$SU_2(\mathbb{C})$ symmetry in quantum spin chain ground states and Haldane's conjecture

TL;DR

This paper proves that certain symmetric quantum spin chain ground states with $SU_2( ext{C})$ symmetry exhibit exponential decay of correlations, confirming aspects of Haldane's conjecture for odd integer spins.

Contribution

It establishes that $SU_2( ext{C})$-invariant pure states with specific symmetries are finitely correlated and have exponentially decaying correlations, and characterizes the ground state of the Heisenberg anti-ferromagnetic model.

Findings

Ground states are finitely correlated with exponential decay of correlations.

Unique low temperature ground state exists for odd integer spins.

Ground state characterized by Clebsch-Gordon inter-twinning isometry.

Abstract

In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever $d$ is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.