Despite intensive efforts for more than a quarter century investigating high-temperature superconductivity in cuprates and related substances, no universally accepted theory exists about the microscopic pairing mechanism. Various experimental findings are explained by different theories. No consensus is achieved whether strong electronic correlations play a key role, or the electron-phonon interaction is strong and crucial for high-temperature superconductivity.
Some key findings for the phenomenon of high-temperature superconductivity in cuprates are the existence of two-dimensional layered CuO2 structures. The electron system is strongly correlated. In the normal state, these systems shows antiferromagnetic exchange correlations at low doping and anomalous behavior of transport properties. There are strong doping effects on the critical temperature for the onset of superconductivity and other properties. A d-wave pairing symmetry has been observed, also the occurrence of a d-wave like pseudogap in the single particle excitation spectrum below another critical temperature . In contrast to normal superconductors, the coherence length in high-temperature superconductors is very short. Isotope effects are established when 16O is replaced by 18O that depend strongly on doping. Many other properties are measured.
To understand the influence of strong electronic correlations, fundamental models such as the Hubbard and the model have been studied intensively, see  and references given therein. These simple models contain terms representing hopping () of electrons between sites of the lattice and their on-site Hubbard repulsive interaction . A spin-dependent Heisenberg near-neighbor antiferromagnetic exchange interaction is described by the parameter J. Within the model, a short-range spin independent attractive interaction V between the tight-binding electrons on the near-neighbor sites of the lattice is introduced in a phenomenological way to stabilize the d-wave pairing.
Recently, the strengths of interaction have been determined for some cuprates (YBaCuO, LaSrCuO) within the t-U-V-J model . The extended Hubbard Hamiltonian, Eq. (1) in that work, contains the parameters for that values were given that reproduce some of their properties. In particular, angle-resolved photoemission spectroscopy and inelastic neutron scattering resonance are considered in a weak-coupling Fermi-liquid theory to determine the values of these parameters, using a Green's function approach. An open question is wether these parameter values can be obtained from ab-initio calculations using, for instance, density-functional theory.
Moreover, there are other experimental observations including isotope effects, pump-probe and tunnelling spectroscopies, normal state diamagnetism and magnetic quantum oscillations that favor a strong electron - phonon interaction in cuprates and related oxides. An outstanding issue to be settled is the origin of a pseudogap phase, see Fig. 1 . It is accepted that these materials are in the strong coupling regime so that the pairs have a relatively short coherence length. A signature of strong coupling is also the formation of a pseudogap that indicates that precritical fluctuations are already present above the critical temperature. Phenomena such as preformed Cooper pairs, nanometer scale charge inhomogeneity, anomalous transport properties, doping effect (doping-induced evolution from a non-Fermi liquid to Fermi liquid behavior) are under discussion. There are alternative stripes models that also explain the incommensurate peaks observed in inelastic neutron scattering resonance. Recently [3-5] it was claimed that the formation of a pseudogap in cuprate high temperature superconductors can be understood within a Fluctuating bond model, where the nonlinear coupling between the vibrational coordinates of the oxygen atoms perpendicular to the CuO2 plane (this degree of freedom is denoted as vibrator) and the electron system is of relevance, distinguishing it from the linear electron-lattice coupling in conventional Bardeen-Cooper-Schrieffer superconductivity. Performing ab initio molecular dynamics calculations, it has been found that adding electrons to the antibonding Cu-O-Cu level will eliminate bond strength leading to off-axis potential energy minima for the oxygen atom. An effective Hamiltonian has been given that contains a nonlinear electron-vibrator coupling as well as a quartic anharmonicity term, but is based on an Einstein model that ignores inter-vibrator interactions. Within this model, different experimental findings with respect to the pseudogap formation have been interpreted.
To give a description of the relevant microscopic processes, a model Hamiltonian is considered that contains the contribution of the lattice, the electrons, and the electron-lattice interaction. At the site i on the square Cu lattice in the CuO2 plane (), see Figs. 1, 2, we denote with the creation operator of an electron in the orbital. The bond from lattice site i to is mediated by the orbital of the oxygen atom O, the bond from lattice site i to is mediated by the orbital of the oxygen atom O. The displacement (e.g. in z direction) of O is , the displacement of O is . Introducing the two-dimensional vector for the vibrational displacements we can describe the nonlinear vibrational part of the lattice Hamiltonian as
a next-neighbor () term is added that prefers for a staggered ordering. The electron contribution is given by the tight-binding model as
that contains the superexchange hopping term t between the Cu atoms via the intermediate O atoms as well as the screened Coulomb interaction between the electrons given by , see .
The most interesting part is the interaction between the electrons and the lattice. According to the Fluctuating bond model [3-5], electrons in the upper part of a band are placed on antibonding orbitals that increase the length of the bond, leading to the buckling Cu-O-Cu bond, as characterized by the vibrational displacement u. The hopping matrix element between O p orbital and Cu d orbital is reduced, . As shown in Refs. [3-5], this effect is described by
with . In a mean-field approximation, the quadratic term in the vibrator displacement u is strongly influenced by the occupation number of the bond so that a double-well potential can arise, see Fig. 2.
It was possible to evaluate the parameters of the Fluctuating bond model  from ab initio calculations, and different effects, in particular isotope effect and properties of the pseudogap phase, have been derived. Electron-lattice interaction, nonlinear forces and soliton formation are discussed in two-dimensional systems like cuprates, see the recent work  and references given therein. In , a phenomenological approach is used where the length of the Cu-O bond is assumed to depend on the ionization state of both atoms.
Is there a two-electron bound state that may form a quantum condensate? The Fluctuating bond model [3-5] claims: yes! But these calculations have to be improved taking into account more details such as the inter-vibrator interaction between different lattice points.
Recently, the problem of two electrons in a nonlinear one-dimensional lattice has been investigated , and the formation of a bound state was found for appropriate parameter values. Compared with the Fluctuating bond model, inter-site coupling and lattice polarization is included. The solitonic solutions describe a rather extended deformation of the lattice. Bound electron pairs (singlet state) are formed with an extended wave function so that the Coulomb repulsion is counterbalanced. The coexistence of both deformed regions of the lattice due to electron pairs that give areas of local oriented phases and normal regions is an interesting aspect of such an approach.
Coming back to the model , the empirical determination of the parameters of a purely electronic Hamiltonian can reproduce some experimental facts, but the microscopic origin of these interaction terms remains unclear. In particular, the ad hoc introduction of the V term may mimic the electron-lattice interaction. Also the ab initio calculation of parameters like U is not discussed in  so that the origin of high-temperature superconductivity remains nature's great puzzle.
The author thanks M. G. Velarde, W. Ebeling, and A. P. Chetverikov for interesting discussions.