  One of the key motivations for the development of atomically resolved
spectroscopic imaging STM (SI-STM) has been to probe the electronic structure
of cuprate high temperature superconductors. In both the d-wave superconducting
(dSC) and the pseudogap (PG) phases of underdoped cuprates, two distinct
classes of electronic states are observed using SI-STM. The first class
consists of the dispersive Bogoliubov quasiparticles of a homogeneous d-wave
superconductor. These are detected below a lower energy scale |E|={\Delta}0 and
only upon a momentum space (k-space) arc which terminates near the lines
connecting k=\pm({\pi}/a0,0) to k=\pm(0, {\pi}/a0). In both the dSC and PG
phases, the only broken symmetries detected in the |E|\leq {\Delta}0 states are
those of a d-wave superconductor. The second class of states occurs at energies
near the pseudogap energy scale |E| {\Delta}1 which is associated
conventionally with the 'antinodal' states near k=\pm({\pi}/a0,0) and k=\pm(0,
{\pi}/a0). We find that these states break the 90o-rotational (C4) symmetry of
electronic structure within CuO2 unit cells, at least down to 180o rotational
(C2) symmetry (nematic) but in a spatially disordered fashion. This
intra-unit-cell C4 symmetry breaking coexists at |E| {\Delta}1 with
incommensurate conductance modulations locally breaking both rotational and
translational symmetries (smectic). The properties of these two classes of |E|
{\Delta}1 states are indistinguishable in the dSC and PG phases. To explain
this segregation of k-space into the two regimes distinguished by the
symmetries of their electronic states and their energy scales |E| {\Delta}1 and
|E|\leq{\Delta}0, and to understand how this impacts the electronic phase
diagram and the mechanism of high-Tc superconductivity, represents one of a key
challenges for cuprate studies.
