Abstract
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 K(t) analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, t_ Th. We obtain a striking dependence of t_ Th on the spatial dimension d and size of the system. For d>1, t_ Th is finite in the thermodynamic limit and set by the inter-site coupling strength. By contrast, in one dimension t_ Th diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form.