Emergence of charge density wave domain walls above the superconducting dome in TiSe$_2$

Superconductivity (SC) in so-called ``unconventional superconductors'' is nearly always found in the vicinity of another ordered state, such as antiferromagnetism, charge density wave (CDW), or stripe order. This suggests a fundamental connection between SC and fluctuations in some other order parameter. 1$T$-TiSe$_2$ is a prototypical CDW material in the transition-metal dichalcogenide family and was previously shown to exhibit SC when the CDW is suppressed by hydrostatic pressure or intercalation of Cu atoms. Here, we present detailed high pressure x-ray scattering study on 1$T$-TiSe$_2$. We found that the CDW phase of 1$T$-TiSe$_2$ is completely suppressed on the application of hydrostatic pressure and established the existence of a quantum critical point (QCP). Unexpectedly, we observed a weakly first order, incommensurate CDW phase, suggesting the presence of a Lifshitz tricritical point somewhere above the superconducting dome. Our study suggests that SC in 1$T$-TiSe$_2$ may not be directly connected to the QCP of the CDW order, but to the formation of CDW domain wall.