our results, compare them with cuprate high–temperature superconductors, and give an opinion of the outlook in Sect. 4.3.

4.1 Elementary Spin Excitations in the Normal State

of Sr2RuO4

4.1.1 Importance of Spin–Orbit Coupling

First, we show in Fig. 4.1 the results for the Lindhard response functions for the di erent bands, calculated using their tight–binding dispersions without taking into account the spin–orbit coupling using (2.34) and (2.35). As expected, owing to the pronounced nesting of the xz and yz bands, the susceptibilities of those bands display peaks at Qi = (2π/3, 2π/3), while the xy band does not show any significant feature. On the other hand, the response of the xy band is enhanced owing to the presence of the van Hove singularity close to the Fermi level. It becomes clear that the features observed by INS relate mainly to the magnetic response of the xz and yz bands. However, the present results cannot account for the observed magnetic anisotropy, since both the longitudinal and the transverse components of the total spin susceptibility are the same if the term which transfers the anisotropy from the orbital subspace into the spin subspace is not taken into account. In order to do this, we include a spin–orbit coupling as we discussed in Chap. 2. Using in addition the random–phase approximation for each band (see Chap. 2 for

Fig. 4.1. Calculated real part of the Lindhard spin susceptibility χl0(ω = 0) in the normal state of Sr2RuO4 along the path (0, 0) → (π, 0) → (π, π) → (0, 0) in the first two–dimensional square BZ for the kz = 0 dispersion of the three di erent orbitals (xz, yz, and xy) crossing the Fermi level. Owing to the nesting of the xz and yz bands, their susceptibilities show an enhancement at the incommensurate antiferromagnetic wave vector Qi = (2π/3, 2π/3). Note that the response of the xy band is more isotropic but significantly larger than that in the normal metal owing to the nearness of the van Hove singularity.

180 4 Results for Sr2RuO4

Fig. 4.2. Results for the real part of the out–of–plane (i.e. longitudinal, solid curve) and in–plane (i.e. transverse, dashed curve) magnetic response of an RuO2 plane to Hz and Hxy , respectively. Reχ(q, ω = 0) was calculated within the RPA including spin–orbit coupling, using U = 0.505 eV and thyb = 0, along the path (0, 0) → (π, 0) → (π, π) → (0, 0) within the first Brillouin zone at a temperature T = 100 K using (2.45) and (2.46). (Note that χzz and χ+− refer to the response of the projection of the orbitals dxy , dxz , and dyz on the xy plane).

details) we have calculated the longitudinal and transverse components of the total susceptibility in the RuO2 planes.

In Fig. 4.2, we show the results for the static limit of the real part of the

U = 0.505 eV obtained using (2.45) and (2.46). Note the important di erence between the two components. Most importantly, the IAF fluctuations at Qi = (2π/3, 2π/3) are present in the case of xz and yz bands only in the longitudinal components of the spin susceptibility, and not in the transverse components. This is connected to the fact that matrix elements of type uk and vk are important because they suppress transition between “+” and “−” bands for the transverse susceptibilities. The transverse susceptibility is larger than the longitudinal one at small values of q, indicating ferromagnetic fluctuations. These point mainly in the RuO2 plane. On the other hand, the longitudinal component shows a structure at the IAF wave vector indicating a direction of the IAF fluctuations perpendicular to the RuO2 plane.

We also want to emphasize that our results are in accordance with earlier estimates made by Ng and Sigrist [30], with one important di erence. In addition to what was done in that work, we have included, in accordance with the mixing of the spin and orbital degrees of freedom, the orbital contribution to the magnetic susceptibility χ (i.e. we have used the g-factor for the vertex function). For example, owing to lz and l+ and (l−) vertices at Qi = (2π/3, 2π/3), χzz is a ected by a factor of 2 from spin–orbit coupling. Moreover, in [30], it was found that the IAF fluctuations are slightly enhanced in the longitudinal components of the xz and yz bands in comparison with

Fig. 4.3. Temperature dependence of the imaginary part of the spin susceptibility including spin–orbit coupling (but no hybridization), divided by the spin fluctuation energy ωsf and summed over q. Note, as in Fig. 4.2, that the notation zz and +− refers to the out–of–plane (solid curve) and in–plane (dashed curve) components of the RPA spin susceptibility calculated for a single RuO2 plane. In the inset we show the corresponding frequency dependence of Im χRP A(Qi, ω) at the IAF wave vector Qi = (2π/3, 2π/3). The results for the out-plane component (solid curve) are in quantitative agreement with INS experiments [2].

the transverse component. In our case there are roughly no IAF fluctuations in the transverse component of the spin susceptibility. Furthermore, by taking into account the correlation e ects within the RPA, we have shown that the IAF fluctuations will be further enhanced in the z direction.

This is illustrated further in the inset of Fig. 4.3, where we present the results for the frequency dependence of the imaginary parts of the total susceptibilities at Qi = (2π/3, 2π/3) and a temperature T = 20 K. The longitudinal component has a peak at approximately ωsf = 6 meV, in quantitative agreement with experimental data from INS [2]. On the other hand, the transverse component is featureless, showing the absence of IAF spin fluctuations. This also indicates that the IAF fluctuations are aligned perpendicular to the RuO2 plane.

In order to see the temperature dependence of the magnetic anisotropy induced by the spin–orbit coupling we display in Fig. 4.3 the temperature de-

identical, since thermal e ects wash out the influence of the spin–orbit interaction. With decreasing temperature, the magnetic anisotropy arises and, at low temperatures, we find the important result that the out-of-plane component χzz is about two times larger than the in-plane one (χzz > χ+−/2).

In the next step, the susceptibility matrix [χij ] was calculated, where i, j refer to the hybridized bands. Our results for χαα and χββ are shown in Fig. 4.5a. For comparison, χγγ is also displayed again. In Fig. 4.5b, our results for the cross–susceptibilities (i.e. the non diagonal elements of [χij0 ]) are shown. As already mentioned, we find the important result that all elements

Fig. 4.5. (a) Calculated momentum dependence of Re χαα0 (dashed curve) and Re χββ0 (dotted curve) obtained using (2.35) and hybridized bands. For comparison, Re χγγ (solid curve) is also displayed. (b) Calculated momentum dependence of the corresponding cross–susceptibilities.