  We investigate ground- and excited-state properties of the deformed Fredkin
spin chain proposed by Salberger, Zhang, Klich, Korepin, and the authors. This
model is a one-parameter deformation of the Fredkin spin chain, whose
Hamiltonian is $3$-local and translationally invariant in the bulk. The model
is frustration-free and its unique ground state can be expressed as a weighted
superposition of colored Dyck paths. We focus on the case where the deformation
parameter $t>1$. By using a variational method, we prove that the finite-size
gap decays at least exponentially with increasing the system size. We prove
that the magnetization in the ground state is along the $z$-direction, namely
$\langle s^x \rangle =\langle s^y \rangle=0$, and show that the $z$-component
$\langle s^z \rangle$ exhibits a domain-wall structure. We then study the
entanglement properties of the chain. In particular, we derive upper and lower
bounds for the von Neumann and R\'enyi entropies, and entanglement spectrum for
any bipartition of the chain.
