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We introduce a class of quantum circuits that are unitary along three distinct “arrows of time.” These dynamics share some of the analytical tractability of “dual-unitary” circuits, while exhibiting distinctive and richer phenomenology. We find that two-point correlations in these dynamics are strictly confined to three directions in (1+1)-dimensional spacetime—the two light cone edges, δx=±vδt, and the static worldline δx=0. Along these directions, correlation functions are obtained exactly in terms of quantum channels built from the individual gates that make up the circuit. We prove that, for a class of initial states, entanglement grows at the maximum allowed speed up to an entropy density of at least one half of the thermal value, at which point it becomes model-dependent. Finally, we extend our circuit construction to 2+1 dimensions, where two-point correlation functions are confined to the one-dimensional edges of a tetrahedral light cone—a subdimensional propagation of information reminiscent of “fractonic” physics.

We present a general symmetry-based framework for obtaining many-body Hamiltonians with scarred eigenstates that do not obey the eigenstate thermalization hypothesis. Our models are derived from parent Hamiltonians with a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as degenerate multiplets that transform as irreducible representations of the symmetry (“tunnels”). We show that large classes of perturbations break the symmetry, but in a manner that preserves a particular low-entanglement multiplet of states, thereby giving generic, thermal spectra with a shadow of the broken symmetry in the form of scars. The generators of the Lie algebra furnish operators with “spectrum-generating algebras” that can be used to lift the degeneracy of the scar states and promote them to equally spaced “towers.” Our framework applies to several known models with scars, but we also introduce models with scars that transform as irreducible representations of symmetries such as SU(3) and q-deformed SU(2), significantly generalizing the types of systems known to harbor this phenomenon. Additionally, we present examples of generalized Affleck-Kennedy-Lieb-Tasaki (AKLT) models with scar states that do not transform in an irreducible representation of the relevant symmetry. These are derived from parent Hamiltonians with enhanced symmetries, and bring AKLT-type models into our framework.