Order, disorder and chaos in $\mathrm{2D}$ lattice of coupled Sinai billiards

Transport properties of a new kind of ballistic electron billiards — two-dimensional ($\mathrm{2D}$) lattice of Sinai billiards coupled through quantum point contacts — are experimentally studied. This lattice is peculiar by simultaneous existence of the effects inherent to single Sinai billiards or quantum dots, and the features reflecting lattice properties of system. Magnetotransport measurements give very pronounced commensurability peak even if the conductivity of the lattice $G\ll e^2/h$. Consequently it preserves the properties of ballistic regular structure at these conductivity states. On the other hand, the gate voltage dependencies of $G$ show that the system behaves as percolation one. In weak magnetic fields negative magnetoresistance (NMR) is observed. It is described by theory of chaotic weak localization developed for case of single ballistic cavity. This NMR increases in going from $G>e^2/h$ to $G\ll e^2/h$. Thus, $\mathrm{2D}$ lattice of coupled Sinai billiards is a unique system where coexistence of order, disorder and chaos are clearly demonstrated.