arXiv:1310.1886

Translation invariant pure state on $\clb=\otimes_{\IZ}M_d(\IC)$ and its split property

Anilesh Mohari

Published Oct 7, 2013• 0 citations

Abstract

We prove that a real lattice symmetric reflection positive translation invariant pure state of $\clb=\otimes_{\IZ}M_d(\IC)$ is a split state if its two points spatial correlations functions decay exponentially.

DOI: 10.48550/arxiv.1310.1886

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arXiv:1310.1886

Translation invariant pure state on $\clb=\otimes_{\IZ}M_d(\IC)$ and its split property

Anilesh Mohari

Published Oct 7, 2013• 0 citations

arXiv HTML PDF

Abstract

We prove that a real lattice symmetric reflection positive translation invariant pure state of

\clb=\otimes_{\IZ}M_d(\IC)

is a split state if its two points spatial correlations functions decay exponentially.

DOI: 10.48550/arxiv.1310.1886

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Translation invariant pure state and its split property

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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 statesWe 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

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Dissipative dynamics for quantum spin systems on a lattice

Adam W Majewski, Robert Olkiewicz, Bogusław Zegarliński

Journal of Physics A Mathematical and GeneralFeb 27, 199824 citations

We prove that a real lattice symmetric reflection positive translation invariant pure state of

\clb=\otimes_{\IZ}M_d(\IC)

is a split state if its two points spatial correlations functions decay exponentially.

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

Spontaneous

SU(2)

symmetry breaking in ground state of quantum spin chain

d

is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnetic model with

{1 \over 2}

-odd integer spin degrees of freedom admits spontaneous symmetry breaking at its ground states.

SU_2(\mathbb{C})

symmetry in quantum spin chain ground states and Haldane's conjecture

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 $SU_2(\IC)$.