Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases

TL;DR

This paper analyzes the critical states of integrable antiferromagnetic spin chains, revealing their classification via SO(3) Wess-Zumino-Witten models and their correspondence to two-dimensional topological phases with distinct excitations.

Contribution

It introduces a classification of quantum critical states of integer spin chains using SO(3) Wess-Zumino-Witten models and links them to two-dimensional topological phases.

Findings

Integer spin-S chains are characterized by SO(3) level-S Wess-Zumino-Witten models.

Even-S chains have gapless bosonic excitations; odd-S chains have both bosonic and fermionic excitations.

Critical states correspond to boundary states of two distinct 2D topological phases.

Abstract

We examine the effective field theory of the Bethe ansatz integrable Heisenberg antiferromagnetic spin chains. It shows that the quantum critical theories for the integer spin-S chains should be characterized by the SO(3)level-S Wess-Zumino-Witten model, and classified by the third cohomology group $H^{3}(SO(3),Z)=Z$. Depending on the parity of spin S, this integer classification is further divided into two distinct universality classes, which are associated with two completely different conformal field theories: the even-S chains have gapless bosonic excitations and the odd-S chains have both bosonic and fermionic excitations. We further show that these two classes of critical states correspond to the boundary states of two distinct topological phases in two dimension, which can be described by two-dimensional doubled SO(3) topological Chern-Simons theory and topological spin theory, respectively.