  The charge density wave (CDW) in solids is a collective ground state
combining lattice distortions and charge ordering. It is defined by a complex
order parameter with an amplitude and a phase. The amplitude and wavelength of
the charge modulation are readily accessible to experiment. However, accurate
measurements of the corresponding phase are significantly more challenging.
Here we combine reciprocal and real space information to map the full complex
order parameter based on topographic scanning tunneling microscopy (STM)
images. Our technique overcomes limitations of earlier Fourier space based
techniques to achieve distinct amplitude and phase images with high spatial
resolution. Applying this analysis to transition metal dichalcogenides provides
striking evidence that their CDWs consist of three individual charge
modulations whose ordering vectors are connected by the fundamental rotational
symmetry of the crystalline lattice. Spatial variations in the relative phases
of these three modulations account for the different contrasts often observed
in STM topographic images. Phase images further reveal topological defects and
discommensurations, a singularity predicted by theory for a nearly commensurate
CDW. Such precise real space mapping of the complex order parameter provides a
powerful tool for a deeper understanding of the CDW ground state whose
formation mechanisms remain largely unclear.
