Hilbert-Space Fragmentation of an Embedded Fredkin Spin Chain

As a result of emerging constraints, we show that the Hilbert space fragments into many disconnected sectors in an embedded Fredkin model. Closely related to the frustration-free Fredkin model with spin moves given by the control-SWAP operation. The model we study embeds the zero-energy ground into the middle of the spectrum resulting in a dramatic effect on the Hilbert space structure. This embedding induces a spectral-reflection symmetry that accompanies the U(1) charge and domain wall conservation. The rich structure is developed through an effective model which is a constrained 1D Fermi-Hubbard model at infinite-U. Using this formalism we show that certain emergent subsectors display non-trivial dynamics including chaotic with many-body scarring, as well as, Bethe-ansatz solvable. The structure of the initial state is shown to result in a wide class of dynamics.