The physics of correlated electron systems can be strongly influenced by Van Hove singularities (VHSs) in the vicinity of the Fermi energy (E_F). In the Ruddlesden-Popper (RP) series of strontium ruthenates, Sr_n+1Ru_nO_3n+1, near- E_F VHSs are thought to drive a host of complex emergent phenomena including enhanced superconductivity and metamagnetism. The tri-layer compound, Sr₄Ru₃O₁₀, hosts a ferromagnetic ground state, with magnetisation aligned parallel to the c-axis. Interestingly, an in-plane metamagnetic transition has been observed whose physical origin is not well understood, although it has been postulated that the transition is related to VHSs situated close to E_F. I will show our recent work combining angle-resolved photoemission spectroscopy and scanning tunnelling microscopy to identify and classify a rich hierarchy of VHSs located within 30 meV of the Fermi level in Sr₄Ru₃O₁₀. ^[1] Most importantly, we find that a spin-orbit induced hybridisation between a heavy spin-majority and light spin-minority band mediates the formation of a saddle point VHS right at the Fermi level. We further identify a delicate interplay between the magnetisation direction and spin-orbit coupling, demonstrating a new mechanism through which field-driven Lifshitz transitions may arise.

 [1] Carolina A. Marques, Philip A.E. Murgatroyd, et al., “Spin-orbit coupling induced Van Hove singularity in proximity to a Lifshitz transition in Sr₄Ru₃O₁₀”,  arXiv:2303.05587, Oct. 2023.