Quantum Geometry of Finite XY Chains: A Comparison of Neveu-Schwarz and Ramond Sectors

This paper presents a geometrical analysis of finite length XY quantum chains. We begin by examining the ground state and the first excited state of the model, emphasizing the impact of finite size effects under two distinct choices of the Jordan Wigner transformation: the Neveu Schwartz (NS) and Ramond (R) sectors. We explore the geometric features of the system by analyzing the quantum (Berry) curvature derived from the Fubini Study metric, which is intimately connected to the quantum Fisher information. This approach uncovers a rich interplay between boundary conditions and quantum geometry. In the gamma h parameter space, we identify distinct sign changing arcs of the curvature, confined to some region. These arcs mark transitions between the NS and R sectors, indicating fundamental changes in the structure of the fermionic ground state. Remarkably, the number of such transition lines increases with system size, hinting at an emergent continuum of topological boundary effects in the thermodynamic limit. Our findings highlight a novel mechanism where boundary conditions shape quantum geometric properties, offering new insights into finite size topology and the structure of low dimensional quantum systems.