Title: Exact zero modes in twisted Kitaev chains,
Abstract: We study the Kitaev chain under generalized twisted boundary conditions, for
which both the amplitudes and the phases of the boundary couplings can be tuned
at will. We explicitly show the presence of exact zero modes for large chains
belonging to the topological phase in the most general case, in spite of the
absence of "edges" in the system. For specific values of the phase parameters,
we rigorously obtain the condition for the presence of the exact zero modes in
finite chains, and show that the zero modes obtained are indeed localized. The
full spectrum of the twisted chains with zero chemical potential is
analytically presented. Finally, we demonstrate the persistence of zero modes
(level crossing) even in the presence of disorder or interactions.