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The influence of quantum lattice fluctuations on theone-dimensional Peierls instability

Claude Bourbonnais, Laurent G. Caron

To cite this version:Claude Bourbonnais, Laurent G. Caron. The influence of quantum lattice fluctuations onthe one-dimensional Peierls instability. Journal de Physique, 1989, 50 (18), pp.2751-2765.�10.1051/jphys:0198900500180275100�. �jpa-00211099�

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The influence of quantum lattice fluctuations on the one-dimensional Peierls instability

Claude Bourbonnais (*) and Laurent G. Caron

Centre de Recherche de Physique des Solides, Département de Physique, Université de

Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1

(Reçu le 31 mai 1989, accepté le 21 juin 1989)

Résumé. 2014 L’influence réciproque des fluctuations quantiques du réseau et du systèmeélectronique pour le modèle du crystal moléculaire unidimensionnel est analysée à partir d’uneapproche d’intégrale de parcours. Avec l’aide du groupe de renormalisation, il est démontré

comment on génère microscopiquement le développement haute température de type Ginzburg-Landau-Wilson quantique. Utilisant un découplage à une boucle pour l’interaction mode-modedu champ de phonon, on montre comment les fluctuations quantiques du réseau favorisent lasuppression de l’instabilité de Peierls. Dans le cas d’une bande demi-remplie, on analyse le casd’électrons avec ou sans spins. L’effet de l’interaction direct électron-électron est considéré et unecomparaison avec les résultats Monte Carlo de Hirsch et Fradkin sur le même modèle est aussiprésentée.

Abstract. 2014 The interplay between quantum lattice and electronic fluctuations in one-dimension-al molecular crystal electron-phonon system is analyzed by a path integral approach. By means ofa high temperature renormalization group method, it is shown how a quantum Ginzburg-Landau-Wilson functional of the phonon field can be generated. Using a single-loop decoupling for themode-mode interaction of the phonon field, it is shown how quantum lattice fluctuations leads toa continuous suppression of the Peierls instability. In the half-filled band case, a detailed analysisis made for electrons with an without spins. The effect of direct electron-electron interaction isconsidered and a comparaison with the Monte Carlo results of Hirsch and Fradkin for the samemodel is made.

J. Phys. France 50 (1989) 2751-2765 15 SEPTEMBRE 1989,

Classification

Physics Abstracts05.30C - 71.38

1. Introduction.

The occurence of the Peierls instability [1] in various type of low dimensional conductors iswell known to give rise to many spectacular cooperative phenomena. This is particularly truefor example for all the rich phenomenology shown by charge density wave [1] and conducting

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180275100

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polymers [2] materials. In general most of the basic analysis were done in the so-calledadiabatic approximation in which the ionic mass M is so large compared to the one ofelectrons that quantum effects on the Peierls distortion can be considered as essentiallyinexistant. For the Su-Schrieffer-Heeger (SSH) [3] and the molecular crystal (MC) [4] modelsHirsch and Fradkin [5] used the Monte Carlo technique to investigate in details the stability ofthe dimerized ground state against the zero point motion of the lattice. In the half-filled bandcase, they found that the Peierls distortion is decreased by quantum lattice fluctuations. Forspinless electrons, there is a critical mass M below which there is no dimerization while forelectrons with spins, the ground state was found to be dimerized for all M > 0. Various

analytical approaches have been used to calculate the first quantum corrections to the meanfield approximations [6] while many others [7-12] discussed the possibility of a completesuppression of the Peierls instability. Here one should mention the 1/N (N being the numberof fermion field components) field theoretical approach to quantum fluctuations for the noninteracting SSH model by Schmeltzer et al. [6].

Bychkov et al [7a] and Gor’kov and Dzyaloshinskii [7b] pointed out several years ago thatthe occurrence of the Peierls instability, in the spirit of the Landau mean field theory, is

meaningfull whenever retardation effects in the phonon induced interaction between

electrons inhibit the quantum interference between 2 kF electron-hole (Peierls) and thesuperconducting (Cooper) channels of correlations. This was found to occur for 2 Ir TOMF wowhere TMF is the mean field ordering temperature and w o is the bare phonon frequency.Otherwise, for 2 -uT9 lù 0’ the retardation was considered as irrelevant so that one is left

