Irreversible work versus fidelity susceptibility for infinitesimal quenches

We compare the irreversible work produced in an infinitesimal sudden quench of a quantum system at zero temperature with its ground state fidelity susceptibility, giving an explicit relation between the two quantities. We find that the former is proportional to the latter but for an extra term appearing in the irreversible work which includes also contributions from the excited states. We calculate explicitly the two quantities in the case of the quantum Ising chain, showing that at criticality they exhibit different scaling behavior. The irreversible work, rescaled by square of the quench's amplitude, exhibits a divergence slower than the fidelity susceptibility one. As a consequence, the two quantities obey also different finite-size scaling relations.