In this work, we analyzed the spectral form factor (SFF) for two classes of quantum Ising models. For an ensemble of Hamiltonians, the SFF is defined by the Fourier transform of the eigenvalue density-density correlation function. In random systems, this quantity can be a useful probe for the emergence of quantum chaos - perhaps the most famous SFF is that of the SYK model, where connections can be made between the behavior of the SFF and certain observables in quantum gravity.

We first studied the SFF of the 1D transverse field Ising model at criticality as well as for small deviations away from criticality, for which we are able to obtain analytic results that agree well with numerics. At criticality, we find that the SFF is periodic in time. This is a generic feature of any rational CFT; the periodicity arises from a regularity in the energy spectrum implied by the existence of a finite number of primary operators. Of course, rational CFTs can still host non-trivial SFFs if the periodicity is large enough.

We also study the SFF of a disordered 1D Ising model at criticality, which is known to flow to a infinite-randomness fixed point. We find that the SFF behavior corresponds to Poissonian statistics. In some sense, this is always guaranteed so long as our model admits a free fermion representation, which ours does through a Jordan-Wigner transformation. However, it is non-trivial to reconcile this with the behavior of the infinite-randomness fixed point, which is that the eigenstates become extended at low energies and hence should exhibit eigenvalue repulsion. We argue that a careful analysis of the finite-size scaling of the energy gap prevents this delocalization from manifesting itself as eigenvalue repulsion in the spectrum.