Topological Lifshitz transitions and native quenched disorder in $1T$-TiSe$_2$

Spatially inhomogeneous electronic states are expected to be key ingredients for the emergence of superconducting phases in quantum materials hosting charge-density-waves (CDWs). Prototypical materials are transition-metal dichalcogenides (TMDCs) and among them, 1$T$-TiSe$_2$ exhibiting intertwined CDW and superconducting states under Cu intercalation, pressure or electrical gating. Although it has been recently proposed that the emergence of superconductivity relates to CDW fluctuations and the development of spatial inhomogeneities in the CDW order, the fundamental mechanism underlying such a phase separation (PS) is still missing. Using angle-resolved photoemission spectroscopy and variable-temperature scanning tunneling microscopy, we report on the phase diagram of the CDW in 1$T$-TiSe$_2$ as a function of Ti self-doping, an overlooked degree of freedom inducing CDW texturing. We find an intrinsic tendency towards electronic PS in the vicinity of Fermi surface (FS) hot spots, i.e. locations with band crossings close to, but not at the Fermi level. We therefore demonstrate an intimate relationship between the FS topology and the emergence of spatially textured electronic phases which is expected to be generalizable to many doped CDW compounds.