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

Zhao Zhang, Israel Klich

Published 2017-02-12Version 1

We introduce a general 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 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 when 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 $n \log (D/2)$ where $D$ is the dimension of the local spin Hilbert space, and $n$ is the half-chain length. We also prove an upper bound on the spectral gap, scaling as $\Delta=O((2s)^nt^{-n^2/2})$, similar to a recent a result about the deformed Motzkin model, albeit derived in a different way.