  The Fredkin model describes a spin-half chain segment subject to three-body,
correlated-exchange interactions and twisted boundary conditions. The model is
frustration-free, and its ground state wave function is known exactly. Its
low-energy physics is that of a strong xy ferromagnet with gapless excitations
and an unusually large dynamical exponent. We study a generalized spin chain
model that includes the Fredkin model as a special tuning point and otherwise
interpolates between the conventional ferromagnetic and antiferromagnetic
quantum Heisenberg models. We solve for the low-lying states, using exact
diagonalization and density-matrix renormalization group calculations, in order
to track the properties of the system as it is tuned away from the Fredkin
point; we also present exact analytical results that hold right at the Fredkin
point. We identify a zero-temperature phase diagram with multiple transitions
and unexpected ordered phases. The Fredkin ground state turns out to be
particularly brittle, unstable to even infinitesimal antiferromagnetic
frustration. We remark on the existence of an "anti-Fredkin" point at which all
the contributing spin configurations have a spin structure exactly opposite to
those in the Fredkin ground state.
