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

Abstract

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 $c\ge 2$. 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 $8ns{t}^{-n^2/3}$ for $t>1$.

15 pages, 6 figures. arXiv admin note: text overlap with arXiv:1609.09160

References in corpus (3)

• Quantum phase transition from bounded to extensive entanglement entropy in a frustration-free spin chain
• Violation of Cluster Decomposition and Absence of Light-Cones in Local Integer and Half-Integer Spin Chains
• Entanglement and correlation functions of the quantum Motzkin spin-chain

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