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PEIERLS INSTABILITIES I. SCREENING AND
RETARDATION EFFECTS
S. Barišić
To cite this version:
PEIERLS INSTABILITIES
I. SCREENING AND RETARDATION EFFECTS
S. BARISIC
Institute of Physics of the University, Zagreb, Croatia, Yugoslavia
Abstract. — A simple tight-binding model for the quasi-one-dimensional conductor is described. The electrons of the one-dimensional band are coupled to three-dimensional phonons and interact through repulsive Coulomb forces. The limit 2 nT > coD is examined, in which the retardation effects
in the electron-electron phonon mediated interaction are important. They suppress the phonon mediated BCS superconductivity. The only instability which can then occur at finite temperature is the CDW instability, accompanied by the lattice deformation. In this case the relevant phonon mediated interaction is unretarded. If it dominates the appropriately screened Coulomb forces, the transition is of the conventional, Peierls type.
1. Introduction. — Two different approaches to the physics of the (quasi)-one-dimensional conductors have been intensively developed over the last decade. The first, microscopic many-body approach started with the work of Bychkov, Gor'kov and Dzyaloshin-skii in 1966 [1]. This approach has reached a high degree of sophistication, after being reformulated in terms of the renormalization group differential equa-tions [2]. It is based on the observation that in the one-dimensional electron system interacting via the unre-tarded short range forces the magnetic (SDW), Cooper pairing and the dielectric (CDW) correlations can be exceptionally strong, and possibly occur simul-taneously. It was also pointed out [1] that the CDW instability is accompanied by the lattice distortion, when, beside their Coulomb interactions, electrons are linearly coupled to phonons. The same singularity occurs then in the CDW and the phonon correlation function.
The other line of approach, which was initiated by the early works on the Peierls instability [3, 4] and Frohlich conductivity [5] received a renewed attention upon realizing that the Labbe-Friedel electron-pho-non model [6] with Coulomb forces [7, 8] quite succes-full in the A-15 compounds, can be conveniently extended [9, 10] to cover the field of the more recently discovered chain conductors. The activity in this direction culminated after Bardeen's suggestion [11] that the increase of conductivity observed in TTF-TCNQ close to the structural phase transition might be due to the Frohlich superconductivity mechanism.
This approach emphasizes the structural aspect, i.e. works with the CDW/structural order parameter only. The fluctuation [12-15] and conductivity pro-blems [5, 16-20] are treated here within the Ginz-burg-Landau-like theories. In particular, this for-mulation explained successfully [10, 21-24] the set of structural instabilities in TTF-TCNQ.
The fundamental question which arises then is why the theory with only the structural order para-meter can deal reasonably well with the physics of the organic conductors and platinum salts. Some indication as to the answer came upon realizing [8] that due to retardation effects, only the phonons with frequencies larger than roughly 2 T can build the BCS superconductivity. The cut-off frequency 2 T was later shifted to 2 nT [25]. This would rule out the phonon-mediated BCS correlations in the tempe-rature range interesting for organic conductors and Pt-salts, where the respective anomalies occur around 50 and 100 K, while the Debye temperature is of the order of 100 K. Indeed, the retardation effects were consistently treated within the many-body parquet theory by Gor'kov and Dzyaloshinskii [26] who argued that for T > <oD the electron-phonon parquet
theory reduces to the Peierls mean-field theory. However, when Coulomb forces were added, the theory again became complex, allowing for the mix-ing of instabilities in the one-dimensional regime.
It was further suggested [27] that the instability retains its simple structural/CDW character provided that the bare Coulomb couplings are sufficiently
Résumé. — Un modèle simple pour un conducteur quasi-unidimensionnel en liaisons fortes est présenté. Les électrons d'une bande unidimensionnelle sont couplés aux phonons tridimensionnels et interagissent entre eux par des forces répulsives de Coulomb. La limite 2 nT > a>D est examinée,
où interviennent les effets de retard dans l'interaction électron-électron médiée par les phonons. Ils suppriment la supraconductivité du type BCS, induite par les phonons. L'instabilité qui peut se produire à une température élevée est alors celle de l'onde de charge, accompagnée par une défor-mation du réseau. Dans ce cas, l'interaction par les phonons est non retardée. Si elle domine l'inter-action Coulombienne écrantée, il se produit la transition de Peierls conventionnelle.
