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

Abstract

We establish dimerization in $O(n)$-invariant quantum spin chains with large enough $n$, in a much larger region of the phase diagram than previous results. This entails identifying two distinct infinite volume ground states which are gapped, are translations by one unit of each other, and which both have exponentially decaying correlations. We show these states are the limits of finite-volume, finite temperature states as the graph tends to $\mathbb{Z}$ and temperature tends to zero in any order. 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.

34 pages, 11 figures