Revisiting Quasiparticle Scattering Interference in High-Temperature Superconductors: The Problem of Narrow Peaks

We revisit the interpretation of quasiparticle scattering interference in cuprate high-$T_c$ superconductors. This phenomenon has been very successful in reconstructing the dispersions of $d$-wave Bogoliubov excitations, but the successful identification and interpretation of QPI in scanning tunneling spectroscopy (STS) experiments rely on theoretical results obtained for the case of isolated impurities. We introduce a highly flexible technique to simulate STS measurements by computing the local density of states using real-space Green's functions defined on two-dimensional lattices with as many as 100,000 sites. We focus on the following question: to what extent can the experimental results be reproduced when various forms of distributed disorder are present? We consider randomly distributed point-like impurities, smooth "Coulombic" disorder, and disorder arising from random on-site energies and superconducting gaps. We find an apparent paradox: the QPI peaks in the Fourier-transformed local density of states appear to be sharper and better defined in experiment than those seen in our simulations. We arrive at a no-go result for smooth-potential disorder since this does not reproduce the QPI peaks associated with large-momentum scattering. An ensemble of point-like impurities gets closest to experiment, but this goes hand in hand with impurity cores that are not seen in experiment. We also study the effects of possible measurement artifacts (the "fork mechanism"), which turn out to be of relatively minor consequence. It appears that there is an unknown mechanism at work which renders the QPI peaks much sharper than they are based on present theoretical understanding.