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

Abstract

We introduce a multi-parameter deformation of the Fredkin spin chain whose ground state is a weighted superposition of Dyck paths, depending on a set of parameters ti along the chain. The parameters are introduced in such a way to maintain a frustration-free system while allowing the exploration of 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 law. The volume entropy obtained for half a chain is 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 phase, scaling as , similar to a recent a result about the deformed Motzkin model (Levine and Movassagh 2017 J. Phys. A: Math. Theor. 50 255302), 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 , which provides an example of a system with bounded entanglement entropy and a vanishing spectral gap.