  The most general and versatile defining feature of quantum chaotic systems is
that they possess an energy spectrum with correlations universally described by
random matrix theory (RMT). This feature can be exhibited by systems with a
well defined classical limit as well as by systems with no classical
correspondence, such as locally interacting spins or fermions. Despite great
phenomenological success, a general mechanism explaining the emergence of RMT
without reference to semiclassical concepts is still missing. Here we provide
the example of a quantum many-body system with no semiclassical limit (no large
parameter) where the emergence of RMT spectral correlations is proven exactly.
Specifically, we consider a periodically driven Ising model and write the
Fourier transform of spectral density's two-point function, the spectral form
factor, in terms of a partition function of a two-dimensional classical Ising
model featuring a space-time duality. We show that the self-dual cases provide
a minimal model of many-body quantum chaos, where the spectral form factor is
demonstrated to match RMT for all values of the integer time variable $t$ in
the thermodynamic limit. In particular, we rigorously prove RMT form factor for
odd $t$, while we formulate a precise conjecture for even $t$. The results
imply ergodicity for any finite amount of disorder in the longitudinal field,
rigorously excluding the possibility of many-body localization. Our method
provides a novel route for obtaining exact nonperturbative results in
non-integrable systems.
