p !hˆ ~p !Cˆ m8 ~p !#3S A~p ! C~p !2 E~p !2C~p !2 B~p ! F~p !2E~p !2F~p !2D~p !D S Fˆ ~2p !hˆ ~2p !Cˆ m8 ~2p !D1ikF2 E d2p~2p!2Uˆ (0)21~p !Cˆ 8(0)~p !d2~p !, ~7!03340where we have defined Cm8 5Cm1sm andA(p),B(p),C(p),D(p),E(p), and F(p) are potential-dependent functions ~see Ref. 20 for more details!.At this point, we see that the generating functional can beformally expanded in powers of (M2smemne2ikFxGn), incomplete analogy with the usual procedure employed in thepath-integral bosonization of (111) massive QFT8s.21,22 Infact, the x dependence of this perturbative parameter, to-gether with the appearance of Cm8 (x) in the bosonic actionare two of the new features of the present computation. Aslong as these functions are well behaved, one can assume theexistence of every term in the corresponding series.From now on, we will specialize the computation to thecase s150. This allows us to define M 6(x)5M (x)2s0(x)e72ikFx and one can then perform the above-mentioned expansion of Z taking M 6(x) as perturbative pa-rameters. As explained in Ref. 20, one can show that thesame expansion can be obtained by starting from a purelybosonic nonlocal extension of the sine-Gordon model givenbyL85 12 ~]mw!2112E d2y ]mw~x ! d (m)~x ,y !]mw~y !1Fm ]mw212b2~a1~x ! eibw(x)1a2~x ! e2ibw(x)!,~8!where Fm(x) represents a couple of classical functions to berelated to the Cm8 ’s and d (m)(x ,y) are two bilocal functionsthat will be associated to the electron-electron potentials ~asimilar nonlocality in the kinetic term was considered in thestudy of the influence of LR correlations in the metal-insulator transition23!. b is a constant and a6(x) are func-tions that can be considered as extensions of the parametera0 used by Coleman.24 Indeed, for d (m)505Fm and a15a25a05constant , the model above coincides with theusual sine-Gordon model. In the present approach, the quan-tities a6(x) are related to M 6(x), which are in turn con-nected to the strength of the scatterer. Let us stress that in ourformulation, it is straightforward to consider a nonpointlikeimpurity @a6(x)ÞVd(x2d)# . However, in order to illus-trate our method, in this report we will consider the usualcase of a completely localized impurity. For this particularcase, Eq. ~8! contains the same terms that can be derivedfrom standard bosonization ~see, for instance, Ref. 12!.Now, going to momentum space and employing standardprocedures to evaluate each VEV, the partition function Z8corresponding to this generalized sine-Gordon model coin-cides with Z provided that the following three relations hold:2p2p@p02Uˆ (1)~p !1p12Uˆ (0)~p !#1p25b22@p21dˆ (0)~p !p021dˆ (1)~p !p12#, ~9!2-2BRIEF REPORTS PHYSICAL REVIEW B 64 033402a6~x !b25M 6~x !5M ~x !2s0~x !e72ikFx, ~10!2iCˆ m8 ~2p !emnpn2p@p02Uˆ (1)~p !1p12Uˆ (0)~p !#1p25bFˆ m~2p !pmp21dˆ (0)p021dˆ (1)p12 .~11!Therefore, we have obtained an equivalence between the par-tition functions Z and Z8 corresponding to the nonlocalThirring and sine-Gordon models with extra interactions de-fined above. This means that we can use Z8 together with theabove conditions in order to compute the charge-density inthe Luttinger liquid in the presence of impurities. Indeed, asa result of this bosonization technique, we can evaluate^r(x)& through functional derivation of Z8 instead of Z. In sodoing we obtain:^r~x !&5K iAp ]xw~x !1cos@2Apw~x !22kFx#L ~12!where the VEV is taken with respect to the Lagrangian den-sity L8@s050# obtained from Eq. ~8! after using Eqs. ~9!,~10!, and ~11! and setting s050. Note that we have also setb52Ap .Let us remark that there is an additional contribution toEq. ~12!, coming from the functional derivative of the nor-malization constant N8@Cm8 # with respect to sm . Since thisquantity is a constant, its only effect is to shift the back-ground value of the charge density. For this reason we havejust disregarded it.Now we return to our main goal, that is to use the path-integral framework depicted above in order to obtain an ex-plicit formula for the charge density in a Luttinger liquidwith arbitrary electron-electron and electron-impurity inter-actions. When one imposes these conditions in L8@s050# ,one gets a Lagrangian density that has an undefined parity asa function of w . However it is much simpler to work with aneven Lagrangian since, in this case, all VEV’s of odd func-tions of w will vanish. It is easy to see that the translationw(x)→w(x)1 f (x) yields an even Lagrangian Leven8 pro-vided that the classical function f (x) is tx independent andits gradient satisfies:]xf ~x !12pE dy U~x2y!]yf ~tx ,y!1 iAp V d~x2d !50.~13!We then get^r~x !&5iAp]x f 1cos@A4p f ~x!22kFx#3^cos@A4pw~x !#&Leven8, ~14!where cos@A4p f (x)22kFx# is called the Friedel oscillationand A(x)5^cos@A4pw(x)#&Leven8is the corresponding enve-lope.03340Let us point out that Eq. ~13! has been previously found inRef. 