Doping-induced nematic and stripe orders within the charge density wave state of TiSe$_2$

In this work, we present a theory to address conflicting experimental claims regarding the charge density wave (CDW) state in TiSe$_2$, including whether there is a single or multiple CDW transitions and whether threefold rotation symmetry ($C_3$) is broken. Using a continuum $\boldsymbol{k}\cdot\boldsymbol{p}$ model coupled to the CDW order parameter, we show how commonplace conduction band doping induces a nematic transition from a $C_3$-symmetric $3Q$ CDW to a $C_3$-breaking $3Q$ CDW, which is favored by the large ellipticity of the conduction bands of TiSe$_2$. We also find that a $1Q$ stripe CDW is generically stabilized for sufficiently high electron doping. We then show how both stripe and nematic CDW states emerge self-consistently from a minimal interacting tight-binding model, for both positive and negative initial gaps. Our theory provides a new scenario in which, as temperature is lowered, a second $C_3$-breaking transition may occur or not depending on the doping level, potentially explaining the experimental variability. These predictions can be further verified with a variety of probes including transport, photoemission and tunneling.