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We study the stationary points of what is known as the lattice Landau gaugefixing functional in one-dimensional compact U(1) lattice gauge theory, or asthe Hamiltonian of the one-dimensional random phase XY model in statisticalphysics. An analytic solution of all stationary points is derived for latticeswith an odd number of lattice sites and periodic boundary conditions. In thecontext of lattice gauge theory, these stationary points and their indices areused to compute the gauge fixing partition function, making reference inparticular to the Neuberger problem. Interpreted as stationary points of theone-dimensional XY Hamiltonian, the solutions and their Hessian determinantsallow us to evaluate a criterion which makes predictions on the existence ofphase transitions and the corresponding critical energies in the thermodynamiclimit.

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian