The gap of Fredkin quantum spin chain is polynomially small

We prove a new result on the spectral gap and mixing time of a Markov chain with Glauber dynamics on the space of Dyck paths (i.e., Catalan paths) and their generalization, which we call colored Dyck paths.

Let $2n$ be the number of spins. We prove that the gap of the Fredkin quantum spin chain Hamiltonian, is $\Theta (n^{-c})$ with $c \geq 2$. Our results on the spectral gap of the Markov chain are used to prove a lower bound of $O(n^{-15/2})$ on the energy of first excited state above the ground state of the Fredkin quantum spin chain.We prove an upper bound of $O(n^{-2})$ using the universality of Brownian motion and convergence of Dyck random walks to Brownian excursions. Lastly, the ‘unbalanced’ ground state energies are proved to be polynomially small in $n$ by mapping the Hamiltonian to an effective hopping Hamiltonian with next nearest neighbor interactions and analytically solving its ground state.

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Received 25 November 2016

Published 9 August 2018