with an effective direct electron-electron problem for which the quantum interferencementionned above can not be neglected. In such a case the existence of a Peierls orderparameter in the Landau sense would be lost. Later on, similar arguments have been used inthe frame work of two cut-off renormalization group procedures [9-11] and many of theresults were found to agree qualitatively [10] with the numerical simulations [5].The sharp cut-off procedure is rather crude however, since it tells nothing about the

mechanism by which the Landau Peierls order parameter is suppressed by quantumfluctuations and if the latter induce a smooth decrease or not of the dimerization with

w o as shown by numerical simulations. In this work we would like to study these problems inmore details for the case of the 1D half-filled band molecular crystal model for whichnumerical simulations have been performed [7].

In section 2 we start our analysis with a path integral formulation of the partition functionfor the 1D MC model in the case of interacting spinless electrons. In section 3, a Kadanoff-Wilson type of renormalization group approach, reminiscent of the one used for quasi-one-dimensional conductors [13], is applied to the electronic degrees of freedom in a band-widthcut-off scheme [14]. This naturally leads to the generation of the quantum Ginzburg-Landau-Wilson type functional for the intramolecular phonon field. In section 4, we analyze at theone loop level the quantum effects of the mode-mode coupling (anharmonic) terms on thePeierls softening temperature TMF. By assuming a proportionality relation between

TMF and the dimerization at T = 0, a comparison is made with the results of numerical

simulations [7a, c]. In section 5 our procedure is extended to electrons with spins and wediscuss the differences with the spinless case. In section 6 we summarize our work and

conclude.

2. Functional integral formulation.

The total hamiltonian of the one-dimensional MC model [4] for interacting spinless electronsis written as :

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The electronic part H, is given by :

where t is the electronic hopping and V is the nearest-neighbour electron-electron interactionwith ni = ai ai , i being the site index. The molecular degrees of freedom are described by :

Here P is the momentum and 0 the intramolecular displacement. K is the elastic constant andM is the ionic mass. The electrons are coupled to the molecular displacement through :

where À is the electron-lattice interaction. In the following, we will be interested in the half-filled band case. The continuum version of the model for which the electronic part reduces tothe Tomanaga-Luttinger model [15], is given by :

where eP(K) = vF(PK - KF) is the linearized electronic spectrum for right (p = + ) and left(p = - ) moving electrons with VF = 2 td as the Fermi velocity. L is the length of the system(L = Nd). In the following we will take d = 1 for the lattice constant. In the g-ologynotation, g2 = 2 V is the forward scattering amplitude of the electron-electron interaction.

Following the standard procedure for the path integral formulation of the partition functionfor fermion [16, 13] and boson [17] fields, we can write in Fourier space :

with the measure

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S [Ji *, ip, 0 ] is the full Euclidian action in terms of the anticommuting Grassman

( t/J) and c number field variables (cp) for electronic and molecular degrees of freedomrespectively. The corresponding free field (quadratic) parts of S namely SO[tp *, Il] andSO[o ] are characterized by the bare propagators Do(,w,,) M- 1(,W 2 + w 2)-l with

úJ ° = 0 as the characteristic molecular frequency, Gop (K, £ô m)(i’,, - VF(PK - kF»- and w m = 2 7TmT andw, = (2 n + 1) 7TT. The quartic fermion termcorresponds to SI whereas for the electron-molecular interaction part, Sà [ip *, 0 ], the tworelevant contributions come from the 0 modes near ± 2 KF and q = 0.

3. Renormalization group approach.

The influence of electronic degrees of freedom on the molecular vibrations will be studiedthrough a perturbative renormalization group approach. Exploiting the fact that the

« t/J 4» field theory for the electronic part of the partition function is marginal in onedimension [13] (E = 1 2013 D), we can apply a Kadanoff-Wilson type of transformation of thepartition function for the gi variables. Here, it is the de Broglie characteristic length

v

03BE~F/T for electrons that plays the role of a correlation length [13]. The perturbation theoryin terms of g2 and À is regularized at high energy by a band-width cut-off Eo = 4 t = 2 EF,EF = VF kF being the fermi energy.

In this band-width cut-off scheme, putting gi (*) -+ (*) + Ji (*) for each ip field that enters inS we get :

Fig. 1. - 1 D linearized electronic spectrum. The shaded regions of width -1 E,, (f ) df represent the outerenergy shells states to be integrated in the partial trace operation (9). Eo [Eo is the bare (scaled) bandwidth.