PEIERLS INSTABILITIES. I. SCREENING AND RETARDATION EFFECTS C2-263
small. Previously however, little attention was attach- ed to the values of the bare Coulomb couplings in the many-body differential approach, because they play there the rather secondary role of the boundary conditions. The backward Coulomb scattering can be reasonably small [lo] with respect to the phonon induced coupling constant, but the bare forward ~ o u l o m b scattering includes [28] the rather complex long-range screening. It is not a priori clear that this screening is very efficient, because the plasma fre- quency in quasi-one-dimensional systems can be quite low in the large part of the phase space. The dielectric constant for small wave-vectors is measured in optical experiments [29]. Figure 1 shows the reflec- tance of TTF-TCNQ vs frequency of the incoming radiation. While the curve which corresponds to the electric field E parallel to the chain axis b exhibits the well-defined plasma edge, there is no low frequency screening at all when E is perpendicular to b : The electrons are constrained to move along the chains. We conclude that the plasma frequency decreases when the plasmon propagates perpendicularly to the chains. As is well known, there is no screening above the plasma frequency. Moreover, the longitu- dinal reflectance in the infrared range is somewhat lower than expected for the metallic regime [29]. This latter fact will be ignored in the present quali- tative discussion of the screening.
with a particular emphazis on the structural proper- ties, while the conductivity problem will be only briefly mentioned.
2. Model. - 2.1 ELECTRON SPECTRUM. - The resistivity measurements in KCP and TTF-TCNQ suggest that the electron spectrum is very nearly one-dimensional. Our main interest here is the cal- culation of the structural properties. The couplings which give rise to the three-dimensional nature of those properties are different from the interchain hopping which governs the transverse conductivity, and with this in mind we shall neglect this latter. Concerning the one-dimensional band structure, the effective mass approximation can be used in the tight- binding limit, when the band is nearly empty or almost full [8, 271. Restricting for simplicity the dis- cussion to the single set (KCP) of chains, we take
where t is the hopping integral between the sites at distance dll.
2.2 ELECTRON-PHONON COUPLING. - In this,
tight-binding picture [8], the electron-phonon coupl- ing arises from the variation of overlap integrals with the change of the inter-atomic (inter-molecular) relative position produced by deformation. When the atoms (or molecules) have their symmetry axis coinciding;, with the', chain axis, only the longitu- dinal component of their displacement couples to electrons through the matrix element
FIG. 1. - Reflectance of TTF-TCNQ for the electric field parallel and perpendicular to the chain axis (after Ref. [29]). The former
exhibits the plasma edge.
This sets the context of the present review. We shall start by reviewing briefly the tight-binding electron- phonon model. Further, the many-body theory for the model will be discussed and the phonon self- energy will be rederived in the limit 2
nT
> oD.Next, the metallic screening of the forward Coulomb interaction will be briefly examined. Two extreme limits of the strong and weak Coulomb scattering will then be distinguished. Finally, the conditions for the validity of the Peierls theory with only the structural order parameter will be given.
The theories with only the structural order para- meters will be reviewed in the companion paper 1301,
where q, t measures the derivative of t with respect to dl,. Eq. (2) is appropriate for Pt d,, band in KCP. In contrast, the planar TTF and TCNQ molecules are stacked in a jish-bone way to form the chains of the crystal. A straight-forward generalization of eq. (2) then shows that the transverse displacements of these molecules are also coupled to electrons. The analysis of the full symmetry problem [23, 31, 321 is however beyond the scope of the present qualitative conside- rations; for which eq. (2) is sufficient.
linear sequence of dipoles, corresponding to overall E F neutrality, i.e. it vanishes as [lo]
FIG. 2. - The band structure (a) in absence of deformation and (b) in presence of the long wave-length deformation.
and (2). We note in this respect that t is calculated with all sites equally occupied. Thus, the Coulomb interaction of the displaced charges has to be added to the interaction (2). t in eqs. (1) and (2) is a short- range property related to the non-linear atomic screening. The charges redistributed by the coupling (2) over the sites as in figure (3), interact to the lowest order by the unscreened (i.e. long range) Coulomb forces. In the tight-binding limit the short range electron-phonon interaction (2) is associated with the unscreened Coulomb interaction U(q) [8]. This property of tight-binding contrasts sharply with the nearly free electron limit [33], where the unscreened Coulomb interaction goes with the long-range elec- tron-phonon interaction. As a physical consequence, the Coulomb coupling between the CDW's on the neighbouring chains is considerably stronger in the tight-binding limit.
