Finite-size Gap, Magnetization, and Entanglement of Deformed Fredkin Spin Chain

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.