17. As shown by these authors, in the short-range case ithas the solution f 5constant}U whose only effect is to adda constant phase in the cosine term associated to the Friedeloscillation. A nontrivial phenomenon takes place for LR po-tentials, since the cosine ceases to be a periodic function.This nonperiodicity effect, although weak at large distances,could eventually be observed in carbon nanotubes.25From now on we shall focus our attention on the compu-tation of the envelope of the oscillation. Since Leven8 is notexactly solvable, we shall employ the well-known self-consistent harmonic approximation,14 which amounts to re-placing Leven8 byLSCHA512 ~]mw!211pE dy ]xw~tx ,x! U~x2y! ]yw~tx ,y!1m~V ! d~x2d !2 w2, ~15!where m(V) is a constant, related to the impurity strength, tobe variationally determined. The precise relationship be-tween m(V) and V was obtained in Ref. 12. For instance, inthe strong-scattering limit, when V is much larger than acertain bandwidth, one has simply m5V ~See also, Ref. 17!.Let us now consider the computation of A(x) using thisapproximation. Performing a translation in the field w(x)→w(x)1a(x), with a(x) a classical function, we findA(x)5exp iApa(x).Going to momentum space we see that the Fourier trans-form of a(x) satisfies an integral equation whose solution isaˆ ~p !52iApp21p12 2U~p1!pe2ipmxmS 12meip1rI~p0 ,r !p1mI~p0 ,0 ! D ,~16!withI~p0 ,r !5E0‘dq1cos~q1r !p021q12S 11 2U~q1!p D, ~17!where we defined r5ux2du.The envelope of the Friedel oscillation then readsA~r !5exp21pE2‘‘dp0S I~p0 ,0 !2 mI2~p0 ,r !p1mI~p0 ,0 ! D ,~18!which is our main formal result. Indeed, formulas ~17! and~18! give an analytical expression ~exact within the Gaussianapproximation! for A(r) as a function of both the electron-electron potential and the variational parameter m(V). Sincethe self-consistent harmonic approximation seems to fail forweak impurity strength, due to the neglect of interwelltunneling,12 we restrict our analysis to the strong impurityregime. We also consider a large distance approximation ofEq. ~17! which consists of inserting 1/r as infrared cutoff.Let us call Ir(p0 ,r) the integral ~17! regulated in this way.2-3BRIEF REPORTS PHYSICAL REVIEW B 64 033402We will examine, as examples, two specific short-range andCoulomb potentials. This, in turn will allow us to illustratehow to use our general formula ~2! for other cases. More-over, since these problems were previously considered inRefs. 12 and 17 by using standard ~operational! bosoniza-tion, our computation will give an independent confirmationby means of a different approach. First of all, we note that itis convenient to split out the two terms of the exponentialfactor on Eq. ~18!, such that A(r)5exp(T1W)(r).For the simple contact potential U(q1)5U5constant, wegetT~r !5ln~Lr !2g, ~19!andW~r !5g exp~mg2r ! Ei~2mg2r !2g exp~2g ! Ei~22g !,~20!where L is an ultraviolet cutoff. We have also introduced theinteraction constant g5(112U/p)21/2. Taking into accountthe asymptotic behavior of the exponential integral functionEi for mg2r@1, one obtainsA~r !5C~g ,L! ~2g r !2g expS 2 1m g r D , ~21!which coincides with Refs. 12 and 17 under the same re-gime.In the Coulombian case, one has U(uxu)5U/Auxu21b2,whose Fourier transform is U(q1)52UK0(b q1), where bplays the role of a lattice spacing. Inserting this expression inT and W, and considering the same regime as before we findthat W vanishes and03340T~r !52p2U SA12 4Up ln b2r2A12 4Up ln bL2 D , ~22!which yieldsA~r !5C8~g ,b ,L!expS 2ApU ln rb D . ~23!Again, this behavior is equal to the one previously found inRefs. 12 and 17.In summary, we have described an alternative bosoniza-tion approach to the computation of charge-density fluctua-tions. This technique, previously originated in the context ofQFT’s, parallels, in the path-integral framework, the opera-tional schemes usually employed in condensed-matter appli-cations. In particular, we have computed the envelope ofFriedel oscillations in a simple version of the TL model witha nonmagnetic impurity. By combining that bosonizationprocedure and the self-consistent harmonic approximation,we were able to express the envelope of the oscillations as afunction of the electron-electron interaction @see Eqs. ~17!and ~18!#. Finally, as a consistency check of this formal re-sult, and in order to illustrate our method, we considered thelong-distance regime for contact interactions and Coulombpotentials. Our results are in agreement with Refs. 12 and 17.This work was partially supported by the Consejo Nacio-nal de Investigaciones Cientı´ficas y Te´cnicas ~CONICET!,Argentina. We are grateful to Anı´bal Iucci for useful discus-sions. 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Friedel oscillations in a Luttinger liquid with long-range interactions

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