In order to discuss this effect in more detail, let us consider the electrostatic potential created by the qll CDW at the distance R, from the chain. As shown in figure 3 there are two regimes, R,
2
2 n:/qll.Concerning the energy of two CDW's on neighbour- ing chains at R, = d,, it is clearly minimum when the CDW's are in opposition of phase, as shown in the insert of figure 3. This fact is accounted for. by the phase factor cos q, d, and the full Coulomb matrix element becomes [lo]
U , =
UP
+
Ui cos q, d, (4)2 n: 2 71
in the limit
-
< dl. For R, <-
the potential411 411
at R, is that of the charge near this point (cf. Fig. 3), i.e.
in the limit of small q.
2.4 ELASTIC COUPLINGS. - In the present, tight- binding formulation [8] all the renormalization of the phonon frequencies, except the one associated with the couplings (2), (4) or (5) should be incorporated into the unrenormalized phonon frequencies oo(q). The renormalization due to the couplings (2), (4), (5) is important in the vicinity of 2 k,. 2 k , is rather large with respect to the width of the Kohn anomaly so that the elastic constants (i.e. the q x 0 limit of coo(@) give reliable information about oo(q). oo(q) is related to the cohesive properties of the crystal with exclusion of the 2 k , lattice-CDW effect. The measurement of the elastic constants has shown that in KCP oo(q) x o,(qll) i.e. that the chains are elastically weakly coupled. In sharp contrast, the crystal of TTF-TCNQ is harder in a and c directions than in the chain b direction, i.e. the elastic coupling between the chains is strong [34]. This is particularly surprising for the c direction, where the equally charged molecules are the first neighbours, as shown in figure 4a. In fact, the stabi- lity along this direction cannot be understood within the point charge model. This model would favor, by the usual Madelung argument, the alternation of the oppositely charged molecules in both a and c direc- tions. The latter configuration, observed in HMTTF-
FIG. 3. -TWO regimes for Coulomb interchain coupling : (a)
R , < 2 nlqli and ( b ) R, > 2 x/qI1.
FIG. 4. - The a-c plane configuration in (a) TTF-TCNQ and
PEIERLS INSTABILITIES. I. SCREENING AND RETARDATION EFFECTS C2-265 TCNQ is shown in figure 4b, while TTF-TCNQ
prefers the configuration shown in figure 4a. This suggests that the transition from the HMTTF- TCNQ structure to the TTF-TCNQ structure results from the delicate balance between the Madelung energy and an energy term which favors the configura- tion of figure 4a [34]. It was pointed out by Friedel [35] that any short range two body attractive force can serve ,this purpose : one goes from (4b) to (4a) by replacing two short Q-F bonds by a Q-Q short bond and a F-F short bond, all other things being (roughly) equal. Denoting by a the property characterizing the given body we find the corresponding energy variation associated with the attractive two body forces to be
Indeed (4a) is the preferred configuration. Physically, the quantities aQ,,, characterizing TCNQ and TTF (or HMTTF) molecules, are likely to be the mole- cular polarizabilities. Then eq. ( 6 4 describes the variation of the van der Waals energy. In comparison to eq. (6a), the variation of the nearest neighbour Madelung energy is
EM
-
- 2 QQ QF - (Q4+
QF2)
=which, as expected favors the configuration (4b) (Note that QQ = Q,). The van der Waals forces are of much shorter range then the direct Coulomb inter- action. This feature is in a qualitative agreement with the fact that the replacement of TTF by the larger molecule of HMTTF leads to the configuration (4b). However, our considerations are certainly quite over- simplified, because they are based on multipole expansions, valid only when the distances between the multipoles are larger then the dimension of the multipoles (molecules) themselves. This condition is not satisfied in TTF-TCNQ, and similar organic conductors.
The above discussion shows that even the simplest cohesive properties of TTF-TCNQ cannot be explain- ed within the single band model of eqs. (1) and (2), since the molecular polarizability mechanism involves at least wo states for each TTF or TCNQ molecule.
f
There are some reasons to believe that the full expla- nation of four anomalies in TTF-TCNQ also requires additional degrees of freedom to those introduced by eqs. (1) and (2). But the simplified model can satis- factorily explain the three dominant transitions in TTF-TCNQ 1301. The strong elastic interchain coupl- ing in TTF-TCNQ will only be symbolized here, and this by an oo(q) depending strongly upon all three components of q.
