  We study gapless quantum spin chains with spin 1/2 and 1: the Fredkin and
Motzkin models. Their entangled groundstates are known exactly but not their
excitation spectra. We first express the groundstates in the continuum which
allows for the calculation of spin and entanglement properties in a unified
fashion. Doing so, we uncover an emergent conformal-type symmetry, thus
consolidating the connection to a widely studied family of Lifshitz quantum
critical points in 2d. We then obtain the low lying excited states via
large-scale DMRG simulations and find that the dynamical exponent is z = 3.2 in
both cases. Other excited states show a different z, indicating that these
models have multiple dynamics. Moreover, we modify the spin-1/2 model by adding
a ferromagnetic Heisenberg term, which changes the entire spectrum. We track
the resulting non-trivial evolution of the dynamical exponents using DMRG.
Finally, we exploit an exact map from the quantum Hamiltonian to the
non-equilibrium dynamics of a classical spin chain to shed light on the quantum
dynamics.
