Electronic interactions, charge conjugation symmetry breaking, and phonon dynamics in an extended SSH-model

Synthetic Metals, 27 (1988) A 3 3 - A 4 0

ELECTRONIC BREAKING,

A33

INTERACTIONS, AND

PHONON

CHARGE

CONJUGATION

DYNAMICS

IN AN EXTENDED

SYMMETRY SSH-MODEL

Johannes VOIT, Theoretische Physik 1 and BIMF, Universit~it Bayreuth, D-8580 Bayreuth, Germany

ABSTRACT We present a new weak-coupling approach to the SSH model including interactions for electrons both on sites and bonds as well as phonon dynamics. We find a nonvanishing Umklapp scattering coupling bond and site charges. The influence of these electronic interactions on dimerization and its dependence on the finite phonon frequency is discussed.

INTRODUCTION The basic physics of polyaeetylene [1] is nowadays believed to be contained in the Hamiltonian of the SStI-model [2] describing a one-dimensional (1D) tight-binding model of uncorrelated electrons whose hopping amplitude is modulated by acoustic phonons. One important extension of this model consists in the inclusion of electron-electron interactions often parametrized by an (extended) Hubbard model with an electronic on-site repulsion U (and a nearest neighbor interaction V). An established result of many studies was that the addition of weak electronic interactions increases the dimerization [3,4,5]. It therefore came as a surprise that recent work [6,7] claimed that important terms in the electronic interactions were missing and that upon their inclusion, dimerization would be decreased by electronic correlations. Another point of debate is the influence of quantum fluctuations of the phonons on the dimerization. Several authors[g/ concluded that the low energy properties of the SSH model with finite ionic mass were determined by the Gross-Neveu (w2kr ~ oo) limit. However, it is clear that for low enough phonon frequencies, a crossover to effective semiclassical (Peierls) behaviour must occur. Accurate conditions for this crossover as well as its possible dependence on electronic interactions are not known today. Moreover, many of the previous studies of the influence of electronic interactions were performed in the adiabatic limit for the phonons. Despite the low phonon frequencies in real systems, justifying in principle a mean field theory, it is not clear to what extend the above results hold when the phonon dynamics is retained. This situation calls for an approach treating Coulomb and electron-phonon interaction with phonons of arbitrary frequency in a unified manner. It is the purpose of this paper to present preliminary resuits of such a study, basically an extension of previous work [9] to commensurate systems. We work

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A34 in the continuum limit and therefore encounter significantcomplications due to Umldapp processes. Moreover, we are limited to asymptotic low energy properties in weak-coupling systems. The main advantage of the method is to allow a treatment on equal footing of electron-electronand electron-phonon scattering and an explicit discussion of the role of the phonon frequency.

THE MODEL AND ITS SOLUTION The most general model we shall consider in the following is that of Kivelson, Su, Schrieffer,and Heeger (KSSH) [6],supplemented by the kinetic energy of the acoustic phonons, and given by the

Hamiltouian H

HSSH

(1)

=

HSSH + HO~2) + Hi.t ,

=

1 2 -- ~--~[to- a(u,+1 - un)][c~+1,,cn,,+ H.c.] + ~ K ~ ( U n + I -- Un) , n,#

H(ph) n

(l2)

n

p~ 2M

'

H,., =

V(n, ,l p)d,.c).,c,,.,c.,. + n)rr;~ldl),a)al

cLc. . . .

(4)

n~#

Here, c~,o creates an electron at site n with spin s, to is the nearest neighbor hopping element and V(n, m, l,p) the matrix element of the electron-electron interaction. 6/~ is a shift in the chemical potential such that the Fermi energy remains at E F = 0. u, is the displacement of the ion at site n out of its equilibrium position,pn its momentum, and K and M are its spring constant and mass, respectively. & is the electron-phonon coupling constant. Following KSSH, we only keep electronic couplings involving nearby sites,i.e. the U and V of the extended Hubbard model, and W

and X

introduced by KSSH. W e now go over to the continuum limit and perform a bosonization transformation [9]. For the extended Hubbard model, we arrive at the standard interaction terms denoted by gl... g4 [10]. As noted by KSSH, the main effect of W is to increase gz and decrease g3. There are, however, two new terms: one is parallel-spinUmklapp scattering arising from V and W . Moreover, by carefully evaluating the X-term in the boson representation,we find the following nonvanishing Umklapp process

