84 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

structure in our theory and, in addition, contains fingerprints of the pairing interaction.

2.5 Other Scenarios for Cuprates: Doping a Mott Insulator

Although our generalized Eliashberg theory is quite successful in describing the elementary excitations and their interdependence with spin excitations in holeand electron-doped cuprates, it has a few weak points. The main problem of our approach is the fact that for x → 0 (where x is the doping) one does not obtain a Mott insulator. Physically speaking, our theory profits from the vicinity to an antiferromagnetic phase (because paramagnons are present), but ignores the fact that cuprates are doped Mott insulators. Therefore, in this subsection, we shall sketch other theoretical scenarios based on doping a Mott insulator and contrast them with our theory. Note that the above criticism does not hold for Sr2RuO4 because it is a good Fermi liquid, i.e. a more weakly correlated material, and, as discussed in connection with Fig. 1.9, it is in the vicinity of a ferromagnetic metallic phase (except when Sr is replaced by Ca; Ca2SrO4 is a Mott–Hubbard insulator).

2.5.1 Local vs. Nonlocal Correlations

In 1989, Metzner and Vollhardt [131] showed in a pioneering work that the Hubbard model used in our theory undergoes significant simplifications in

the limit of infinite dimensions, i.e. d = ∞. In this limit, provided that the

√

kinetic energy is scaled as 1/ d, the self-energy and vertex functions may be taken to be purely local in space, although they retain a nontrivial frequency dependence. This means that the Hubbard model can be mapped onto a self–consistently embedded Anderson impurity problem, which can then be solved by various many–body techniques [132, 133]. The resulting dynamical mean–field theory (DMFT) is exact in an infinite number of dimensions. Recently, W¨olfle and coworkers extended the DMFT of the t–J model discussed below and also generalized the noncrossing approximation (NCA) [134, 135]. These authors have calculated the single–particle spectral density and other response functions and find good agreement with the properties of underdoped cuprates. Also back in 1989, M¨uller-Hartmann [136] proved the locality of many–body Green’s function perturbation theory and used it in order to derive self-consistent equations for the self-energy in terms of the Luttinger–Ward functional, which he evaluated to various orders in weak-coupling perturbation theory. A similar self-consistent single site theory was developed by Jarrell and coworkers [137] who assumed a purely local self–energy and vertex function even in a finite number of dimensions. The resulting mean–field theory for correlated lattice systems is usually called the dynamical mean–field approximation (DMFA).

How can we approximate a complicated problem with correlated electrons on a lattice to a single–site e ective problem with fewer degrees of freedom? Similarly to the well–known case of the Weiss mean–field theory for magnetism, in which the resultant mean–field equations become exact in the limit where the coordination number of the lattice becomes large, the mean–field description of the Hubbard model (see (2.1)) involves a generalized Weiss function which is a function of time instead of being a single number. Thus we are required to take local quantum fluctuations into account. The corresponding mean-field description involves

Here, the subscript 0 refers to the mean–field and G0−1(τ − τ ) plays the role of the generalized Weiss e ective field. Its physical content is an e ective amplitude for a fermion to be created on an isolated site at time τ (coming from an “external bath”) and to be destroyed at time τ (going back to the bath). As shown by Kotliar and coworkers [133], a closed set of mean-field equations can be obtained from (2.211) and from expressions relating G0 to local quantities computable from Sef f itself. One obtains

where G(iωn) is the on-site interacting Green’s function and R(G) denotes the reciprocal function of the Hilbert transform of the corresponding density of states. Hence, the DMFA approach provides a good possibility for a controlled treatment of the Mott–Hubbard transition. However, as shown by Hettler et al. [138], the DMFA is not a conserving approximation, with violations of the Ward identity associated with current conservation in the equation of continuity for any number of dimensions, including the limit d = ∞. Furthermore, the DMFA does not incorporate the important nonlocal correlations, and hence it is not possible to study dx2−y2 –wave Cooper pairing with it.

To overcome these problems, Jarrell and coworkers developed the so-called dynamical cluster approximation (DCA), which incorporates the nonlocal corrections to the DMFA by mapping the lattice problem onto an embedded cluster of size Nc, rather than onto an impurity problem [138]. These authors have shown further that the DCA is a fully causal approach and that it becomes exact in the limit of large Nc, while it reduces to the DMFA for Nc = 1. Thus Nc determines the order of the approximation in a simple way and provides also a systematic expansion parameter, 1/Nc. Similarly to DMFA, the DCA solution remains in the thermodynamic limit, but the dynamical correlation length is restricted to the size of the embedded cluster, of course.

