Dimerization in $O(n)$-invariant quantum spin chains

We establish dimerization in $O(n)$-invariant quantum spin chains with big enough $n$, in a large part of the phase diagram where this result is expected. This includes identifying two distinct ground states which are translations of one unit of eachother, and which both have exponentially decaying correlations. Our method relies on a probabilistic representation of the quantum system in terms of random loops, and an adaptation of a method developed for loop $O(n)$ models on the hexagonal lattice by Duminil-Copin, Peled, Samotij and Spinka.