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The Kerr rotation in the unconventional superconductor Sr$_2$RuO$_4$

The interpretation of Kerr rotation measurements in the superconducting phase of Sr$_2$RuO$_4$ is a controversial topic. Both intrinsic and extrinsic mechanisms have been proposed, and it has been argued that the intrinsic response vanishes by symmetry. We focus on the intrinsic contribution and clarify several conflicting results in the literature. On the basis of symmetry considerations and detailed calculations we show that the intrinsic Kerr signal is not forbidden in a general multi- band system but has a rich structure in the near infrared regime. We distinguish different optical transitions determined by the superconducting gap (far infrared) and the inter orbital coupling of the normal state (near infrared). We argue that the low frequency transitions do not contribute to the Hall conductivity while only the inter-orbital transitions in the near infrared regime contribute. Finally, we discuss the difficulties to connect the calculations for the optical Hall conductivity to the experimental measurement of the Kerr angle. We will compare different approximations which might lead to conflicting results.