86 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

Within this approach, Maier et al. calculated the mean-field dx2−y2 –wave Tc for Nc = 4 by using the NCA to solve the cluster problem [139]. For U = 12t, they found that Tc has its maximum value of approximately 0.05t 150 K for a doping concentration x = 0.2. It was also found that Tc increases for positive values of the next–nearest–neighbor hopping parameter t , and decreases for negative values of t . Note that this is in agreement with density– matrix–renormalization–group (DMRG) calculations on the t–J model by White and Scalapino who find dx2−y2 –wave pairing for t > 0 [140]. Recently, Jarrell and coworkers used the DCA approach with the Hubbard model in order to analyze ARPES spectra [141], the occurrence of a pseudogap [142], and the role of impurities in dx2−y2 -wave superconductors [141], and extended the DCA in such a way that the FLEX approximation rather than the NCA was employed for the cluster problem [143].

Thus, in short, the current DCA approach is an interesting method that incorporates causal nonlocal corrections to the DMFA in a transparent way. Importantly, at half-filling the DCA yields a T = 0 phase transition and a charge pseudogap accompanied by a non-Fermi-liquid behavior in the thermodynamic limit. For finite Nc, the DCA retains some mean-field character, which emulates the finite coupling of the two-dimensional model to the third dimension and hence emulates the Mermin–Wagner theorem [144, 145]. This is similar to our theory based on the two–dimensional generalized Eliashberg equations (see Appendix A), in which the self-consistent treatment prevents the system from becoming antiferromagnetic as required by the Mermin– Wagner theorem. On the other hand, in our theory fluctuations of the superconducting state are considered in a controlled way (see sect. 2.2.2) yielding a Kosterlitz–Thouless–like transition for Tc. Another advantage of the DCA with respect to our theory is the fact that its mean-field character can gradually be reduced as Nc tends to infinity. On the other hand, within the DCA the very small cluster size of Nc = 4 (which is the minimum for determining dx2−y2 –wave Cooper pairing) is the largest cluster that can be treated so far. Finally, we would like to stress that the DCA yields a crossover from a Fermi surface centered around the (0, 0) point in the Brillouin zone for large doping to a Fermi surface centered around (π, π) for small doping which is in disagreement with ARPES experiments [141]. Thus, we can safely conclude that it would be di cult to find a consistent description of the elementary excitations (i.e. for the kink feature) and their interdependence with spin excitations below Tc (the resonance peak) using the DCA approach. This is one of the main advantages of our theory.

2.5.2 The Large-U Limit

As discussed in connection with Fig. 1.10, the electronic states of the cuprates can be described by a three-band version of the Hubbard model, where in each unit cell one has a Cu dx2−y2 orbital and two oxygen p orbitals (see (1.3)). However, the largest energies in the problem are the correlation energies for

doubly occupying the copper or oxygen orbitals. In the hole picture, the Cu d9 configuration is reflected by an energy level Ed occupied by a single hole with S = 1/2 and the oxygen p orbital is empty of holes and has an energy of Ep. The energy cost for doubly occupying Ed, yielding a d8 configuration, is Ud, which is very large and often considered to be infinity. Thus, following this picture, the lowest–energy excitation is the charge transfer excitation where a hole hops from d to p with amplitude −tpd. If the energy di erence Ep − Ed is su ciently large compared with tpd, the hole will form a local moment on Cu [146]. Essentially, Ep − Ed plays the role of the Hubbard U in a one-band model of a Mott insulator.

In a one-band Mott–Hubbard insulator, in which virtual hopping to doubly occupied states leads to an exchange interaction JS1 · S2 (with J = 4t2/U ), the local moments on nearest–neighbor Cu sites prefer to align antiferromagnetically because spins can virtually hop to an orbital with energy Ed. If one ignores the Up for doubly occupying the p orbital with holes, the exchange integral is given by

Early Raman scattering experiments on two–magnon excitations by Klein and coworkers found an exchange energy of J 0.13 eV ([147] and references therein), and this value has been confirmed by INS experiments, described with a spin wave theory in which additional exchange terms are found [148, 149].

As described in Chap. 1, by focusing on the low-lying singlet excitations, we can make the doped three–band Hubbard model simplifly into an e ective one–band model, using an e ective hopping integral t and an e ective on– site Coulomb repulsion U , which is –within the above picture– of the order of Ep − Ed. In the large–U limit, the Hubbard model that we use in this work (see (2.1)) maps onto the t–J model,

where n = c†iσ ciσ , Si = 1/2c†iασαβ ciβ (σαβ is a vector of Pauli matrices), and P is a projection operator that restricts the Hilbert space by projecting out double occupation. Within the t–J model, the basic physics is seen to be the competition between the energy gain xt (where x denotes the doping concentration) due to mobile holes, and the cost of the exchange energyJ resulting from the disruption of the antiferromagnetic order. If J was small, the cost of the exchange energy could be overcome by delocalization, yielding a conventional Fermi liquid. This seems to be the case for the doped Mott insulator La1−xSrxTiO3. However, in the case of cuprates J is large and one can expect di erent physics. One important idea is the following: