The gap of the area-weighted Motzkin spin chain is exponentially small

We prove that the energy gap of the model proposed by Zhang, Ahmadain, and Klich [1] is exponentially small in the square of the system size. In [2] a class of exactly solvable quantum spin chain models was proposed that have integer spins ($s$), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all $s-$colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system's size ($\sim\sqrt{n}$) for $s>1$. For $s=1$, the violation is logarithmic [3]. Moreover in [2] it was proved that the gap vanishes polynomially and is $O(n^{-c})$ with $c\ge2$. Recently, a deformation of [2], which we call "weighted Motzkin quantum spin chain" was proposed [1]. This model has a unique ground state that is a superposition of the $s-$colored Motzkin walks weighted by $t^{\text{area\{Motzkin walk\}}}$ with $t>1$. The most surprising feature of this model is that it violates the area law by a factor of $n$. Here we prove that the gap of this model is upper bounded by $8ns\text{ }t^{-n^{2}/3}$ for $t>1$.