Comparative study of organic metals and high-T cuprates
The Bechgaard salts and the high T cuprates are described by two and three band models, respectively, with the lowest band (nearly) half filled. In organics the interactions are small, while in cuprates the repulsion on the Cu site is the largest energy. The Mott AF state is stable in undoped materials in both cases. In the metallic phase of cuprates the limit produces a moderate effective repulsion. The theories of the coherent SDW and charge-transfer correlations in the metallic phases of organics and cuprates are thus similar. In (undoped) organics those correlations are associated with commensurate SDW and bond or site modes. The corresponding modes in metallic cuprates are the incommensurate SDW and in particular O/O quadrupolar charge transfer with wave vector . They are enhanced for dopings , which bring the Fermi level close to the van Hove singularity. Strong coupling to the lattice associates the static incommensurate O/O charge transfer with collinear “nematic” stripes. In contrast to organics, the coherent correlations in cuprates compete with local quantum charge-transfer disorder.
The high-T cuprates are usually described by the Emery model em1 () where the role similar to the external dimerization bs7 () in the Bechgaard salts is played by the Cu-O hopping , which puts two oxygens in the CuO unit cell and makes the lowest of three bands half filled. The weak coupling theory at zero doping is then a quite straightforward analogue fr2 (); fr3 (); dz2 (); bs5 () of the 1D theory, provided that the imperfect nesting associated with the O-O hopping is ignored. The appearance of the Mott-AF state is essentially independent of the value of where and are the Cu and O site energies in the hole picture, respectively.
However, while the Hubbard repulsion in the Bechgaard salts is small, in the high-T cuprates the repulsion on the copper site is the largest energy em1 (). It has long been maintained fr2 (); fr3 () that the fundamental question in the high-T cuprates concerns the nature of correlations which reduce . The relevant limit is usually taken starting from the unperturbed state with average Cu-occupation . The lowest order process shown in Fig. 1a then corresponds to the fact that two holes on the neighboring Cu-sites can hop simultaneously to the intermediate O-site, empty at , provided that their spins are opposite. This leads to the superexchange za1 () which is the basis of the models za1 (). Alternatively, one can assume that all holes in the unperturbed metallic state are on the O-sites, i.e., that . When two -holes of opposite spin are crossing the intermediate empty Cu-site one hole hops to the Cu-site when is turned on, while the other has to wait as long as the Cu-site is occupied. The waiting time is of the order of where is the chemical potential of the two holes. The whole process of Fig. 1b consists of two independent hoppings, one per particle, and of the waiting time. Therefore the resulting effective repulsion of two -particles is of the order of
and can be interpreted as a retardation (kinetic) effect.
Ii slave particle theory of the metallic phase
We have carried out the corresponding systematic theory which starts from the unperturbed metallic ground state by using the slave particles. This time dependent diagrammatic approach, of infinite order in the perturbation , requires the use of the spinless fermion-Schwinger boson representation in order to avoid the degeneracy of the unperturbed ground state in the overcomplete slave particle Hilbert space. The advantage of this representation is that the perturbation theory is manifestly translationally invariant at each stage, and ultimately locally gauge invariant. The disadvantage is that the three sorts of particles involved, -fermions, f-spinless fermions and Schwinger´s -bosons are distinguishable, i.e., that the Cu-O anticommutatiom rules are replaced by commutations. The theory has to be therefore antisymmetrized a posteriori. Here, we only quote the results.
First, the omission of the Cu-O anticommutation rules is irrelevant in the lowest order of the Dyson perturbation theory. The reason is that the Cu site is initially empty, while one particle on Cu is required for anticommutation and two for interaction. This makes the expression for physical single-particle propagators strictly equivalent to the result of the hybridized HF theory. The HF theory is expressed in terms of two hybridized propagators, one which starts and finishes with the appropriately weighted propagation on the O-sites ( propagator hereafter) and the other begins and ends on the Cu-sites ( propagator). Both propagators are characterized by the three bands of poles (branches) at mr1 (). This holds irrespectively of the average HF occupation of the Cu-site associated with the chemical potential . On the other hand, the relations required ultimately by the local gauge invariance and are nearly satisfied at only when is small, i.e., the theory then converges quickly. It is therefore important to keep in mind that fr2 (); bs5 () at small for and that finite decreases it mr1 () further. The overall rule of thumb is that as long as the HF chemical potential falls below the vH singularity in the lowest L-band, i.e., as long as where mr1 () is the positive doping required to reach the vH singularity. As easily seen, small corresponds to a weak effective interaction .
