Title: Predicting the temperature-strain phase diagram of VO$_2$ from first principles
Description: Predicting the temperature-strain phase diagram of VO$_2$, including the various structural allotropes, from first principles is a grand challenge of materials physics, and even the phase diagram remains unclear at T = 0K. The coexistence of Peierls and Mott physics suggests that a theory which can capture strong electronic correlations will be necessary to compute the total energies. In order to understand the complex nature of the first-order transition of VO$_2$, we build a minimal model of the structural energetics using the Peirels-Hubbard model and solve it exactly using the Density Matrix Renormalization Group (DMRG) methods demonstrating that the on-site interaction $U$ has a minimal effect on the structural energetics for physical parameters. These results explain the qualitative failures of Density Functional Theory (DFT) and DFT+$U$ for the structural energetics, in addition to the partial success of the unorthodox DFT+$U$ results (i.e. non-spin-polarized and small $U$). It also guides the creation of empirical corrections to the DFT+$U$ functional which allow us to semi-quantitatively capture the phase stability of the rutile and monoclinic phases as a function of temperature and strain. Our work demonstrates that VO$_2$ is better described as a Mott assisted Peierls transition.