,,2gs'x/drain [v~).(z) ] R~'x = ~2~j2

g,,x-X '

vF

to

'

(s)

contrary to the assertions of K S S H that the X-term would not couple Fermi surface states in a halffilled band. Here, vF is the Fermi velocity and a a short distance cutoff. ~ ( z ) is a phase field describing charge (v = p) and spin (v = ~,) fluctuations. Further, the effective interaction coming from (5) in second order perturbation theory has exactly the same form as that due to the ordinary Umklapp scattering indicating that this term must not be neglected. The contribution of X to the coupling constants gl ...g4 does vanish. This term couples charge densities on the bonds and on the sites and breaks the charge conjugation symmetry of (1).

A35 Finally, the electron-phonon Hamiltonian becomes ~l,ep

= __2{'1/dZ sin [~,/-2~,(.)] cos [V/2~.(z)] ~2tcj.(z) ~r~

~1 = - - 4 ~ .

(6)

~2hp(z) is a (real) field describing the ±2kr-components of the displacement field u2kr(z) R e [exp(2//¢Fz)~2&p(z)]. Forward scatteringby acoustic phonons is negligible [11].

RESULTS The model consists,in principle,of two XY-models describing charge and spin degrees of freedom which are coupled through various interactionterms, the most important being electron-phonon interaction. The a-XY-model scales, for spin-isotropicinteractions, along the line X~ = Y~ towards the weak coupling fixed point for repulsive electron-electroninteractionsand into a strong coupling regime for attractive ones corresponding to a gap in the spin density excitations. For repulsive coupling, the p-XY-model is placed on the unstable fixed line (Xp < 0,Yp = 0) while for attractive electronic coupling, it is on the stable counterpart Xp > 0. Scaling into the strong-coupling region with positive (negative) Yp implies bond (site)order in the charge degrees of freedom. The equations can practically be guessed from the previous work on the incommensurate electron-phonon system [9]using symmetry arguments. The following features are noteworthy, however. (i) The phonon contribution to Umklapp scattering (gs) has the opposite sign of the contribution to backward scattering (gl) on account of the nonlocal electron-phonon coupling. This is in contrast to previous work by Caron and Bourbonnais [5]and is the technicalfeature underlying the observation of Hubert [12] that the excitation gap must be independent of the sign of the Hubbard U. (ii)It is not possible to derive consistent scaling equations for the SSH-Hubbard-model discarding parallel-spinUmklapp scattering (ga,ll).Such terms are generated by the renormalization group transformations and thereforemust be included from the outset. Their most important property is to couple charge and spin density fluctuations. However, they are irrelevantfor weak coupling and only become relevant for interactionswhose magnitude is a sizablefractionof the bandwidth. A particularly striking example is the extended Hubbard model on the line U : 2V implying gl,]l= gl,± : gz,± : 0 while g2,ll = g2,± ~ U + 2V = 4V, and gs,li ~ - 2 V . We find that gs,ll goes relevant for 2V ,~ 3. There is a new fixed point (gs,[[ --~ oo, g2,1[ --* oo, g2,± finite) in addition to the usual one (gs, H = 0, g2 finite). It is associated with the first order transition between COW and SOW at intermediate couplings recently found in numerical work [13]. Our results for the extended Hubbard model suggest a picture where there is long range order in an up-spin COW and a down-spin C O W with only very weak interactions between them. (iii) Also the eIectron-phonon interaction couples charge and spin density fluctuations at energy scales above the phonon frequency, i.e. as long as the phonons behave effectively retarded. Contrary to (ii), this coupling is strongly relevant also for weak coupling [9].

