The discovery of high temperature superconductivity in the cuprates in 1986 triggered a spectacular outpouring of creative and innovative scientific inquiry. Much has been learned over the ensuing 28 years about the novel forms of quantum matter that
are exhibited in this strongly correlated electron system. This progress has been made possible by improvements in sample quality, coupled with the development and refinement of advanced experimental techniques. In part, avenues of inquiry have been motivated by theoretical developments, and in part new theoretical frameworks have been conceived to account for unanticipated experimental observations. An overall qualitative understanding of the nature of the superconducting state itself has been achieved, while profound unresolved issues have come into increasingly sharp focus concerning the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the normal state at elevated temperatures. New conceptual approaches, drawing from string theory, quantum information theory, and various numerically implemented approximate approaches to problems of strong correlations are being explored as ways to come to grips with this rich tableaux of interrelated phenomena.
We show that orbital currents in a CuO2 plane, if present, should be described by two independent parity and time-reversal odd order parameters, a toroidal dipole (anapole) and a magnetic quadrupole. Based on this, we derive the resonant X-ray diffra
ction cross-section for monoclinic CuO at the antiferromagnetic wavevector and show that the two order parameters can be disentangled. From our analysis, we examine a recent claim of detecting anapoles in CuO.
Much attention has been given to a possible violation of the optical sum rule in the cuprates, and the connection this might have to kinetic energy lowering. The optical integral is composed of a cut-off independent term (whose temperature dependence
is a measure of the sum rule violation), plus a cut-off dependent term that accounts for the extension of the Drude peak beyond the upper bound of the integral. We find that the temperature dependence of the optical integral in the normal state of the cuprates can be accounted for solely by the latter term, implying that the dominant contribution to the observed sum rule `violation in the normal state is due to the finite cut-off. This cut-off dependent term is well modeled by a theory of electrons interacting with a broad spectrum of bosons.