The ’s ( t/J’s) refer to fermion degrees of freedom located inside (outside) an outer bandenergy shell of thickness EO(f) df. Here EO(f) = Eo e- e and dQ 1 (see Fig. 1). The2

03C8’s represent the degrees of freedom to be integrated for all Matsubara frequenciesúJ n. This integration is made with respect to SO[t*, gi ] by considering SI and S À in (8) asperturbation terms. Keeping the other li’s and the cp’s fixed and using the linked clustertheorem this partial trace integration can be formally written as :

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where the subscript c refers to connected diagrams. The outer energy shell averages

(’ ’ ’ ) are defined by :

First, it is clear from (9) that the outer shell integration will generate corrections to the variousterms already present in S. In the following, we will consider only those corrections thatinvolve logarithmic terms. This is the case for example for the quadratic (free) 0 field term(Fig. 2a) and the 2 kF electron-phonon vertex (Fig. 2b). For the former at úJm = 0, a

straigthforward evaluation of the bubble diagram of figure 2a that involves outer shell

electrons, leads to the following recursion formula for the phonon propagator :

where n ( ) is the electron-phonon vertex part due to the electron-electron interaction. Thelatter gives the recursion formula for the electron-lattice interaction (Fig. 2b) :

Note that the renormalization of the electron-phonon vertex part at q = 0 does not involvelogarithmic corrections and will not be considere here.Now, in the electron-electron vertex part, the first logarithmic contributions from the

2 kF electron-hole and Cooper channels (Fig. 2c) have opposite sign and cancel each other.This leads to the well known result [15]

Fig. 2. - Diagrammatic representation of the first order renormalization for the action So + kF full( - k, dashed) electron lines with oblique bars are in the outer energy shell : a) 0 propagator (wavylines) near 2 kF ; b) electron-phonon vertex ; c) forward scattering electron-electron vertex ; d) the firstlogarithmic contribution to the electron self energy ; e) electronic self energy coming from the

0 field.

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that is, 92(£) is a renormalization invariant. This reflects the fact that the 92 process conservesthe number of particles on each branch ± kF.From (12) and (13), the vertex part is easily integrated to give

with y = g2/2 7rt and n(0) = 1. The 2 kF charge-density-wave response function is related tothe vertex part according to [18] :

with the boundary condition Another important

quantity is the auxiliary charge density wave response function [15] :

which is known to present homogeneity properties.Concerning the renormalization of the one-particle propagator, the first logarithmic

correction comes from the diagram given in figure 2d. After an outer shell evaluation, this

self-energy term is of the order of 1 (g2/-ff2t2 ) di which leads, after an f integration, to a16

power law decrease in temperature of the density of states at the fermi Level [15]. However,the presence of this term is not essential to the present discussion and we will restrict our

perturbative RG analysis to leading logarithmic corrections (first order RG) where such aself-energy term does not appear. The electron-phonon interaction will also lead to self-

energy corrections as shown in figure 2e. It can be easily shown that the integration over the0 field will connect the 0 lines. Together with (11) this can be used to study the effect of the2 kF phonon softening on the electronic density of states.

Besides the intramolecular phonon frequency softening produced by electronic charge-density-wave correlations in (11), the partial trace operation in (9) will also generate new termsthat were not present in the bare action (6). As shown in figure 3 for example, the

renormalization generates a high temperature expansion series for the interaction betweenthe 0 field modes near 2 kF and q - 0 to all orders of perturbation theory. Diagrams whichinvolve only q - 0 external lines are not relevant here since there is no phonon softening in theq - 0 sector. However, collecting those terms with 2 kF lines together with the quadratic termin (6) will lead to a quantum Ginzburg-Landau-Wilson type functional for the Euclidianaction of the 0 field. Up to the fourth order in the * ’s we have :

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Note that the last term gives the first contribution for the interaction between q - 0 andq - 2 kF modes. This corresponds to the second diagram of figure 3a. When evaluated at{ q} = {úJ m} = 0, the quartic term coefficients at the step f are given by :

with C == 7 e (3)/2 ir2 (e (3) = 1.202... ).It is interesting to note that, in contrast to other functional approaches, S [ 0 ] in (17), is

obtained without recourse to a complete integration over electronic degrees of freedom. Thisallows to treat explicitely the interplay between the remaining electronic degrees of freedomand lattice fluctuations.