A36 (iv) As discussed previously, the phonon frequency is the most important parameter controlling the interplay of attractive eleetron-phonon and repulsive Coulomb interaction [9]. This is visible in the scaling equations through an exponential suppression of the contributions of electron-phonon scattering to the parameters describing electron-electron interactions as soon as the electronic energy scale is decreased below the phonon frequency w2kv = 2~/K/M. Below this scale, the electron-phonon interaction behaves effectively unretarded. SSH-model with finite phonon frequencies. The electron-phonon system has a gap in its excitation spectrum at any finite coupling constant 71 and at any phonon frequency. However, our results show a marked crossover between two limiting cases. In a high phonon frequency regime and for relatively weak electron-phonon coupling, we can scale our model consistently onto one containing only instantaneous though renormalized interactions. The electronic gap is then given by A ~ w2~r e x p ( - 1 / ] Yv* 1), where Yp* = - Y * is the effective electron-electron interaction generated once all retardation effects have been scaled out. Consistency requires both electron-phonon and the generated electron-electron interactions to remain small at that energy scale (Y~* < 1, Yp*h= 0 71 Is /xvFPlW~k~,)* < Ee/w2kl,). For frequencies below (or coupling constants above) the values given by this criterion, we determine the gap and the order parameter from the adiabatic limit. Fluctuation effects may then be accounted for by perturbation theory. The consistency condition given clearly marks the crossover criterion between Gross-Neveu (i.e. weakly retarded, weak-coupling) and Peierls (strongly retarded, strong-coupling) behaviour. Our results are in good agreement with previous work [8]. SSH-model with electronic interactions. We first discuss the case of relatively high phonon frequencies. The renormalized propagator of the 2 k F - p a r t of the phonon spectrum is given by [14]

D(2kr q- q) = Do(2kF 4- q) "1-7~Do(2kF "1-q)RBow(2kF "l- q)D(2kF -{- q) , RBOw(2kF q'- q) "~ ABOW(VF I q l) -aB°m , I q I<<( 2kF"

(7) (8)

Do is the bare propagator and RBOW the BOW correlation function. The influence of electronelectron interaction on dimerization is assessed quantitatively through its renormalization of both Yph

and a v o w . In the following Figures, we display the scaling flow of these two quantities as high energy fluctuations are integrated out. Fig. 1 shows the influence of a Hubbard U on Yvh and a v o w for fixed eleetron-phonon coupling and phonon frequency. In agreement with expectation, both Yph and

a v o w are increased by a finite U at any length scale. Moreover, an improved method [16] allows to derive also renormalization equations for the prefactor A v o w in Eq.(8); it is found that ABow, too, increases with increasing U. Thus, although the correlation exponent reaches the value aBow = 2 indicating B O W

long range order at T = 0 independently of U, it is fair to conclude that dimerization

increases with U. Notice that at finite temperature, renormalization would stop at lT = In(E~,/T) leading to an even increased influence of U on dimerization. We do not find a change in these trends

A37 20

.

,

.

,.; i

i

;

•ph10

I I;

OCBOW

5

1

!

2

2

3

l

l

Figure i: Influence of a Hubbard on-site repulsion on the scaling flow of effective electron-phonon interaction Yvh and B O W

correlation exponent aBOW. Yvh(l = 0) = 0.5, w2k~, = O.IEF, (a): U = 0,

(b): U = 0.5, (c): V -- 1, (d): U -- 2 in units of to.

within the domain of validity of our approach, i.e. weak coupling. We have also investigated the influence of a finite nearest-neighbour repulsion V on the dimerization of the system shown in Fig.1 and find that the dimerization is increased by nonzero V. Surprisingly, and in contrast to previous work [4], we do not find a decrease in dimerization for V > it is clear that b o t h gl,± and g3.± change sign for V >

UI2. While

UI2, at the same time the electron-phonon

interaction becomes more relevant with increasing V. However, the value of V where the dimerization starts to decrease, approaches

U/2 with decreasing Yph and w2kj, indicating clearly the importance of

treating the interactions of electrons among themselves and with dynamic phonons on an equal basis. More interesting in view of the recent debate about possibly missing interactions in the SSHHubbard model for a description of e.g. polyacetylene is the influence of the new terms W and X [6,7]. Fig.2 displays the change in scaling behaviour with increasing W. Clearly, dimerization strongly decreases and ultimately is completely destroyed. The curve where aBOW < 0 for all l describes a system with sizable triplet-superconducting fluctuations. The coupling term parametrized by X does not vanish in a haft-filled band but produces an Umldapp scattering. Due to the particular symmetry of this term it is neither renormalized by electronphonon interaction nor does it renormalize that interaction. This scattering always acts in favour of bond order but could itself become relevant only in the presence of interactions favouring site order. As a consequence, it usually has marginal scaling behaviour. This peculiar behaviour results from the fact that all other interactions drive the system away from the B O W - C D W

separation line.

