Criticality without frustration for quantum spin-1 chains

TL;DR

This paper presents the first example of a frustration-free, translation-invariant spin-1 chain with a highly entangled ground state exhibiting critical-like behavior, including logarithmic entanglement scaling.

Contribution

It introduces a novel frustration-free spin-1 chain with a highly entangled ground state and analyzes its critical properties and energy gap.

Findings

Ground state is a superposition of balanced parentheses strings.

Entanglement entropy scales as log(n)/2 + O(1).

Energy gap is polynomial in 1/n.

Abstract

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right parentheses separated by empty spaces. Entanglement entropy of one half of the chain scales as log(n)/2 + O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.