The Cu-O (anti)commutation rules are also irrelevant for two further expressions. leads to the coherent Brinkmann-Rice-like band narrowing and the variation of through . However, unlike in the mean-field mr1 (); ko1 (); ja1 () slave particle approximations, this is accompanied by the incoherent, local, dynamic fluctuations identified as the charge-transfer disorder. Importantly, the disorder falls far from the Fermi level, although there are indications ja1 (); ni1 (); zl1 () that in higher perturbation orders it spreads all over the spectrum.
The interaction appears explicitly for . It introduces the particle-particle and particle-hole correlations in the single particle and propagations. favorizes the coherent particle-hole correlations which appear here as pseudogaps. The slave fermion theory predicts however that the effect of is the same in the singlet and the triplet channels, which is the consequence of the omission of the Cu-O anticommutation rules. The a posteriori antisymmetrization of the theory therefore associates with the singlet (SDW) scattering only.
The relevant SDW hole-electron correlations are associated to lowest order with the bubble, as suggested by Fig. 1b, where appears as the effective interaction between -particles. In contrast, the corresponding small theory fr2 (); dz2 (); bs5 () involves the bubble. However, although the spectral densities of the and propagators are complementary, the poles are the same. The associated elementary intraband bubbles share therefore the properties of the overlap of the vH singularities and of the (imperfect) Fermi surface (FS) nesting, which both favor the coherent SDW fluctuations with a dominant .
As mentioned above, in the vicinity of the vH singularity brings the metallic theory into the intermediate regime with . Various experiments and NQR in particular ku1 () indicate indeed that for the average occupation of the Cu site is close to . Therefore, we associate the transition, between the long- and short-range magnetic order and the metallic phase at finite , to the crossover in between the regime and the finite metallic regime considered here. This latter is characterized by the close competition of the charge-transfer disorder and the coherence effects which is difficult to cover analytically. In this light, we shall identify below the physical content of the important hole-electron correlation functions and determine when their coherent limit is consistent with experiments in the metallic phase.
Iii Coherent e-h correlations and stripes
Let us start with the SDW correlations for . The vH overlap/FS nesting behavior of the elementary intraband particle-hole bubble is not universal. It is however well known that becomes large for small and close to . Taking formally , the log square singularity in occurs dz2 (); fr2 () at due to the vH overlap and to the perfect 2D FS nesting at . Analytical sc1 () and numerical tu1 () calculations show that imperfect nesting associated with finite at produces spikes at incommensurate along the zone main axes. With finite the spikes were also obtained numerically xu1 () for along its diagonals. While it seems well established sq1 () that the spikes at are dominated by the pairing of holes and electrons close to the antinodal and vH points, the pairing which gives rise to the spikes at is not yet determined unambiguously.
The single well-defined collinear leg or the nearly circular is observed clearly by the magnetic neutron scattering tr2 (); st1 (); hi1 (). As a rule, appears for small energy transfers while occurs at high frequencies e.g. in metallic YBCO st1 (). The observation of the strong leading harmonics is consistent with metallic coherency in the propagation of -hybridized particles. There is however also good NMR evidence go3 () that non-magnetic disorder is present in LSCO. It may well correspond, in part, to the local charge disorder, which is dynamic in the present theory but becomes frozen go3 (); go1 () by the strong coupling bs2 () to the heavy lattice.