From the previous discussion it has become apparent that the system practically always dirnerizes for the parameters Yph and wlkp chosen as long as we do not have very specialcombinations of electronic interactions with some of them being extremely strong (e.g.W >> all other coupling constants). This is in agreement with the picture obtained by simple inspection of the scaling equations showing that Y~h

A38 20

a

t

:(c)

'

i

i

2

15

~XBoW 0

5

0

i

-I

i

2

4

2

6

l

4

l

Figure 2: Effect of bond charge repulsion W on dimerization through Yph and aBOW.

Yph(l = O) = 0.5, w~/¢~. = 0.1EF, U = 0.3, (a): W = O, (b): W = 0.1, (c): W = 0.3, (d): W = 0.5 in units of to. is the most relevant variable among all coupling constants. Specifically, dlnYph(f)/dt _> 1 for small f, where the equal-sign applies for the case without electron-electron interaction and the presence of these interactions generally increases the derivative. Electron-electron interactions may themselves become relevant, however with significantly smaller derivatives. Therefore in practice, their relevance implies significant quantitative corrections to the dimerization but only in very special cases qualitative changes to different ground states. With decreasing electron-phonon coupling and, more importantly, increasing phonon frequency, the relative importance of electron-electron and electron-phonon coupling in a v o w m a y change considerably and eventually lead to an undimerized state. This is due to the fact that explicit phonon corrections to the electronic quantities vanish for energy scales below w2k~, and is exemplified in Fig. 2(d), where sufficiently strong W changes the system into a state where triplet-superconducting fluctuations are dominant despite the fact that Yph(t) continues to increase under scaling. With decreasing phonon frequency and for fixed bare electron-phonon coupling constant (in units of EF), dimerization considerably increases. One may then be in a regime where a consistent renormalization group treatment is no longer possible. We use standard crossover scaling arguments [15] to determine both the order parameter 71~o0 and electronic gap A [9] as I+aBOW/3

')'I~PO

~

\ 1 -F c~vow/2/

, A ,.,

--(71~°)~+~a°wa

(9)

where Ctsow now is the BOW correlation function exponent characterizing the electronic system in the absence of electron-phonon interaction, e.g. in the Hubbard model at T = 0: a v o w = 1 for any U > 0. In the extended Hubbard model, we have c~sow = 1 for V < U/2. For V > U/2 we have A ~ e x p ( - 1 / 2 Y ~ ) as in the case without interaction. Similar exponents can be derived in the presence of W depending on the relative magnitude of W, V, and U.

A39 A and 7x~o vary continuously with c~Bow: while A increases monotonously with aBOW, 71~o0 first increases and then starts to decrease as a function of a B o w (and, thereby, of U in a Hubbard model) [5].This behaviour is indeed extremely close to what is observed in Monte Carlo simulations [4]. However, at least at T = 0, such a discussion is misleading: then, a B O W has to be determined from the fixed points of the electronicsystem which, in the case of the Hubbard model do not depend continuously on U. Consequently, the present theory would predict dimerization to be considerably enhanced by U but not vary continuously with U. The same arguments would apply for the extended Hubbard model, giving a decrease in dimerization when V > U/2. The variation of dimerization in the presence of electronicinteractionsin the strong retardation case directlyreflectsthe zero temperature correlationfunctions of the electron-electronHamiltonian considered. W e note, however, that it is only precisely at

T :

0 that these singularitiesarise. At finitebut low temperatures where the exponents

a B O W are slightlyaway from their T = 0 fixed point values due to incomplete scaling and therefore do

depend explicitelyon the electronicinteractions,a continuous dependence of dimerization on electronelectron interactions is obtained. Our results axe valid in the weak-coupling limit.For the Hubbard model, it is clear,however, from a mapping onto a spin-Peierls problem in the limit U ~> 4t that the dimerization must ultimately decrease. W e suspect the crossover to occur for U ..~4t marking the limit between weak-coupling band picture and a strong-coupling atomic limit. However, this parameter region lies outside the domain where our approach is valid; here results of numerical simulations are definitelymore reliable [4].

