  A recent scanning tunneling microscopy (STM) experiment reports the
observation of charge density wave (CDW) with period of approximately 8a in the
halo region surrounding the vortex core, in striking contrast to the
approximately period 4a CDW that are commonly observed in the cuprates.
Inspired by this work, we study a model where a bi-directional pair density
wave (PDW) with period 8 is at play. This further divides into two classes, (1)
where the PDW is a competing state of the d wave superconductor and can exist
only near the vortex core where the d wave order is suppressed, and (2) where
the PDW is the primary order, the so called mother state that persists with
strong phase fluctuations to high temperature and high magnetic field and lies
behind the pseudogap phenomenology. We study the charge density wave structures
near the vortex core in these models. We emphasize the importance of the phase
winding of the d-wave order parameter. The PDW can be pinned by the vortex core
due to this winding and become static. Furthermore, the period 8 CDW inherits
the properties of this winding, which gives rise to a special feature of the
Fourier transform peak, namely, it is split in certain directions. There are
also a line of zeros in the inverse Fourier transform of filtered data. We
propose that these are key experimental signatures that can distinguish between
the PDW-driven scenario from the more mundane option that the period 8 CDW is
primary. We discuss the pros and cons of the options considered above. Finally
we attempt to place the STM experiment in the broader context of pseudogap
physics of underdoped cuprates and relate this observation to the unusual
properties of X ray scattering data on CDW carried out to very high magnetic
field.
