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Journal of Mathematical Physics

Translation invariant pure state on B=⊗j∈ZMd(j)(C) and its split property

Anilesh Mohari

Published Jun 1, 2015• 5 citations
Publisher

Abstract

In this paper, we prove that a real lattice symmetric reflection positive translation-invariant pure state of B=⊗j∈ZMd(j)(C) admits split property, if and only if its two-point spatial correlation functions decay exponentially. We use amalgamated representation of Cuntz algebras to represent two-point spatial correlation functions on an augmented Hilbert space. The underling symmetries and reflection positive property of the pure state make it possible to investigate its split and decaying two-point correlation functions properties as spectral properties of a contractive self-adjoint operator on the augmented Hilbert space. Haag duality property of the pure state is crucially used in the analysis.

DOI
10.1063/1.4922013

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In this paper, we prove that a real lattice symmetric reflection positive translation-invariant pure state of B=⊗j∈ZMd(j)(C) admits split property, if and only if its two-point spatial correlation functions decay exponentially. We use amalgamated representation of Cuntz algebras to represent two-point spatial correlation functions on an augmented Hilbert space. The underling symmetries and reflection positive property of the pure state make it possible to investigate its split and decaying two-point correlation functions properties as spectral properties of a contractive self-adjoint operator on the augmented Hilbert space. Haag duality property of the pure state is crucially used in the analysis.
We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where Md(C)M_d(C)Md​(C) is the set of d×dd \times dd×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)su(2)su(2) invariant when ddd 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
A translation invariant state in quantum spin chain is determined uniquely upto isomorphism by a Markov map on the support projection of an associated Cuntz’s state. We prove that Kolmogorov’s property of the Markov map is a necessary and sufficient condition for such a state to be pure. Kolmogorov’s property naturally give rise to a Mackey’s system of imprimitivity for the group of integers. A duality argument originated from non-commutative probability theory is employed to prove an elegant alternative necessary and sufficient condition for pureness. Main result of this theory made it possible to prove Haag duality property of any translation invariant lattice symmetric pure state. Further such a real state is split if special correlation function decays exponentially. The last statement proves T Matsui’s conjecture on split property for a translation invariant real lattice symmetric pure state.
In this exposition we investigate further the general methodology proposed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain A = ⊗ ZMd, where Md is the matrix of d × d complex matrices. We introduce a notion of quantum detailed balance [Mo1] for a translation invariant state on A and prove that such a pure state is uniformly mixing [BR,Ma2] if and only if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, translation and SU(2) gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. As a consequence of these results we conclude that if the ground states for an integer spin SU(2) invariant (2s + 1 = d) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for integer
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 $\clb$.
SU2(C)SU_2(\mathbb{C})SU2​(C) symmetry in quantum spin chain ground states and Haldane's conjecture
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 ddd 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 $SU_2(\IC)$.