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

Understand

The stationary points of the Hamiltonian H of the classical XY chain with power-law pair interactions (i.e., decaying like r^{-{\alpha}} with the distance) are analyzed.

• For a class of "spinwave-type" stationary points, the asymptotic behavior of the Hessian determinant of H 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 Szeg\"o-type theorem.
• The results serve to illustrate a recently discovered relation between phase transitions and the properties of stationary points of classical many-body Hamiltonian functions.