Fig. 3. - Diagrammatic representation of the partial trace generation of the perturbation series for thecouplings between the 0 modes. a) Mode-mode couplings that involved 2 kF 0 lines. b) Couplings withonly q - 0 external lines.

4. Effect of quantum lattice fluctuations.

4.1 g2 = 0 CASE. - We first consider the situation where there is no electron-electroninteraction namely when 92 = 0. This implies that n (Q ) = 1 and from (15) one has,X (f) = - f/4 7rt. Therefore, from the harmonic term in (17), the mean field (MF) softeningcondition at fo MF = ln E F/eMF for the static (úJm = 0) 2 kF mode becomes :

which leads to the MF temperature :

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Although 1D systems can not sustain long range ordering at finite temperature, T°MF gives acharacteristic energy scale at which the lattice developps strong 2 kF fluctuations. In theadiabatic limit where M --+ oo, the 1D half-filled band MC model develops long range order atT = 0 and the dimerization 6 is known to satisfy the proportionality relation6 = A 4 = A 1.75 k B ItF’ where L1 is the electronic gap for the binding energy of a

2 kF electron-hole pair.

Fig. 4. - Diagrams for the one-loop quantum self-energy corrections to phonon propagator at

q = 2 kF. The double wavy lines represent phonon propagator at úJm =1= 0. a) Contributions coming fromthe 2 kF non thermal modes. The last diagram is present in the half-filled band case only.b) Contribution from the q - 0 non thermal modes.

More generally, it is known from the statistical mechanics of Ginzburg-Landau theory thatfor a one component classical order parameter with T MF =1= 0 one has an ordered state atT = 0 characterized by the T = 0 equilibrium MF value of the order parameter [19]. In thepresent case, however, 4) is not static and quantum fluctuations of the lattice will act to reducethe tendency to dimerization at T = 0. This effect originates from the mode-mode couplingterms in (17) in the presence of non thermal fluctuation modes. In a single looprenormalization scheme, which is reminiscent of what has been used in reference [20] forquantum quasi-1D systems, we connect two of the four non thermal 0 lines in the quarticterms of figure 3a. This corresponds to an integration over these O’s at úJm =1= 0. There are sixcontributions involving 2 kF O’s and which are represented by the diagrams of figure 4a andonly one that connectes- two O’s at q - 0 (Fig. 4b). It is clear that such contributions will lead toa supplementary renormalization of the 2 kF phonon propagator at wm = 0 due to nonthermal fluctuations of the 0 field near q - 2 kF and q - 0. The renormalization softeningcondition at f MF = In (EF/TMF) will then read :

where the prime (double prime) summation is for q - 2 kF(q - 0). At £o,,, --A 0, there will beessentially no softening effect for wo in D(q, wm) (C.F. [15] and [17]) so that in (21) one cantake the bare propagator DO(úJm) which is also independent of q. In a half-filled band case,the prime summation near q = 2 kF is performed over 1/4 of the Brillouin Zone due to theequivalence of the ± 2 kF points while the double prime summation which is centered atq = 0 covers half of the zone. Note that in the spirit of a Ginzburg-Landau-Wilson orderparameter expansion for S, all the temperature dependence of the quartic term in (21) have

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been fixed to the unrenormalized mean field temperature 7omF. After the evaluation of thefrequency summation, we finally get for the renormalized mean field temperature :

From this result, we observe that a non zero phonon frequency ù) () will reduce the MF

temperature and thus in turn the dimerization at T = 0. In the limit of non-adiabaticity forexample, where M --+ 0 (w 0 --> oo ), we have TMF --+ 0 and the dimerization vanishes. This isconsistent with the fact that, in this limit, the electron-phonon system becomes equivalent tothe one of free electrons. In the opposite adiabatic limit however, where M - oo and

w o - 0 one has T MF -+ Tm 0 F namely, there is no quantum renormalization and the electronicgap is well known to satisfy the BCS type of relation : k1 0 = 1.75 TMF = 5( A. In theintermediates cases, the results given in (22) infer a continuous decrease of TMF witho. This differs from the more qualitative criterion [7-11] which states that one has a

