Entropy, gap and a multi-parameter deformation of the Fredkin spin chain

We introduce a multi-parameter deformation of the Fredkin spin $1/2$ chain whose ground state is a weighted superposition of Dyck paths, depending on a set of parameters $t_i$ along the chain. The parameters are introduced in such a way to maintain the system frustration-free while allowing to explore a range of possible phases. In the case where the parameters are uniform, and a color degree of freedom is added we establish a phase diagram with a transition between an area law and a volume low. The volume entropy obtained for half a chain is $n \log s$ where $n$ is the half-chain length and $s$ is the number of colors. Next, we prove an upper bound on the spectral gap of the $t>1, s>1$ phase, scaling as $\Delta=O((4s)^nt^{-n^2/2})$, similar to a recent a result about the deformed Motzkin model, albeit derived in a different way. Finally, using an additional variational argument we prove an exponential lower bound on the gap of the model for $t>1, s=1$, which provides an example of a system with bounded entanglement entropy and a vanishing spectral gap.