Translation invariant pure state and its split property

Details Translation invariant pure state and its split property Translation invariant pure state and its split property

2009-04-14

arxiv.org/abs/0904.2104v4

Mathematics

Operator Algebras

The proof given in the last version for Haag duality needs some additional work and added one more section. Last section adds

Scientific paper

Main motivation of this paper is to study translation invariant pure states on a quantum spin chain with additional symmetry arises in Hamiltonian dynamics in condense matter physics. A translation invariant state in quantum spin chain is determined uniquely up to isomorphism by a Markov map on the support projection of an associated Cuntz's state. We prove that a weaker Kolmogorov's property of the Markov map is a necessary and sufficient condition for such a state to be pure. A duality argument originated from non-commutative probability theory is employed to prove an elegant alternative necessary and sufficient condition for pureness. Kolmogorov's property naturally give rise to a Mackey's system of imprimitivity for the group of integers. However Kolmogorov's property is not necessary for purity. Nevertheless Mackey's system of imprimitivity played a vital role to prove Haag duality for any translation invariant pure state and finds it's intimate connection with criteria on support projection of Cuntz's state. This criteria of purity made it possible to prove that a real refection positive state is split if spatial correlation function decays exponentially. The last statement supports Taku Matsui's conjecture. Studying symmetry we also prove that ground state for a half-odd integer Heisenberg's isospin anti-ferromagnets ($XXX$-model) is not unique.

Affiliated with

Also associated with

No associations

Rating

Translation invariant pure state and its split property does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Translation invariant pure state and its split property, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Translation invariant pure state and its split property will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-98747