3. Many body formulation. - 3.1 GENERAL. -
It is convenient to consider the electron-phonon interaction as giving rise to the phonon mediated 12 Do (Do-bare phonon propagator) electron-elec- tron interaction, in order to stress out the basic analogy of this interaction and the Coulomb (pho- ton mediated) interaction [l, 271. If both, photon- and phonon-electron couplings are assumed to depend only upon the transfer of momentum (as in eqs. (2), (4), (5)) the parallelism is even better [27]. The remaining difference comes from the fact that the bare phonon mediated interaction is retarded [8, 25-28], whereas, for our purposes, the bare Coulomb interaction can be taken as unretarded. The interaction
y which enters the electron-electron many body theory is the sum [27] of those two interactions.
The parallelism between phonons and photons is exhibited by the fact that both the exact phonon propagator D (response to staggered stress) and the CDW density-density correlation function (response to external photon) are given by essentially the same expressions [I, 271, respectively
Here the shaded vertex represents the exact vertex calculated with the combined interaction y. From eq. (7) we draw the important conclusion [l] that the lattice and CDW instabilities go together.
3.2 VERTEX CORRECTIONS. - We proceed by the discussion of the vertex corrections. The low order vertex corrections, such as those shown in figure 5, when calculated with the one-dimensional band (I), have the following special feature : under certain conditions to be specified below, both electron-hole and Cooper diagrams are logarithmically divergent, and have to be treated on equal footing [I]. Usually [I, 2, 261 one distinguishes corrections to the backward
FIG. 5. - Three potentially logarithmic, lowest order corrections to backward scattering; B (bubble), M (Migdal) and C (Cooper) diagrams. Dashed line is Coulomb interaction, wavy line is phonon correlation function D, the electron-phonon coupling I is denoted by a black spot, full line is the k, propagator and dash-dotted line
the given electron line undergoes the 2 k, (backward) momentum change, when passing through the dia- gram, while in the forward scattering the electron line comes out with the same momentum. The three diagrams in figure 5, B, M and C respectively are the lowest, potentially logarithmic corrections to the backward scattering. To the same order the forward scattering is renormalized only by the diagram of the Cooper type.
The electron-hole diagram M and the Cooper diagram C are logarithmic, provided that the internal interaction lines do not carry too much momentum and/or frequency dependence. In contrast to that, the bubble diagram B is always logarithmic, because the interaction lines are not involved in the internal integration. Its logarithmic nature is thus always preserved. In the preceding text we have seen that the phonon mediated and/or Coulomb interaction lines do have an appreciable frequency and/or momentum dependence. It is then our further pur- pose to see whether these dependences are strong enough to modify the logarithmic nature of the diagrams M and C.
3.3 PHONON RETARDATION EFFECTS.
-
We start with the discussion of the frequency dependence of the phonon mediated interaction within the diagram C . It is well known from the conventional theory of superconductivity [8] that the Cooper correlation length is5,
cc v,/T. Therefore two electrons with. velocity v, are correlated over the timeIn this time interval they have to exchange the pho- non in order to make the correlation possible. The time of the phonon flight is 2 nlo, and our condi- tion becomes z > 2 n/wO. It is not satisfied in KCP and TTF-TCNQ for acoustic phonons where o,
<
2 n T in the interesting range of temperatures. In this case C is not logarithmic. In other words thestrong dependence of the phonon propagator Do on the Matsubara frequency
arising from the smallness of o, renders C negligible.
A similar argument is valid for M. -More generally [26], each time the phonon interaction line is involved in an internal vertex integration the corresponding diagram can be neglected in the logarithmic approxi- mation, provided that 2 n T
>
o,. The third term in eq. (7) consists therefore of the diagrams of the type shown in figure 6.FIG. 6. - Logarithmic contribution to the renormalized phonon correlation function D in the limit 2 xT > o,.
The empty vertex represents here the sum of all potentially logarithmic diagrams calculated with the
cal series corresponding to the contributions of the type shown in figure 6 we find from eqs. (7, 8)..
with the phonon self-energy equal to [26]
3.4 COULOMB SCREENING. - Our next step will
consist of analysing [28] the Coulomb vertex in eq. (10). In the diagrams M and C of figure 5 there is a Coulomb line which runs at qll z 0 for arbitrary q,. If eq. ( 5 ) were used for the corresponding matrix element, its q-2 singularity would immediately destroy the loga- rithmic nature of the diagrams under consideration. The existence of the q-2 singularity requires in fact the summation of the non-logarithmic diagrams [36], e.g. of the RPA bubble chain [8, 371 in the q
w
0 limit. This procedure replaces U2 of eq. (5) byin both M and C.
