Complexity and information geometry in spin chains

We study Nielsen complexity and Fubini-Study complexity for a class of exactly solvable one dimensional spin systems. Our examples include the transverse XY spin chain and its natural extensions, the quantum compass model with and without an external magnetic field. We obtain the scaling behaviour of both complexities near quantum phase transitions in the thermodynamic limit, as a function of the system parameters. We provide analytical proofs of these, in an information geometric framework, which verify our numerical analysis. The scaling of the Nielsen complexity with the system size is also established, close to criticality. We also obtain analytic expressions for the Fubini-Study complexity in some special cases for all the models, while a numerical analysis in more generic situations is carried out. Our study clearly demonstrates the differences in the two notions of complexity in quasi-free fermionic systems.