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We derive an exact closed-form expression for fidelity susceptibility ofeven- and odd-sized quantum Ising chains in the transverse field. To this aim,we diagonalize the Ising Hamiltonian and study the gap between its positive andnegative parity subspaces. We derive an exact closed-form expression for thegap and use it to identify the parity of the ground state. We point outmisunderstanding in some of the former studies of fidelity susceptibility anddiscuss its consequences. Last but not least, we rigorously analyze theproperties of the gap. For example, we derive analytical expressions showingits exponential dependence on the ratio between the system size and thecorrelation length.

Exact results for fidelity susceptibility of the quantum Ising model: The interplay between parity, system size, and magnetic field