Superconductivity often emerges as a dome around a quantum critical point
(QCP) where long-range order is suppressed to zero temperature. So far, this
has been mostly studied in magnetically ordered materials. By contrast, the
interplay between charge order and superconductivity at a QCP is not fully
understood. Here, we present resistance measurements proving that a dome of
superconductivity surrounds the charge-density-wave (CDW) QCP in pristine
samples of 1T-TiSe2​ tuned with hydrostatic pressure. Furthermore, we use
quantum oscillation measurements to show that the superconductivity sets in at
a Lifshitz transition in the electronic band structure. We use density
functional theory to identify the Fermi pockets enabling superconductivity:
large electron and hole pockets connected by the CDW wave vector Q​
which emerge upon partial suppression of the zero-pressure CDW gap. Hence, we
conclude that superconductivity is of interband type enabled by the presence of
hole and electron bands connected by the CDW Q​ vector. Earlier
calculations show that interband interactions are repulsive, which suggests
that unconventional s±​ superconductivity is realised in TiSe2​ -
similar to the iron pnictides. These results highlight the importance of
Lifshitz transitions in realising unconventional superconductivity and help
understand its interaction with CDW order in numerous materials.Comment: 21 pages, 5 figure