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The influence of quantum lattice fluctuations on the one-dimensional Peierls instability
Claude Bourbonnais, Laurent G. Caron
To cite this version:
Claude Bourbonnais, Laurent G. Caron. The influence of quantum lattice fluctuations on the one-dimensional Peierls instability. Journal de Physique, 1989, 50 (18), pp.2751-2765.
�10.1051/jphys:0198900500180275100�. �jpa-00211099�
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 système électronique pour le modèle du crystal moléculaire unidimensionnel est analysée à partir d’une approche 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-mode du champ de phonon, on montre comment les fluctuations quantiques du réseau favorisent la
suppression de l’instabilité de Peierls. Dans le cas d’une bande demi-remplie, on analyse le cas
d’électrons avec ou sans spins. L’effet de l’interaction direct électron-électron est considéré et une
comparaison avec les résultats Monte Carlo de Hirsch et Fradkin sur le même modèle est aussi
présentée.
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 of
a 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 the
mode-mode interaction of the phonon field, it is shown how quantum lattice fluctuations leads to a continuous suppression of the Peierls instability. In the half-filled band case, a detailed analysis
is made for electrons with an without spins. The effect of direct electron-electron interaction is considered and a comparaison with the Monte Carlo results of Hirsch and Fradkin for the same
model is made.
Classification
Physics Abstracts
05.30C - 71.38
1. Introduction.
The occurence of the Peierls instability [1] in various type of low dimensional conductors is well known to give rise to many spectacular cooperative phenomena. This is particularly true
for 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
polymers [2] materials. In general most of the basic analysis were done in the so-called adiabatic approximation in which the ionic mass M is so large compared to the one of
electrons that quantum effects on the Peierls distortion can be considered as essentially
inexistant. For the Su-Schrieffer-Heeger (SSH) [3] and the molecular crystal (MC) [4] models
Hirsch and Fradkin [5] used the Monte Carlo technique to investigate in details the stability of
the dimerized ground state against the zero point motion of the lattice. In the half-filled band case, they found that the Peierls distortion is decreased by quantum lattice fluctuations. For
spinless electrons, there is a critical mass M below which there is no dimerization while for electrons 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 mean
field approximations [6] while many others [7-12] discussed the possibility of a complete suppression of the Peierls instability. Here one should mention the 1/N (N being the number
of fermion field components) field theoretical approach to quantum fluctuations for the non
interacting SSH model by Schmeltzer et al. [6].
Bychkov et al [7a] and Gor’kov and Dzyaloshinskii [7b] pointed out several years ago that the 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 the superconducting (Cooper) channels of correlations. This was found to occur for 2 Ir TOMF wo where 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 interference mentionned above can not be neglected. In such a case the existence of a Peierls order parameter in the Landau sense would be lost. Later on, similar arguments have been used in the frame work of two cut-off renormalization group procedures [9-11] and many of the results 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 quantum fluctuations 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 in
more details for the case of the 1D half-filled band molecular crystal model for which numerical simulations have been performed [7].
In section 2 we start our analysis with a path integral formulation of the partition function
for 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-width
cut-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 the
one loop level the quantum effects of the mode-mode coupling (anharmonic) terms on the
Peierls 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 we
discuss 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 electrons
is written as :
The electronic part H, is given by :
where t is the electronic hopping and V is the nearest-neighbour electron-electron interaction with 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 and M 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 to the 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-ology notation, 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 function
for fermion [16, 13] and boson [17] fields, we can write in Fourier space :
with the measure
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 freedom respectively. The corresponding free field (quadratic) parts of S namely SO[tp *, Il] and SO[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 term corresponds to SI whereas for the electron-molecular interaction part, Sà [ip *, 0 ], the two
relevant 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 studied
through 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 one
dimension [13] (E = 1 2013 D), we can apply a Kadanoff-Wilson type of transformation of the
partition function for the gi variables. Here, it is the de Broglie characteristic length
03BE~F/T
v 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 in S we get :
Fig. 1. - 1 D linearized electronic spectrum. The shaded regions of width -1 E,, (f ) df represent the outer energy shells states to be integrated in the partial trace operation (9). Eo [Eo is the bare (scaled) band
width.
The ’s ( t/J’s) refer to fermion degrees of freedom located inside (outside) an outer band
energy shell of thickness EO(f) df. Here EO(f) = Eo e- e and dQ 1 (see Fig. 1). The
2
03C8’s represent the degrees of freedom to be integrated for all Matsubara frequencies
úJ n. This integration is made with respect to SO[t*, gi ] by considering SI and S À in (8) as perturbation terms. Keeping the other li’s and the cp’s fixed and using the linked cluster theorem this partial trace integration can be formally written as :
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 various terms already present in S. In the following, we will consider only those corrections that involve 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. The latter 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 involve logarithmic 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 (wavy lines) near 2 kF ; b) electron-phonon vertex ; c) forward scattering electron-electron vertex ; d) the first logarithmic contribution to the electron self energy ; e) electronic self energy coming from the 0 field.
that is, 92(£) is a renormalization invariant. This reflects the fact that the 92 process conserves the 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 to the 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 16 (g2/-ff2t2 ) di which leads, after an f integration, to a
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 a
self-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 the 0 field will connect the 0 lines. Together with (11) this can be used to study the effect of the 2 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 terms that 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 between
the 0 field modes near 2 kF and q - 0 to all orders of perturbation theory. Diagrams which
involve only q - 0 external lines are not relevant here since there is no phonon softening in the
q - 0 sector. However, collecting those terms with 2 kF lines together with the quadratic term
in (6) will lead to a quantum Ginzburg-Landau-Wilson type functional for the Euclidian action of the 0 field. Up to the fourth order in the * ’s we have :
Note that the last term gives the first contribution for the interaction between q - 0 and
q - 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. This
allows to treat explicitely the interplay between the remaining electronic degrees of freedom
and lattice fluctuations.
