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

TL;DR

This paper proves that the energy gap of a weighted Motzkin quantum spin chain model decreases exponentially with the square of the system size, indicating an extremely small gap in the thermodynamic limit.

Contribution

It establishes an exponential upper bound on the energy gap of the weighted Motzkin quantum spin chain, extending previous polynomial gap results to a model with area-law violation.

Findings

The energy gap is exponentially small in the square of the system size.

The model's ground state exhibits superposition of weighted Motzkin walks.

The gap upper bound is proportional to $8ns t^{-n^2/3}$ for $t>1$.

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 $c\ge2$. 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 $t>1$.