  Chirality is a ubiquitous phenomenon in which a symmetry between left- and
right-handed objects is broken, examples in nature ranging from subatomic
particles and molecules to living organisms. In particle physics, the weak
force is responsible for the symmetry breaking and parity violation in beta
decay, but in condensed matter systems interactions that lead to chirality
remain poorly understood. Here, we unravel the mechanism of chiral charge
density wave formation in the transition-metal dichalcogenide 1T-TiSe2. Using
representation analysis, we show that charge density modulations and ionic
displacements, which transform as a continuous scalar field and a vector field
on a discrete lattice, respectively, follow different irreducible
representations of the space group, despite the fact that they propagate with
the same wave-vectors and are strongly coupled to each other. This
charge-lattice symmetry frustration is resolved by further breaking of all
symmetries not common to both sectors through induced lattice distortions, thus
leading to chirality. Our theory is verified using Raman spectroscopy and
inelastic x-ray scattering, which reveal that all but translation symmetries
are broken at a level not resolved by state-of-the-art diffraction techniques.
