Title: Origin of multiple Lifshitz transitions in the Weyl semi-metal RhSi

Abstract: It is known from density functional theory (DFT) calculations that RhSi has a multifold degenerate Dirac point at the Fermi energy, with the dominant states in the low-energy region displaying mostly Rh $d$ character. Using DFT+U, we calculate the band structure by considering an effective local interaction on the Rh $d$ states, with a realistic effective Hubbard $U_\textrm{eff}=2.5$ eV derived from a constrained random-phase approximation calculation, and find the emergence of a double hump structure close to the Fermi energy.By further deriving a low-energy tight-binding model from our first-principles results, we show that the double hump is a direct consequence of a competition between the Rh $d$-Rh $d$ and Rh $d$-Si $p$ interactions, which differ in their momentum dependence. As a consequence, through an artificial tuning of the energy level of the Si $p$ orbitals this hump structure can be suppressed due to the effectively reduced Rh $d$ -Si $p$ interaction.This peculiar low-energy electronic structure additionally results in that a small hole/electron doping ($\sim$ 0.1 $\%$) can tune the Fermi surface topology, going from closed to open Fermi surfaces, which has dramatic consequences for the thermal transport.