[Paper Review] Translation invariant pure states and its split property

Anilesh Mohari, Cit Campus|arXiv (Cornell University)|Apr 14, 2009

Advanced Operator Algebra Research30 references4 citations

TL;DR

This paper establishes that a translation invariant pure state in a quantum spin chain is uniquely determined by a Markov map on the support projection of an associated Cuntz state. It proves that Kolmogorov’s property of the Markov map is both necessary and sufficient for purity, and uses duality from non-commutative probability to derive an alternative characterization, ultimately proving Haag duality and confirming T. Matsui’s conjecture on the split property under exponential decay of special correlation functions.

ABSTRACT

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.

Motivation & Objective

• To characterize translation invariant pure states in quantum spin chains using Markov maps on Cuntz state support projections.
• To identify necessary and sufficient conditions for purity of such states using Kolmogorov’s property of the Markov map.
• To establish a duality-based alternative condition for purity derived from non-commutative probability theory.
• To prove the Haag duality property for any translation invariant lattice-symmetric pure state.
• To confirm T. Matsui’s conjecture on the split property under exponential decay of specific correlation functions.

Proposed method

• Utilizes a Markov map defined on the support projection of a Cuntz state to characterize translation invariant states in quantum spin chains.
• Applies Kolmogorov’s property of the Markov map as a criterion for purity, establishing it as both necessary and sufficient.
• Employs a duality argument rooted in non-commutative probability theory to derive an alternative necessary and sufficient condition for purity.
• Leverages the Mackey system of imprimitivity for the integer group, arising naturally from Kolmogorov’s property.
• Analyzes correlation functions to determine the split property, showing exponential decay implies splitness.
• Combines structural analysis of Markov maps with symmetry and duality to prove Haag duality for lattice-symmetric pure states.

Experimental results

Research questions

• RQ1 What condition on the Markov map ensures purity of a translation invariant state in a quantum spin chain?
• RQ2 How does Kolmogorov’s property of the Markov map relate to the purity of such states?
• RQ3 Can a duality argument from non-commutative probability yield an alternative characterization of purity?
• RQ4 Does every translation invariant lattice-symmetric pure state satisfy the Haag duality property?
• RQ5 Under what conditions does exponential decay of a special correlation function imply the split property?

Key findings

• Kolmogorov’s property of the Markov map is both necessary and sufficient for a translation invariant state to be pure.
• The duality argument from non-commutative probability provides an elegant alternative necessary and sufficient condition for purity.
• All translation invariant lattice-symmetric pure states satisfy the Haag duality property.
• The split property holds for such real states if and only if a specific correlation function decays exponentially.
• The paper confirms T. Matsui’s conjecture on the split property for translation invariant real lattice-symmetric pure states.
• The Mackey system of imprimitivity for the integers emerges naturally from the Kolmogorov property of the Markov map.