Magneto-optical Sum Rules Close to the Mott Transition

The ac Hall effect can provide valuable insights into the dynamics of an electronic medium. This has recently been demonstrated in the case of high-Tc superconductors [1,2]: Various theoretical models based on different scattering mechanisms agree that the anomalous frequency and temperature dependences of the Hall effect are closely intertwined, but they differ in their predictions about these dependences [3]. So far, experiments cannot discriminate between these models, but they will possibly be able to do so in the future [3]. The magneto-optical response of charge carriers can be probed by the frequency-dependent Hall conductivity, Hall constant, or Hall angle. Recently, a sum rule for the Hall angle has been derived [4] that is similar to the wellknown f-sum rule for the optical conductivity [5]. In this paper, we derive new sum rules for the real and imaginary parts of the two other magnetotransport probes. Such sum rules are useful: First, they help elucidating how the corresponding spectral weight is redistributed upon changing the temperature or the doping level. Second, they provide exact constraints on the interdependence of Hall effect–related quantities and thus help interpreting experimental data. For example, the sum rules for the ac Hall constant relate its low-frequency behavior to its infinite-frequency limit. This can be useful because experimentally, only the microwave domain and the far infrared are attainable sufficiently reliably [2,3], whereas the calculation of the Hall constant simplifies considerably in the high-frequency limit [6]. We shall first derive the sum rules for the Hall conductivity and Hall constant quite generally. Then, to illustrate their application, we shall discuss some aspects of the magneto-optical response of correlated electrons close to the density-driven Mott transition. We start by considering the ac conductivities. In terms of the dissipative part of the current-current correlation function,