dimerized ground state whenever 2 7TT£’F > lùO. Otherwise for 2 7TT£’F : W 0, the effectiveelectron-electron interaction induced by a phonon exchange is considered as non-retarded

and, from the quantum interference between the Peierls and the Cooper channels, the MFtransition temperature never occurs and there is no dimerization. From the presentcalculations, however, it is clear that this quantum interference is not the mechanism by whichthe dimerization is destroyed. The interference is rather a consequence of the absence of apole in D (2 kF, lù m = 0 ) which results from the coupling between thermal and non-thermallattice fluctuation modes (Fig. 4). One must note, however, that in the present scheme of

approximation, it is the ratio (0,9 0 that controls in (22) the population of quantum lattice2 TMFmodes. The present mechanism for the renormalization of TMF bears some similarity with thepolaron problem in the sense that it results from the propagation of electron-hole pairs at2 kF for which each particle of the pair emit and absorb virtual phonons (Fig. 4).A rapid decrease of the dimerization due to quantum lattice fluctuations has been also

obtained by Hirsch and Fradkin [7c] by Monte Carlo simulations on the same model. Infigure 5, we have reported their results for the dimerization ratio 6 (o)(»/,6 (0) as a function ofCo Olt at À IlKt = 1.80. The continuous line represents the present results for the ratio

TMF/eMF as obtained from equations (22) and (20). Both results show a similar dependenceon wo. However, the full comparaison of TMF/ eMF with & /,6 (0) assumes that the sameproportionality relation between both quantities remains the same as a function of

cv o. Note that these results significantly differ from the usual « sharp » cut-off criteria shownby the dashed line. Hirsch and Fradkin also concluded that for a given electron-phononinteraction, there is a finite lùO beyond which the ground state is undimerized. In figure 5, thisoccurs at lùO 2:: 1.5 t. As previously mentioned, however, the present calculations only showthe disappearance of TMF in the non-adiabatic limit only. Therefore, the system should bedimerized at T = 0 whenever lùO remains finite. One must observe here that for Monte Carlosimulations finite size effects can mask the detection of a finite dimerization when the latter

becomes small. This should occur whenever the size of the system becomes larger than thevF

one of a binded 2 k electron-hole pair that is when L « VF [21].~

4.2 92 =F 0 CASE. - We have seen that the presence of repulsive forward scattering processesbetween electrons will promote 2 kF charge-density-wave correlations. This gives rise to a

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Fig. 5. - Renormalized mean field temperature as a function of the phonon frequency for spinlesselectrons. The full circles give the dimerization ratio 5 (wo)lâ (0) obtained by numerical simulations ofreference [5c]. The dashed line gives the dimerization profile resulting from the sharp cut-off criteria at2 7rTO.F Wo.

power like singularity in the 2 kF electronic response function. Using (15), the softeningcondition of the quadratic term of (17) at ev m = 0 and q = 2 kF leads to the adiabatic meanfield temperature :

, Therefore, for g2 > 0, electron-electron interactions will increase T°MF with respect to the non-interacting case in agreement with earlier results [8-11].

Since the electron-electron interaction introduces vertex corrections, the mode-mode

coupling is also enhanced by the effect of g2 (cf. Eq. (12) and Figs. 2b, 3a and 4). Followingthe scheme of approximation given in the preceding section for the quantum effects of quarticterms of (17) the renormalized softening condition at 2 kF and úJ m = 0 in the presence of

g2 now reads :

or

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with

Therefore the variation of the softening temperature TMF with coo now follows a nonuniversal power law decay (see Fig. 6). Here again, it is only in the limit of complete non-adiabaticity where wo - oo, A --+ oo, that there is no softening and then no dimerization

(1’ MF -+ 0). In this limit the partition function is equivalent to the spinless Tomanaga-Luttinger model which is known to be gapless at T -- 0 and g2 : 2 7Tt [15].

Fig. 6. - Renormalized mean field temperature as a function of the phonon frequency for electron withspins in the small and large frequency domain. The full circles are the Monte Carlo results of reference[5c] for the dimerization ratio S (Cùo)/ S (0).

