Lattice fluctuations, not excitonic correlations, mediated electronic localization in TiSe$_2$

TiSe$_2$ is thought to be an insulator with a bandgap of ~0.1eV. It has attracted a much interest because, among of a rich array of unique properties, many have thought TiSe$_2$ is a rare realisation of an excitonic insulator. Below 200 K, TiSe$_2$ undergoes a transition from a high-symmetry ({P-3m1}) phase to a low-symmetry ({P-3c1}) phase. Here we establish that TiSe$_2$ is indeed an insulator in both {P-3m1} and {P-3c1} phases. However, the insulating state is driven not by excitonic effects but by symmetry-breaking of the {P-3m1} phase. In the CDW phase the symmetry breaking is static. At high temperature, thermally driven instantaneous deviations from {P-3m1} break the symmetry on the characteristic time scale of a phonon. Even while the time-averaged \emph{lattice} structure assumes {P-3m1} symmetry, the time-averaged \emph{energy band} structure is closer to the CDW phase -- a rare instance of a metal-insulator transition induced by dynamical symmetry breaking. We establish these conclusions from a high-fidelity, self-consistent form of many body perturbation theory, in combination with molecular dynamics simulations to capture the effects of thermal disorder. The many-body theory includes explicitly ladder diagrams in the polarizability, which incorporates excitonic effects in an \emph{ab initio} manner. The excitonic modification to the potential is slight, ruling out the possibility that TiSe$_2$ is an excitonic insulator. Charge self-consistency is essential distinguish the metallic from insulating state.