Complexity and information geometry in spin chains

Geodesic distances on parameter manifolds of quantum systems lead to the notion of the Fubini-Study complexity, that quantifies the difficulty in creating a target quantum state starting from a reference one. Here, we obtain analytic expressions for the equilibrium ground state complexity in a class of spin systems, in the thermodynamic limit. Starting from the fact that the infinitesimal Nielsen complexity in terms of the Bogoliubov angle is the line element in the parameter space, the lengths of some special geodesics in three exactly solvable spin systems are obtained. Our examples include the transverse XY spin chain and its natural extensions, the quantum compass model with and without an external magnetic field. In all these, the Fubini-Study complexity is contrasted with the Nielsen complexity, which is related to the finite difference of Bogoliubov angles between an initial and a target state, for the quadratic Hamiltonians that we consider. The first derivative of the complexity diverges at quantum phase transitions in all cases.