The behaviour of eRPA(q, o ) for small qll is shown
in figure 7. One of two characteristic frequencies in E~~~ is the plasma frequency [37]
It becomes small inthe large q, limit, in agreement with the reflectance data of figure 1, described in sec- tion 1.
I
I STATICPEIERLS INSTABILITIES. I. SCREENING AND RETARDATION EFFECTS C2-267
The second characteristic energy in is the elec- tron-hole excitation threshold energy v, q I 1 . In figure 7 we distinguish therefore three regimes in the behavior of the dielectric constant. In effect the existence of the soft plasmon gives rise to the large portion of figure 7 in which there is no screening at all, i.e. where cRpA a 1.
We note further that the electron Green functions in the integrands of M and C do not introduce any new characteristic energy, since the corresponding electron excitation energies
5
equal u~ q l l , which already appeared in cRPk Thus figure 7 also represents the appropriate integration map. We see that we have to distinguish first v, q l l>
T from v, qll<
T and then, in the latter case, o,>
T from cop<
T (by the assumption co: > T ) .In the case v, q l l < T , o, < T, we can neglect v, q l l with respect to the Matsubara frequencies
on = i (2 n
+
1) nT in the electron Green functions, .which thus contribute only a T-' factor. From figure 7the integration is
It is in fact the appearance of the unusual log T term of the Coulomb origin in eq. (13) which led us [28] to the careful examination of the screening effects in the logarithmic theory. We see however that the second, qll integration is such that it just eliminates the log T dependence, resulting in the temperature independent log
-
term. The similar discussion [28]vc
of the range v, qll < T , op
>
T also yields a nonlo- garithmic contribution.The integration range v, qll
>
T contributes a log T term o i the usual origin. Indeed, in this range the RPA screening is static, and we can neglect the qll dependence of02.
The number of the Matsubara frequencies i(2 n+
1) nT, negligible with respect to5
is t / T and then G o-
5 - I . The dominant contribu- tion of the u, qll > T range is therefore1 i "
-
TFTO,
-
-
a U2 log.5,
C > T T ~ ~ (14)
as expected.
In conclusion, the logarithmic nature of the one- dimensional Coulomb interaction (log q l l d l l ) does not introduce the additional log T tzrms in the logarithmic theory for o,O
>
T in spite of the exis- tence of the soft plasmon o,. The usual log TITF terms are to be associated with the statically screen- ed02.
First order scatterings. - Collecting the loga- rithmic contributions of B, M and C with appro-
priate numerical coefficients, we find the lowest order correction to the backward scattering [26]
and, analogously
for the forward scattering. These corrections are certainly sufficient in the high temperature regime log TF/T x 0, but not at arbitrary T
<
TF.3.5 PARQUET THEORY. - If the term log TF/T,
small at T a TF, is formally replaced by Ax, the eqs. (15) transform into the coupled differential equations [2, 261, ,which can scale U , and U , into the
limit of large log T,/T. Indeed, it can be seen that the solution of these, renormalization group-like equa- tions, with physical U1 and
o2
of eqs. (4), (5) and (1 l ) taken for boundary conditions, coincides with the result of the parquet summation [I, 261 of electron- hole and Cooper diagrams, with all cross terms includ- ed. When the vertex is known the self-energy (10) can also be found [2].Strong Coulomb coupling. - The phonon self-
energy in eq. (10) is in fact the purely Coulomb CDW correlation function, calculated in the logarith- mic approximation and multiplied by Z 2 . If the electron-phonon coupling I is weak, the Coulomb correlation function must be large for the temperatures where the self-energy (10) is of the order of D i l . The singular behaviour of the CDW correlation func-
tion is therefore required in the interesting temperature range. In this small I regime the Coulomb effects are dominant, while the electron-phonon coupling I produces only the unessential corrections consistent with the requirement that the CDW and lattice insta- bilities occur together.