CONCLUSIONS The SSH model extended to include various types of interactions shows an extraordinary richness of physical behaviour and phase transitions. The phonon frequency is an important parameter controlling the interplay of attractive electron-phonon and repulsive electron-electron interactions and therefore enhancement or suppression of dimerization. We have centered our present discussion on the influence of various types of electron-electron interactions: while U and V increase dimerization, the recently proposed bond charge interactions ~V (X) depress (enhance) diinerization. In a situation U > X V = W [6], dimerization decreases in general with respect to the noninteracting case. The condition V >> V~ is necessary for a dimerization enhancement [7] depending, however, sensitively on a~2kr. Eventually, one may even obtain a CDW instead of a BOW. An important effect of the X-term has not been discussed in the present paper. It contributes a scale-dependent renormalization to the velocity rp of the collective charge density fluctuations which ,,Mike all other known contributions [9] depends explicitly on the sign of X. These effects should be visible in the thermodynamic properties of the system which generally depend on rp [9]. I wish to acknowledge useful discussions with H. Bflttner. K. Fesser, H. ]. Scbulz, and V. Waas.

A40

REFERENCES [1] For reviews on recent progress, see T. A. Skotheim (ed.), Handbook of Conductin~ Polymers, Marcel Dekker Inc., New York, 1986; S. Roth and H. Bleier, Adv. Phys. 36 (1987) 385. [2] W.-P. Su, J. It. Schrieffer, and A. J. Heeger, Phys. Rev. Left. 42 (1979) 1698, Phys. Itev. B 22 (1980) 2099, and 28 (1983) 1138. [3] P. Horsch, Phys. Rev. B 24 (1981) 7351; D. Baeriswyl and K. Maki, Phys. Itev. B 31 (1985) 6633; S. Kivelson and D. E. Helm, Phys. Itev. B 26 (1982) 4278. [4] S. Mazumdar and S. N. Dixit, Phys. Itev. Lett. 51 (1983) 292; J. E. H_irsch, Phys. Itev. Lett. 51 (1983) 296; D. K. Campbell, T. A. DeGrand, and S. Mazumdar, Phys. Rev. Lett. 51 (1983) 1717. [5] L. G. Caron and C. Bourbormais, Phys. Hey. B 29 (1984) 4230; B. Horovitz and J. SSlyom, Phys. Rev. B 32 (1985) 2681. [6] S. Kivelson, W.-P. Su, J. It. Schrieffer, and A. J. Heeger, Phys. Itev. Lett. 58 (1987) 1899, and 60 (1988) 72; C. Wu, X. Sun, and K. Nasu, Phys. Itev. Lett. 59 (1987) 831. [7] For a dissenting point of view, see D. Baeriswyl, P. Horsch, and K. Maki Phys. Itev. Lett. 60 (1988) 70, and J. T. Gammel and D. K. Campbell ibid. 71. [8] E. Fradkin and J. E. Hirsch, Phys. Rev. B 27 (1983) 1680; D. Schmeltzer, R. Zeyher, and W. Hanke, Phys. Rev. B 33 (1986) 5141. [9] J. Volt and H. J. Schulz, Phys. Itev. B 34 (1986) 7429, 36 (1987) 968, and 3_.77(1988); J. Volt, Synth. Met. 27 (1988) A41 (these Proceedings). [10] J. S61yom, Adv. Phys. 28 (1979) 201. V. J. Emery, in J. T. Devreese and V. E. van Doren (eds.) Hi~ltly Conductin~ One-Dimensional Solids, Plenum Press, 1979, p.247. [11] J. Volt and H. J. Schu.lz, Molec. Cryst. Liq. Cryst. 119 (1985) 449. [12] L. Hubert, Phys. Itev. B 36 (1987) 6175. [13] J. E. Hirsch, Phys. Rev. Lett. 53 (1984) 2327; B. Fourcade M. Sc. thesis, University of Montreal, 1984, (unpublished). [14] We reserve the abreviation CDW for a charge density wave centred on the sites and use BOW ( : b o n d order wave) for one centred on the bonds. [15] M. P. M. den Nijs, Phys. Rev. B 23 (1981) 6111. [16] J. Voit, to be published.