5. Electrons with spins.

For electrons with spin one half, the Euclidian action will conserve the same form except thatthe fermion fields are replaced by where a = t , 1 refers to the spin orientation. In thefollowing, we consider the case with no direct electron-electron interaction. We first note that

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because of the spin summation in the diagrams of figure 2a, the unrenormalized mean fieldtemperature is higher and is given by :

In the same way there will be an extra factor of two for the fermion loops of figure 3. Asidefrom these factors, the renormalization of TMF by quantum lattice fluctuations looks at firstsight similar to the one given in equation (22) for spinless electrons. There is a fundamentaldifference however which comes from the effect of the integration over the non-thermalO’s (Fig. 4). The latter will generate effective electron-electron interactions by the exchangeof phonons at ev m =1= 0 which turn out to have non trivial effects only for electrons with spins.In the action S, the generated interaction can be written in the general form :

In the adiabatic limit where M- 1 ---> 0, úJo -. 0, we first observe that this interaction vanishesfor all úJ m =F 0. For non zero w o, rewriting L as L - L ’ we remark that the second term

Cl)m:#=O Cl)m Cl)m=O

at wm = 0 will give rise to a repulsive local electron-electron interaction with no frequencytransfer. In the spirit of the RG procedure of the section 3, this coupling is marginal(dimensionless) and one should expect logarithmic corrections. In the g-ology decomposition

of (27) this will be the case for the backward and Umklapp

processes. From the partial trace operation however only the 2 kF,N

wm = 0 bubble insertions will contribute to the first order renormalization of glPh + 03Ph. Atthe step f, one has :

These couplings decrease with f and are therefore irrelevant at low temperature.The couplings coming from the complete frequency sum are attractive and the contributions

of interest will come from the frequency dependent backward (glPh(co,,», forward

(g2Ph(úJm)) and umklapp (93Ph(-.» scattering processes.At f = 0, they are given by :

In the non adiabatic limit where M - 0 (.w 0 --+ oo ), one has g ph = 92Ph = 93Ph = - À 2/2 K 0and there is no retardation. For such a situation, the inequality glph - 2 g2Ph:> 1 g3Ph 1 prevailsand one has a power law divergence for the 2 kF charge density wave and superconductingresponse functions as T - 0 [15]. As we will see, this is consistent with the fact that in thislimit quantum corrections of the type given in figure 4 will completely inhibit the Peierlsinstability. In the case of finite ionic mass, when we express the phonon propagator atf = 0 as a power series, we have for the giph :

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We note that the retarded coupling terms (n :0 0 ) are no longer marginal but are irrelevant.The presence of w 2 nimplies that these couplings will scale as 2 nfor oo (T --+ 0Therefore, for non zero wo and f > 1, it is the unretarded contribution of giph that willcharacterize the correlations of the electron gas. It follows that charge density wavecorrelations at 2 kF will be involved in the phonon softening (Fig. 2a). For

glph - 2 92Ph - 1 93Ph 1 and large f one should therefore use [15] for the vertex part in (14)n (f) ’" e ’le, with y = 1. The size of the expansion parameter 2 TT T in (30) will determine the

wo

amplitude of retardation transients. The unrenormalized softening increases in the nonadiabatic limit and instead of (26) one has :

which makes sense for 2 w Tfi « Cùo and f 1. The quantum corrections to TMF will be alsoaffected by the vertex part according to the diagrams of figure 4. Taking into account thesummation over spin degrees of freedom, one gets for the renormalized f’ MF :

where A (À, (ùo) is given by (25) and is evaluated at JOMF for y = 1. Therefore, for sizeablewo, the decay of TMF with wo is no longer exponential but rather follows a power law (seeFig. 6). The latter is reminiscent of the one obtained in section 4b for interacting spinlesselectrons.For small (ùo, retardation effects become more important and the enhancement of

n (f ) decreases in order to have 0 when wo - 0. Consequently at small

0, the decrease of TMF with cv o will be given by equation (22) in which "MF is given by (26)and an extra spin factor of two is added to the argument of the exponential. Whenw o increases the 2 kF fluctuations build up in n and the TMF profile must evolve to the onegiven in (32) at sizeable w 0. This change should occur in the 2 7TTF ’" (ùo domain asillustrated in figure 6. This sharply contrasts with the case of spinless electrons discussedsection 5.1 (cf. Fig. 5). Indeed, there are no effects on n (f ) coming from the unretardedcouplings (29) at finite (ù 0 since all these couplings vanish for spinless electrons.The decay of the dimerization & (£ô (» with w o at T = 0, here assumed to be proportional to

