Hopf algebras and solvable unitary circuits

Abstract: Exactly solvable models in quantum many body dynamics provide valuable insights into many interesting physical phenomena, and serve as platforms to rigorously investigate fundamental theoretical questions. Nevertheless, they are extremely rare and existing solvable models and solution techniques have serious limitations. In this paper we introduce a new family of exactly solvable unitary circuits which model quantum many body dynamics in discrete space and time. Unlike many previous solvable models, one can exactly compute the full quantum dynamics initialized from any matrix product state in this new family of models. The time evolution of local observables and correlations, the linear growth of Renyi entanglement entropy, spatiotemporal correlations, and out-of-time-order correlations are all exactly computable. A key property of these models enabling the exact solution is that any time evolved local operator is an exact matrix product operator with finite bond dimension, even at arbitrarily long time, which we prove using the underlying (weak) Hopf algebra structure along with tensor network techniques. We lay down the general framework for the construction and solution of this family of models, and give several explicit examples. In particular, we study in detail a model constructed out of a weak Hopf algebra that is very close to a floquet version of the PXP model, and the exact results we obtain may shed light on the phenomenon of quantum many body scars, and more generally, floquet quantum dynamics in constrained systems.