The singular behaviour of the CDW correlation function and of other potentially singular correlation functions, for the SDW and the Cooper pairing, has been extensively examined, using the differential form of eqs. (15) [2, 26, 38-40].
here by the simple RPA approximation. The contri- tures [39, 401. When temperature decreases, CDW butions of M and C (second and third term in the crosses over into the three-dimensional regime and first eq. (15)) cancel each other in the one dikensional subsequently condenses. This is not inconsistent limit. For Uil > 0 the effective couplings (16) are with the enhanced magnetic properties, 4 k , anomaly finite at any T . This is not the case for U)I
<
0, where and the phase transitions observed in some chain the corrections beyond the logarithmic approxima- conductors.tion [2] are important. ~ o w & e r , it is unlikely that the bare Coulomb backwiird's~atterin~ is negative and here (i.e. in the T > 0,/2 n limit) we shall limit our attention only to the positive intrachain backward scattering.
The eKective couplings have to be further appro- priately inserted [2] into the correlation functions. This leads to the phase diagram [2] shown in figure 8. The line UlI = 2 U $ separates the region of the giant
BCS fluctuations from the region where the strong SDW and CDW-2
k,
fluctuations coexist on the equal footing. Under certain conditions they might be accompanied by the 4k,
CDW anomaly [41, 391.FIG. 8. - The T = 0 singular behaviour of the Coulomb corre- lation functions (after Ref, 121):
The transition temperature for the purely (repul- sive) Coulomb problem vanishes already in this, logarithmic, approximation, because the effective couplings (16) are finite at arbitrary T. The inser- tion [26] of the Coulomb CDW correlation function into the coupled electron-phonon problem, accord- ing to eq. (lo), leads however to the small but finite transition temperature. This indicates that the loga- rithmic approximation is not entirely adequate in the one-dimensional electron-phonon problem.
Interchain coupling. - Physically, the finite value
of the transition temperature is expected to arise from the addition of the interchain coupling U:. It has the most important effect [38-401 on the Cou- lomb CDW response function, owing to the direct [lo] Coulomb coupling of CDW's. In contrast SDW's and Cooper pairing remain one-dimensional [38-401. Hence, the only phase transition which can occur at a finite temperature is the Coulomb CDW conden- sation, slightly shifted by the electron-phonon coupl- ing, through eq. (10).
Usually the particular attention is devoted to the range U ;
<
2 Ug. Some workers believe that thisrange might be relevant for the organic chain conduc- tors. In this range the strong one-dimensional SDW
3.6 PERTURBATION LIMIT. - Nevertheless, since
chain conductors exhibit strong structural anomalies, the limit in which the electron-phonon coupling I plays only a minor role does not seem entirely jus- tified. In other words V = 12/Mog is probably not negligible in comparison to U's. It is therefore inte- resting to consider also the limit of the strong elec- tron-phonon coupling [27, 281. The real situations seem to be intermediate between this limit and the limit considered above.
The second term in eq. (10) can be viewed as an expansion in powers of n, U log T F / T (c.f. eq. (15)), and is therefore negligible with respect to the first term in the temperature range above
Therefore, the second term is small at the lattice instability temperature T,O, obtained by equating the first term of eq. (10) to
D;
provided that V >
I
UI.
The logarithmic approxi- mation reduces to the standard mean-field Peierls theory if T > 0,/2 n and V >1
U1
[27, 281. The small corrections arising from the Coulomb coupling are unimportant for the mean-field theory. But this theory is unstable with respect to the interaction of fluctuations, using the Ginzburg-Landau language [12- 151, or with respect to the phonon self-energy correc- tions in the diagrammatic language [2]. In this context the Coulomb interchain coupling might be quite important. It can lead to the phase transition at T p > T M , i.e. justify the single order parameter theory in the whole temperature range (or at least above T,).The condition
PEIERLS INSTABILITIES. -1. SCREENING AND RETARDATION EFFECTS C2-269
when the instability occurs at q, = zld,. It can be 4. Conclusion. - We have discussed the logarithmic
of the order of d,. many-body theory of the electrons coupled via
In the opposite situation,
to,
< d,, the pseudogap phonons and Coulomb interactions. We have empha- opens in the electron spectrum well above Tp. [19, 201. sized that there are physically different limits of this It reduces the screening of U,. This problem currently theory and derived the sufficient conditions under under investigation, is usually ignored in the current which the microscopic theory reduces to the .Ginzburg- Ginzburg-Landau type of theories. Landau type of theory, based on the structural orderparameter. These theories are reviewed in the compa- nion paper [30].
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