TMF, is therefore expected to follow similar profiles. Such a difference with the spinless caseagrees with the Monte Carlo simulations of Hirsch and Fradkin [5c] which show a muchslower decay of 6 with (ùo for electrons with spins. A comparaison between these simulationsand the present calculations for TMF/"MF for small and sizeable wo is given in figure 6 forÀ = 0.9/B/2. A possible source of disparity at large w 0 may originate from the proportionalityrelation between TMF and 6 which is here assumed to be constant but which is more likely tovary with cv o when the electron gas becomes highly correlated at high wo and largef. One must also note that the amplitude of À chosen for the simulations is not small but israther of medium strength. This is near the limit of validity of our perturbation RG approach.

6. Summary.

In summary we have applied a Kadanoff-Wilson type of renormalization group approach to afunctional integral formulation of the partition function for the 1D electron-phonon problem.

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In the half filled band case for the molecular crystal model, successive integrations ofelectronic degrees of freedom lead to the generation of a quantum functional of the molecularfield. In the case of spinless electrons a quantum renormalization of the 2 kF phonon self-energy at the one-loop level leads to a significantly altered softening condition at the meanfield ordering temperature TMF. The mechanism by which TMF is lowered originates from thecoupling, through electrons, of 2 kF molecular vibrational modes with to the non thermal onesnear 2 kF and q - 0.For non interacting spinless electrons, this leads to a rapid, exponential like decrease of

TMF and of the one-component Peierls static order parameter as a function of the baremolecular frequency. When direct electron-electron interaction is included in terms offorward scattering amplitude, charge-density-wave correlations give rise to a non universalpower law-decrease of TMF with wo. At the one loop level, the mechanism of supression ofTMF is akin to a polaron like problem in which the 2 kF electron-hole pair propagation is

dressed by a quantum harmonic molecular field. For the electronic self energy and the

electron-phonon vertex part, the scheme of approximation presented here leads to quantumcorrections a M- 1/2 at large M which in turn become extremely large in the non-adiabaticlimit, in agreement with the Migdal theorem.

It has been found that, for electrons with spins, the Peierls instability is also affected byquantum lattice fluctuations, but there results a much slower decay of TMF with

wo as compared to the spinless case. This comes from quantum fluctuations which induceeffective electron-electron interactions by the exchange of non thermal phonons. The nonretarded parts of these effective interactions turn out to be relevant as f --+ 00 ( T - 0) therebyinducing charge-density-wave fluctuations which are involved in the full evaluation of

TMF. The importance of these correlations depends on the smallness of the ratio2 7T T / úJ 0 which determines the amplitude of the retardation transients that scale to zero asoo.

In all cases, the Peierls instability was found to be completely suppressed in the non

adiabatic limit only wo - oo. In contrast to the usual criterion for the non existence of aPeierls instability for the 2 -rTo MF « WO sector [7-11], the instability always occurs whenretardation effects are not strickly absent.Assuming a proportionality relation between TMF and the dimerization s at T = 0, the

results obtained in this work are in harmony with those obtained from Monte Carlosimulations on the same model. For non interacting spinless electrons, however, Hirsch andFradkin concluded that there is a critical molecular frequency above which the ground state isundimerized. At very small 8 , it is possible that the finite length of the system inhibited theformation of electron-hole pairs with binding energy 2 a oc TMF [21]. One should mentionhowever that Hirsch and Fradkin deduced the absence of a dimerized ground state beyond afinite wo from a perturbation theory in the strong electron-phonon coupling sector [5c].As a final remark we would like to emphasize that results presented here are also quite

relevant to the non-half filled band (incommensurate) case. Aside from a different value ofemF and the absence of certain type of diagrams (e.g. third diagram of Fig. 4a), the latticefluctuations will depress the MF equilibrium value of the amplitude of the order parameterwhich has a non zero value below TMF. For non zero úJo, phase fluctuations of the orderparameter are well known to destroy long range order at T = 0.

Acknowledgments.

The authors would like thank Dr. J. Voit for important discussions at the early stage of thiswork. Fruitfull discussions with Prof. A. Bjelis are also acknowledged.

