The gap of Fredkin quantum spin chain is polynomially small

TL;DR

This paper establishes that the spectral gap of the Fredkin quantum spin chain decreases polynomially with system size, providing bounds on the energy of excited states and analyzing ground state energies through mappings to effective Hamiltonians.

Contribution

It proves the polynomial decay of the spectral gap in the Fredkin quantum spin chain and derives bounds on excited state energies using probabilistic and analytical methods.

Findings

Spectral gap of Fredkin chain is Θ(n^{-c}) with c ≥ 2.

Lower bound of O(n^{-15/2}) on first excited state energy.

Upper bound of O(n^{-2}) on excited state energy.

Abstract

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. The proof uses the comparison theorem of Diaconis and Saloff-Coste and our previous results. Let $2n$ be the number of spins. We prove that the gap of the Fredkin quantum spin chain Hamiltonian [6, 20], is $\Theta(n^{-c})$ with $c\ge2$. 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.