Stationary-point approach to the phase transition of the classical XY chain with power-law interactions.

The stationary points of the potential energy function V of the classical XY chain with power-law pair interactions (i.e., interactions decaying like r{-α} with the distance) are analyzed. For a class of "spinwave-type" stationary points, the asymptotic behavior of the Hessian determinant of V is computed analytically in the limit of large system size. The computation is based on the Toeplitz property of the Hessian and makes use of a Szegö-type theorem. The results serve to illustrate a recently discovered relation between phase transitions and the properties of stationary points of classical many-body potentials. In agreement with this relation, the exact phase transition potential energy of the model can be read off from the behavior of the Hessian determinant for exponents α between zero and one. For α between one and two, the phase transition is not manifest in the behavior of the determinant, and it might be necessary to consider larger classes of stationary points.