spin σ on a square lattice, and U denotes the e ective Coulomb repulsion. niσ = c†iσ ciσ is the density for spin σ. In addition to the usual hopping integral t describing nearest neighbors, we add also a second–nearest–neighbor hopping integral t . The sums i, j are performed taking second–nearest– neighbors into account in this way.

Simply speaking, the one–band Hubbard model tries to mimic the presence of the charge-transfer gap ∆ by means of an e ective value of the Coulomb repulsion U . Thus, in the case of hole doping, the oxygen band becomes the lower band of the model. Note that in the strong–coupling limit it can be shown that the Hubbard model reduces to the t–J model. However, during this procedure, additional terms such as −(1/4)ndi ndj appear spontaneously, which have not received much attention, and their importance is unclear. Usually, they are excluded from numerical studies.

In short, we believe that the one–band Hubbard model is more than just an appropriate starting point for building up an electronic theory of Cooper pairing within a CuO2 plane. The main ingredients are present in this model: kinetic energy vs. potential energy, a strong repulsive (mainly on-site) interaction describing the physics in the vicinity of a Mott–Hubbard transition, and also itinerant carriers, which are experimentally observed in the CuO2 planes and which can easily condense into Cooper pairs below Tc.

In order to obtain a unified theory for both hole–doped and electron– doped cuprates, it is tempting to use the same Hubbard Hamiltonian, taking into account, of course the di erent dispersions for the carriers [101]. As mentioned above, in the case of electron doping the electrons occupy copper d–like states of the upper Hubbard band, while the holes are related to oxygen–like p states, yielding di erent energy dispersions, which we shall use in our calculations. Then, assuming similar itinerancy of the electrons and holes, the mapping onto an e ective one–band model seems to be justified. We consider U as an e ective Coulomb interaction. Throughout this article we shall work within the grand canonical limit, with a chemical potential µ describing the band filling. The parameters t and t will be employed to describe the normal–state energy dispersion measured in ARPES experiments, and a rigid–band approximation for all doping concentrations is assumed. This will be discussed in Chap. 2.

1.4.3 Spin Fluctuation Mechanism for Superconductivity

Before we illustrate how singlet pairing in high-Tc cuprates and triplet pairing in Sr2RuO4 are possible, let us remind the reader of some generalities. In connection with the general phase diagram for hole–doped cuprates, Fig. 1.4, we have discussed the occurrence of (mainly antiferromagnetic) spin fluctuations in the paramagnetic metallic state above Tc. These excitations can be measured in INS experiments, for example [43]. However, they do not appear as an additional excitation in the Hubbard Hamiltonian. Instead, these spin excitations are generated by itinerant carriers in the system and are mainly

24 1 Introduction

of two–dimensional character, and thus are more robust than the long-range 3D N´eel state. To study fluctuations in the paramagnetic state beyond the mean-field level it is convenient to employ the random–phase approximation (RPA). In particular, the spin-wave-like excitations can be obtained from the retarded transverse spin susceptibility

A similar expression holds for the longitudinal spin susceptibility χzz . The RPA result for the transverse spin susceptibility then reads [102]

q

where χ0 refers to the Lindhard function and has to be calculated from the single-particle Green’s function G (and thus from the elementary excitations) of the system using the Hubbard Hamiltonian. z denotes a complex frequency. Note that G can also be simply related to the sublattice magnetization. Thus, we shall assume in the following that an e ective perturbation series for the description of spin fluctuations is valid, and we shall sum the corresponding ladder and bubble diagrams up to infinite order at the RPA level. This procedure, for both singlet and triplet Cooper pairing, will be discussed in detail in the next chapter.

Next, we want to define the term “unconventional” and to investigate how a transition temperature Tc of the order of 100 K for hole–doped cuprates might occur as a result of a purely electronic (i.e. repulsive) mechanism. For this purpose, let us consider the simplified weak–coupling gap equation for singlet pairing (T = 0),

which is a self–consistency equation for the superconducting order parameter ∆(k) in momentum space, where k is defined in the first Brillouin zone. Vsef f (k − k ) represents the e ective two–particle pairing interaction in the singlet channel and is, to a good approximation, proportional to χRP A if Cooper pairing due to spin fluctuations is present. The energy

E(k) = ∆2(k) + 2(k) corresponds to the dispersion relation of the Bogoliubov quasiparticles (i.e. the Cooper pairs), where (k) denotes the dispersion of the electrons in the normal state. In the BCS theory, Vsef f < 0 is taken as a constant and therefore one obtains a solution for ∆(k) of (1.11) which is structureless in momentum space.2

2Of course, owing to retardation e ects, the summation in (1.11) runs over an energy shell of order ωD ; however, the arguments given above remain valid if one integrates over the whole Brillouin zone.

Fig. 1.12. Fermi surface of the one-band Hubbard model close to half-filling, for a square Brillouin zone. The nesting vector Q connects di erent parts of the Fermi surface where the dx2−y2 -wave order parameter has opposite sign. Thus the gap equation can be solved for a repulsive pairing potential. Along the diagonal lines, the corresponding gap has nodes (i.e. ∆(k) vanishes).