Fig. 3. - Diagrammatic representation of the partial trace generation of the perturbation series for the
couplings between the 0 modes. a) Mode-mode couplings that involved 2 kF 0 lines. b) Couplings with only 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-electron interaction 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) softening
condition at fo MF = ln E F/eMF for the static (úJm = 0) 2 kF mode becomes :
which leads to the MF temperature :
Although 1D systems can not sustain long range ordering at finite temperature, T°MF gives a
characteristic energy scale at which the lattice developps strong 2 kF fluctuations. In the adiabatic limit where M --+ oo, the 1D half-filled band MC model develops long range order at T = 0 and the dimerization 6 is known to satisfy the proportionality relation
6 = 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 from
the 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 that
for a one component classical order parameter with T MF =1= 0 one has an ordered state at
T = 0 characterized by the T = 0 equilibrium MF value of the order parameter [19]. In the
present case, however, 4) is not static and quantum fluctuations of the lattice will act to reduce the tendency to dimerization at T = 0. This effect originates from the mode-mode coupling
terms in (17) in the presence of non thermal fluctuation modes. In a single loop
renormalization scheme, which is reminiscent of what has been used in reference [20] for
quantum quasi-1D systems, we connect two of the four non thermal 0 lines in the quartic
terms of figure 3a. This corresponds to an integration over these O’s at úJm =1= 0. There are six contributions involving 2 kF O’s and which are represented by the diagrams of figure 4a and only one that connectes- two O’s at q - 0 (Fig. 4b). It is clear that such contributions will lead to a supplementary renormalization of the 2 kF phonon propagator at wm = 0 due to non
thermal fluctuations of the 0 field near q - 2 kF and q - 0. The renormalization softening
condition 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 be
essentially no softening effect for wo in D(q, wm) (C.F. [15] and [17]) so that in (21) one can
take 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 the
equivalence of the ± 2 kF points while the double prime summation which is centered at q = 0 covers half of the zone. Note that in the spirit of a Ginzburg-Landau-Wilson order parameter expansion for S, all the temperature dependence of the quartic term in (21) have
been fixed to the unrenormalized mean field temperature 7omF. After the evaluation of the
frequency 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 for example, where M --+ 0 (w 0 --> oo ), we have TMF --+ 0 and the dimerization vanishes. This is consistent with the fact that, in this limit, the electron-phonon system becomes equivalent to
the 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 electronic gap is well known to satisfy the BCS type of relation : k1 0 = 1.75 TMF = 5( A. In the
intermediates cases, the results given in (22) infer a continuous decrease of TMF with
o. 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 effective electron-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 MF
transition temperature never occurs and there is no dimerization. From the present calculations, however, it is clear that this quantum interference is not the mechanism by which
the dimerization is destroyed. The interference is rather a consequence of the absence of a
pole in D (2 kF, lù m = 0 ) which results from the coupling between thermal and non-thermal lattice 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 TMF
modes. The present mechanism for the renormalization of TMF bears some similarity with the polaron problem in the sense that it results from the propagation of electron-hole pairs at
2 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. In
figure 5, we have reported their results for the dimerization ratio 6 (o)(»/,6 (0) as a function of
Co 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 dependence
on wo. However, the full comparaison of TMF/ eMF with & /,6 (0) assumes that the same proportionality 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 shown
by the dashed line. Hirsch and Fradkin also concluded that for a given electron-phonon interaction, there is a finite lùO beyond which the ground state is undimerized. In figure 5, this
occurs at lùO 2:: 1.5 t. As previously mentioned, however, the present calculations only show
the disappearance of TMF in the non-adiabatic limit only. Therefore, the system should be dimerized at T = 0 whenever lùO remains finite. One must observe here that for Monte Carlo simulations 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 the
vF
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 processes between electrons will promote 2 kF charge-density-wave correlations. This gives rise to a
Fig. 5. - Renormalized mean field temperature as a function of the phonon frequency for spinless
electrons. The full circles give the dimerization ratio 5 (wo)lâ (0) obtained by numerical simulations of reference [5c]. The dashed line gives the dimerization profile resulting from the sharp cut-off criteria at 2 7rTO.F Wo.
power like singularity in the 2 kF electronic response function. Using (15), the softening
condition of the quadratic term of (17) at ev m = 0 and q = 2 kF leads to the adiabatic mean
field 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). Following
the scheme of approximation given in the preceding section for the quantum effects of quartic
terms of (17) the renormalized softening condition at 2 kF and úJ m = 0 in the presence of g2 now reads :
or
with
Therefore the variation of the softening temperature TMF with coo now follows a non
universal 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 with
spins 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 that the fermion fields are replaced by where a = t , 1 refers to the spin orientation. In the
following, we consider the case with no direct electron-electron interaction. We first note that
