Geometric Aspects of Quantum Spin States

TL;DR

This paper explores the geometric properties of quantum spin states in spin chains using a functional integral approach, revealing insights into phenomena like dimerization, spectral gaps, and correlation decay, and relating these to random cluster models.

Contribution

Introduces a geometric integral representation for quantum spin chains that clarifies the nature of ground states and phase phenomena, connecting spin systems to random cluster models.

Findings

Absence of Neel order in certain regimes

Conditions for spectral gap existence

Dichotomy between slow correlation decay and symmetry breaking

Abstract

A number of interesting features of the ground states of quantum spin chains are analized with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU($2S+1$)-invariant quantum spin-$S$ chains with the interaction $-P^{(0)}$, where $P^{(0)}$ is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of N\'eel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the $-P^{(0)}$ spin-$S$ systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.