The Emery model also encompasses bs2 () the Cu/O and O/O charge transfers and various bond fluctuations within the unit cell. In the limit the D symmetry classifies gz1 (); ts1 (); ku3 () those correlations in A, B and B irreducible representations, the bond fluctuations being involved directly ku3 () into the Raman responses. The elementary bubbles associated with those correlations differ through coherence factors in their numerators while their respective poles and integration ranges are the same ku3 (). The B modes are dominated ku3 () respectively by the contributions from the main axes or diagonals of the CuO zone. They have their own small structures tu1 (); xu1 () (e.g. the elementary O/O bubble is logarithmically singular fr2 (); bs2 () at ). In addition, while the B and B intracell modes are decoupled from the transfers of the total CuO charge among the distant unit cells tu1 (); bs2 (); ku3 (), the coherent Cu/O charge transfer is accompanied tu1 (); ku3 () by the intercell charge transfer. Only the latter is subject to the long range Coulomb screening/frustration tu1 (); go1 (); ku3 (); ki1 ().
The small behavior of the charge transfer correlations is however of secondary interest if the SDW is taken sc1 () as the dominant fluctuation. Then, according to Fig. 2, two SDW´s enhanced at either couple to the intracell fluctuations and then to phonons, or else directly to phonons. In the case of linear coupling of Fig. 2a the corresponding dominant is small and satisfies the physically important relation
Alternatively, as in Fig. 2b, two such SDW´s can also couple quadratically bs2 () to tilts of the CuO octahedra at . When two SDW´s enhanced at close to are associated with the pairing they drive ku3 (), directly or via the B O/O charge transfer, the LTT/(e-e) deformations at . Analogously, if two SDW´s enhanced at close to are associated with the pairing they drive ku3 () the B O-O bond fluctuations and the LTO/(e+e) modes at .
The tilts and the modes are in addition entangled pu1 () by the ionic forces, namely the LTT and LTO tilts are accompanied, respectively, by the homogeneous e-e and e+e shears of the CuO planes. Those entangled single leg deformations thus lift, through the electron-phonon couplings bs2 (), the degeneracy of two O/O sites bs2 () or of four bonds within the CuO unit cell by dimerizing them two by two.
The stability of the striped structures can be investigated using the related sc1 (); bs2 () Landau functionals. E.g., the entangled collinear modes are related in this way to the collinear nematic static stripes ki1 (); em4 () usually characterized by . It appears thus quite clearly that the coherent O/O charge transfer, coupled to the lattice, is an essential ingredient of the collinear stripes ki1 () in the metallic phase. This agrees with observations tr2 (); st1 (); gz1 (); ts1 (); bi1 (); sg1 (). The collinear SDW and O/O charge transfer correlations get enhanced in metallic lanthanum cuprates for dopings . The spikes sq1 () at in then explain the "nematic" version of Eq. (2) observed tr2 (); gz1 () in the metallic phase. The corresponding LTO/LTT lattice instability is predicted bs2 () and observed ax1 (); kr1 () to be of the first order in LBCO for . Those effects are attributed here to the vH overlap/FS nesting, while the commensurability is expected to play only the secondary role.
In contrast to lanthanum cuprates the Fermi level in the optimally doped YBCO and BSCO, as measured by ARPES, falls mr1 () well below the vH point, i.e., . The collinear SDW-O/O stripes enhanced by the vH singularities are therefore not expected to occur in metallic YBCO and BSCO. Indeed, at larger , the collinear stripe structure is replaced ts1 () by the fourfold commensurate stripes in "checkerboard" configuration ts1 (); ha1 (), which restores the D symmetry.
The salient new feature of the present low-order large analysis is thus that the coherent, weak, incommensurate collinear sq1 () SDW and O/O correlations appear to get enhanced with respect to the charge disorder if the vH singularities are reached in1 () by doping . This explains the long standing puzzle ax1 () why the enhanced magnetic/charge coherence occurs twice in LBCO as a function of doping, once as the long- or short-range fr3 (); lo1 () AF order at and then again as the incommensurate SDW at single for finite positive , accompanied by the static O/O charge transfer coupled to the staggered LTT tilting of the CuO octahedra.
Acknowledgements.Invaluable discussions with J. Friedel, L.P. Gor’kov, I. Kupčić, D.K. Sunko and E. Tutiš are gratefully acknowledged. This work was supported by the Croatian Government under Projects No. and No. .
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