Let us now investigate the case of singlet pairing and how it is possible to solve (1.11) with a repulsive pairing potential. Owing to the fact that Vsef f > 0, one might naively think that there exists no solution; however, this is wrong. If one takes into account that Vsef f (k − k ) might have a strong momentum dependence, it is easily seen that indeed the gap equation has a solution. This can be recognized from Fig. 1.12, where we show the first Brillouin zone and a Fermi surface which corresponds to the half–filled case of the two-dimensional one–band Hubbard model. This Fermi surface in fact resembles the measured Fermi surface for the high–Tc superconductor La2−xSrxCuO4. For simplicity, we assume also an underlying tetragonal symmetry of the crystal. In order to illustrate the following argument, a possible superconducting order parameter (dx2−y2 –wave symmetry, with positive and negative signs) is also displayed. If one now assumes that the e ective pairing interaction has a strong momentum dependence, for example a large peak at the antiferromagnetic wave vector qAF = Q = (π, π), Vsef f (k − k ) connects di erent parts of the Fermi surface where the order parameter has opposite signs! Thus, one indeed finds a solution of equation (1.11). The simplest solution is the dx2−y2 –wave gap (∆(k) = ∆0 [cos kx − cos ky ] /2), which has nodes along the diagonals. Note, however, that for an e ective pairing interaction which was structureless in momentum space, such a solution of the gap equation for a d–wave order parameter would not be possible. In fact, an order parameter which belonged to an anisotropic s–wave representation (possi-

26 1 Introduction

bly with nodes) would not satisfy the pairing condition mentioned above either. In order to solve (1.11) for a pairing interaction which is peaked at (π, π), one definitely needs an order parameter that changes sign. Therefore, we see that the superconducting gap has less symmetry than the underlying Fermi surface. In such a situation where, in addition to a broken gauge (U (1)) invariance due to the occurrence of superconductivity, a further symmetry is broken (in our case invariance under a rotation of 90 degrees), we define the situation as “unconventional”. Notice that this definition implies neither an electronic pairing mechanism nor a correspondingly large value of Tc. Finally, let us briefly mention that the arguments, given above are weak-coupling arguments which may be reformulated in the strong–coupling limit of the pairing process. However, it will turn out that, although lifetime e ects of the electrons will lead to a renormalization of the quasiparticles, the weak-coupling arguments given above remain valid.

In the case of triplet pairing, one has to solve the corresponding gap equation

0, i.e. an attractive pairing interaction in momentum space. This is in analogy to phonons, where the minus sign in front of the right-hand side of (1.12) is also canceled, which makes a solution of the gap equation relatively easy. In particular, if the transferred momentum q = k − k is small, one obtains a p–wave symmetry of the superconducting order parameter, for example

Note that the condition q ≈ 0 is indeed fulfilled in the case of superfluid 3He, which it is close to a ferromagnetic transition. Simply speaking, a similar situation is present in the case of Sr2RuO4 (see its phase diagram in Fig. 1.9), which makes p–wave symmetry the most probable candidate for the order parameter.

In order to investigate triplet pairing in Sr2RuO4 in more detail, we show in Fig. 1.13 its corresponding Fermi surface topology, obtained using the three–band Hubbard Hamiltonian discussed earlier in this chapter. However, for simplicity, we discuss here only the γ–band (which has a high density of states). Of course, the other bands (α and β) and their consequences will be analyzed in detail later. For the moment and for simplicity, let us discuss only the γ-band in order to discriminate between Sr2RuO4 and cuprates. A closer inspection of (1.13) shows that |∆p|2 has no nodes; however, Re ∆p (and also Im ∆p) has a nodal line also displayed in Fig. 1.13. This has important consequences if nesting is present: the summation over k in the first BZ is dominated by the contributions due to Qpair and those due to a smaller

Fig. 1.13. Calculated Fermi surface (FS) topology for Sr2RuO4 and symmetry analysis of the superconducting order parameter ∆. The real part of the p–wave order parameter has a node along kx = 0. The plus and minus signs and the dashed lines refer to the sign of the momentum dependence of ∆. α, β, and γ denote the FS of the corresponding (hybridized) bands.

wave vector qpair . Thus, we obtain approximately the following for the γ– band contribution (l = f or p):

where the sum is over all contributions due to Qi and qpair . The wave vectors Qpair bridge portions of the FS where Re ∆p has opposite signs. Because the smaller wave vector qpair bridges areas on the Fermi surface with both the same sign and opposite signs, its total contribution is almost zero, i.e.

Thus, we find a gap equation where ∆l is expected to change its sign for an attractive interaction. This is not possible! In other words, if the corresponding pairing interaction were to have nesting properties similar to those of cuprates, i.e. a peak of χRP A at q ≈ Qpair , Cooper pairing and a solution of (1.13) would not be possible, because Vtef f < 0. In this case, the nesting properties would suppress a p–wave and favor an f –wave, i.e.

Like the dx2−y2 –wave order parameter in cuprates, the f –wave symmetry also has nodes along the diagonals.

To summarize this subsection, we have demonstrated on general grounds that, if Cooper pairing via spin fluctuations is present, the underlying Fermi