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

TL;DR

This paper proves dimerization in certain $O(n)$-invariant quantum spin chains for large $n$, showing two distinct ground states with exponential decay of correlations, using probabilistic loop representations.

Contribution

It introduces a novel probabilistic approach to establish dimerization in quantum spin chains with large $n$, extending methods from loop models.

Findings

Existence of two distinct ground states with translation symmetry breaking

Ground states exhibit exponential decay of correlations

Dimerization occurs in a large part of the phase diagram for large $n$

Abstract

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.