Abstract
The intrinsic geometry of wavevector diagrams describes electronic or photonic transport at a given energy level. Lifshitz transition is an intriguing example of the topological transition in wavevector diagrams, which plays a critical role in abnormal transport with enhanced magnetoresistance or superconductivity. Here, we develop the spatial analogy of the Lifshitz transition, which provides a comprehensive topological perspective on transverse-spin interface states. We establish the excitation conditions of transverse-spin interface states, which require the "Lifshitz interface" - the interface between different topologies of wavevector diagrams - along with the gap in wavevector diagrams. Based on the detailed analysis of this topological phenomenon with respect to the dimensionality and gaps of wavevector diagrams across the Lifshitz interface, we show distinct parity of transverse spins and power flows in transverse-spin modes. The unique symmetry of interface states realizing Abraham-spin-momentum locking represents the gauge induced by the Lifshitz interface, which provides a novel insight into the Abraham-Minkowski controversy.