Scars and Fragmented Hamiltonians from Automaton circuits

Classical cellular automata can be recast as Floquet unitary circuits, and have been recently studied for hosting a variety of interesting dynamical properties. These circuits naturally exhibit Hilbert space fragmentation, with subspaces comprising of orbits of computational basis states that are cycled through with successive applications of the unitary circuit. It is natural to ask if Hamiltonians exhibiting a similar fragmented structure can be derived from such automaton circuits. In this work, we investigate this question in detail and show how certain restrictions on the local unitaries composing these circuits allow for a convergent BCH expansion which further allows us to define effective local Hamiltonians that exhibit exact and approximate scars, as well as weak and strong fragmentation.