Winding number excitation detects phase transition in one-dimensional XY model with variable interaction range

We numerically study the critical behavior of the one-dimensional XY model of the size N with variable interaction range L. As expected, the standard local order parameter of the magnetization is shown to well detect the mean-field-type transition which occurs at any nonzero value of L/N. The system is particularly interesting since the underlying one-dimensional structure allows us to study the topological excitation of the winding number across the whole system even though the system shares the mean-field transition with the globally coupled system. We propose a nonlocal order parameter based on the width of the winding number distribution which exhibits a clear signature of the transition nature of the system.