| Gaia: | We study spectral statistics in spatially extended chaotic quantum many-body systems, using simple lattice Floquet models without time-reversal symmetry. Computing the spectral form factor KðtÞ analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, tTh. We obtain a striking dependence of tTh on the spatial dimension d and size of the system. For d > 1, tTh is finite in the thermodynamic limit and set by the intersite coupling strength. By contrast, in one dimension tTh diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form. Lastly, our Floquet model exhibits a many-body localization transition, and we discuss the behavior of the spectral form factor in the localized phase.
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