Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_1$-$J_2$ Chain

TL;DR

This study reveals a novel gapped topological order coexisting with valence-bond formation in a frustrated ferromagnetic $J_1$-$J_2$ chain, demonstrating quantum fluctuations can induce topological phases in 1D magnets.

Contribution

It provides the first example of a frustrated spin chain where quantum fluctuations induce a gapped topological order alongside valence-bond solid formation.

Findings

Existence of a finite spin gap for $J_2/|J_1|>1/4$

Ground state is a valence bond solid with third-neighbor singlet dimers

The VBS exhibits Affleck-Kennedy-Lieb-Tasaki-type topological order

Abstract

Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin-$\frac{1}{2}$ ferromagnetic ($J_1$) Heisenberg chain with next-nearest-neighbor antiferromagnetic ($J_2$) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at $J_2/|J_1|>1/4$ and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.