  We show that the d-wave ordering in particle-hole channels, dubbed d-wave
checkerboard order, possesses important physics that can sufficiently explain
the scanning tunneling microscopy (STM) results in cuprates. A weak d-wave
checkerboard order can effectively suppress the coherence peak in the
single-particle spectrum while leaving the spectrum along the nodal direction
almost unaffected. Simultaneously, it generates a Fermi arc with little
dispersion around the nodal points at finite temperature that is consistent
with the results of angle-resolved photoemission spectroscopy (ARPES)
experiments in the pseudogap phase. We also show that there is a general
complementary connection between the d-wave checkerboard order and the
pair-density-wave order. Suppressing superconductivity locally or globally
through phase fluctuations should induce both orders in underdoped cuprates and
explain the nodal-antinodal dichotomy observed in ARPES and STM experiments.
