Translation invariant pure state and its split property

TL;DR

This paper establishes the Haag duality for translation invariant pure states on infinite tensor products of matrix algebras, characterizes when such states are pure, and explores their symmetry properties and implications for quantum spin models.

Contribution

It proves Haag duality for translation invariant pure states, provides a criterion for purity with symmetry, and links decay of correlations to the split property, with applications to spin models.

Findings

Haag duality holds for translation invariant pure states.

Exponential decay of correlations implies the split property.

Certain symmetric pure states cannot exist for even-dimensional spins.

Abstract

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where $M_d(C)$ is the set of $d \times d$ dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and $su(2)$ invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states