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References

[1] FRIEDEL J., Low Dimensional Conductors and Superconductors, Eds. D. Jerome and L. G. Caron,NATO Adv. Study Inst. Ser. B (Plenum, New York) 155 (1987) 503 ;

FEINBERG D. and FRIEDEL J., J. Phys. France 49 (1988) 485 ;FEINBERG D. and FRIEDEL J., Low Dimensional Electronic Properties of Molybdenum Bronzes

and Oxydes, Ed. Schlenker (Reidel, Dordrecht) to be published.[2] Yu Lu, Solitons and Polarons in Conducting Polymers (World Scientific, Singapore) 1988.[3] SU W. P., SCHRIEFFER J. R. and HEEGER A. J., Phys. Rev. B 22 (1980) 2099 ;

see also, BARISIC S., Phys. Rev. B 5 (1972) 932 ; ibid (1972) 941.[4] HOLSTEIN T., Ann. Phys. 8 (1959) 325.[5] a) HIRSCH J. E. and FRADKIN E., Phys. Rev. Lett. 49 (1982) 402 ;

b) FRADKIN E. and HIRSCH J. E., Phys. Rev. B 27 (1983) 1680 ;c) HIRSCH J. E. and FRADKIN E., Phys. Rev. B 27 (1983) 1680.

[6] NAKAHARA M. and NAKI K. , Phys. Rev. B 25 (1982) 7789 ;SU Z. , WANG Y. and Yu L., Mol. Cryst. Liq. Cryst. 118 (1985) 75 ;SCHMELTZER D., J. Phys. C 18 (1985) L-1103 ;SCHMELTZER D., ZEIHER R. and HANKE W., Phys. Rev. B 33 (1986) 5141.

[7] a) BYCHKOV Y. A., GOR’KOV L. P. and DZYALOSHINSKII I. E., Sov. Phys. JETP 23 (1966) 489 ;b) GOR’KOV L. P. and DZYALOSHINSKII I. E., Sov. Phys. JETP 40 (1974) 198.

[8] BARISIC S. and BRAZOVSKII S. A., Recent Developments in Condensed Matter Physics, Ed. J.Devreese (Plenum, N.Y.) 1 (1981) 137.

[9] GREST G. S., ABRAHAMS E., CHUI S.-T., LEE P. A. and ZAWADOWSKI A., Phys. Rev. B 14 (1976)1225 ;

CHUI S. T., RICE T. M. and VARMA C. M., Solid State Commun. 15 (1974) 155.[10] CARON L. G. and BOURBONNAIS C., Phys. Rev. B 29 (1984) 4230.[11] VOIT J. and SCHULZ H. J., Phys. Rev. B 37 (1988) 1068.[12] AUBRY S. and QUEMERAIS P., preprint ; ZHENG H., FEINBERG D. and AVIGNON M., Phys. Rev. B

39 (1989) 940.[13] BOURBONNAIS C., Mol. Cryst. Liq. Cryst. 119 (1985) 11 ;

BOURBONNAIS C., thesis, Université de Sherbrooke III-374 (1985), unpublished ;BOURBONNAIS C. and CARON L. G., Physica B 143 (1986) 450 ; Europhys. Lett. 5 (1988) 209.

[14] A short version of this work was presented at the International Conference on Science and

Technology of Synthetic Metals (ICSM’88), Santa Fe, NM, June 26-July 2, 1988, SyntheticMetals 27 (1988) A27.

[15] EMERY V. J., Highly Conducting One-dimensional Solids, Eds. J. T. Devreese and R. P. Evrardand V. E. Van Doren (Plenum N.Y.) 1979 ;

SOLYOM J., Adv. Phys. 28 (1979) 201.[16] OHNUKI Y. and KASHIMA T., Prog. Theor. Phys. 60 (1978) 548 ;

SOPER D., Phys. Rev. D 18 (1978) 4590.[17] FEYNMANN R. P. and HIBBS A. R., Quantum Mechanics and Path Integrals (McGraw Hill, New

York) 1965.[18] DZYALOSHINSKII I. E. and LARKIN A. I. , Sov. Phys. JETP 38 (1972) 202.[19] SCALAPINO D. J., SEARS M. and FERRELL R. A., Phys. Rev. B 6 (1972) 3409.[20] BOURBONNAIS C. and CARON L. G., Phys. Rev. B 29 (1984) 5011.[21] BOURBONNAIS C., NÉLISSE H., REID A. and TREMBLAY A.-M., Phys. Rev. B (1989) to be

published.