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arXiv:0904.2104

Translation invariant pure state and its split property

Anilesh Mohari

Published Apr 14, 2009• 0 citations
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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

DOI
10.48550/arxiv.0904.2104

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Spontaneous SU(2)SU(2)SU(2) symmetry breaking in ground state of quantum spin chain
In this paper we prove that there exists no translation invariant pure state of $\clb=\otimes_{\mathbb{Z}}M_d(\IC)$ that is real, lattice symmetric, reflection positive with a twist and SU(2) invariant when ddd is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnetic model with 12{1 \over 2}21​-odd integer spin degrees of freedom admits spontaneous symmetry breaking at its ground states.
We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2$, where $/!M_d(\IC)$ is the set of d×dd \times dd×d dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$.
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On the uniqueness of translation-invariant ground states of the S=1S=1S=1 Heisenberg and JJJ--DDD antiferromagnetic chains
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