  The cuprate high-temperature superconductors are known to host a wide array
of effects due to interactions and disorder. In this work, we look at some of
the consequences of these effects which can be visualized by scanning tunneling
spectroscopy. These interaction and disorder effects can be incorporated into a
mean-field description by means of a self-energy appearing in the Green's
function. We first examine the quasiparticle scattering interference spectra in
the superconducting state at optimal doping as temperature is increased.
Assuming agreement with angle-resolved photoemission experiments which suggest
that the scattering rate depends on temperature, resulting in the filling of
the $d$-wave gap, we find that the peaks predicted by the octet model become
progressively smeared as temperature is increased. When the scattering rate is
of the same order of magnitude as the superconducting gap, the spectral
function shows Fermi-arc-like patterns, while the power spectrum of the local
density of states shows the destruction of the octet-model peaks. We next
consider the normal state properties of the optimally-doped cuprates. We model
this by adding a marginal Fermi liquid self-energy to the normal-state
propagator, and consider the dependence of the QPI spectra on frequency,
temperature, and doping. We demonstrate that the MFL self-energy leads to a
smearing of the caustics appearing in the normal-state QPI power spectrum as
either temperature or frequency is increased at fixed doping. The smearing is
found to be more prominent in the MFL case than in an ordinary Fermi liquid. We
also consider the case of a marginal Fermi liquid with a strongly
momentum-dependent self-energy which gives rise to a visible "nodal-antinodal"
dichotomy at the normal state, and discuss how the spectra as seen in ARPES and
STS differ from both an isotropic metal and a broadened $d$-wave
superconductor.
