Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:math-ph/0703013 · Reviews in Mathematical Physics

ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

Michael Keyl, Taku Matsui, Dirk-M. Schlingemann, Reinhard F. Werner

Published Jul 1, 2008• 7 citations
arXivHTMLPDF

Abstract

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.
DOI
10.1142/s0129055x08003377

Related papers

ENTANGLEMENT, HAAG-DUALITY AND TYPE PROPERTIES OF INFINITE QUANTUM SPIN CHAINS

Michael Keyl, Taku Matsui, Dirk-M. Schlingemann +1 more

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

arXiv:math-ph/0604071•Reviews in Mathematical Physics•Oct 1, 2006•36 citations

An entropic characterization of Haag duality

Ruizhi Liu, Lauritz van Luijk

Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.

quant-pharXiv:2609.13550•Sep 15, 2026•0 citations

Haag duality for Kitaev’s quantum double model for abelian groups

Leander Fiedler, Pieter Naaijkens

We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

arXiv:1406.1084•Reviews in Mathematical Physics•Oct 1, 2015•34 citations

Local topological order, Haag duality, and reflection positivity

Pieter Naaijkens, David Penneys, Daniel Wallick

In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for abstract quantum spin systems which allow one to access topological order via a boundary algebra construction. Using the LTO axioms, we produced a canonical pure state on the quasi-local algebra, which gives a net of von Neumann algebras associated to a poset of cones in $\mathbb{R}^n$. In this article, motivated by [arXiv:2509.23734], we introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory. We prove this axiom is satisfied for all known topologically ordered commuting projector models. We thus get an independent proof of Haag duality for the Levin-Wen string net models originally proved in [arXiv:2509.23734]. We also give a reflection positivity axiom for LTOs, connecting to the recent article [arXiv:2510.20662]. We again prove this axiom is satisfied for all known topologically ordered commuting projector models about some $\mathbb{Z}/2$-reflection symmetry.

math-phCross-listedarXiv:2605.10693•May 12, 2026•0 citations

Translation invariant pure state on $\otimes_{\IZ}/!M_d(\IC)$ and Haag duality

Anilesh Mohari

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 $d \times d$ 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 $\clb$.

arXiv:1309.7607•Sep 29, 2013•0 citations

Translation-Invariant Pure States with Exotic Half-Chain Factors

Ruizhi Liu, Jinmin Yi

We construct translation-invariant pure quasifree fermionic states on the one-dimensional lattice whose half-chain von Neumann algebras are hyperfinite factors of type $\mathrm{II}_\infty$ or $\mathrm{III}_λ$ for any $0<λ\leq 1$. Via the Jordan-Wigner-Araki transformation, these yield translation-invariant pure spin-chain states with the same half-chain factors, demonstrating that such factors need not be of type I or III. Each state is the unique ground state of a gapless, translation-invariant parent Hamiltonian with superpolynomially decaying interactions. For any fixed factor type, the interval entanglement entropy can exhibit widely varying growth rates, from arbitrarily slow divergence to nearly power-law growth, showing that entropy scaling does not determine the factor type. In contrast, we prove that gauge-invariant quadratic Hamiltonians with finitely many flavours and exponentially decaying hopping exhibit a dichotomy: their ground states either satisfy an area law with a type I half-chain factor, or have logarithmic entropy growth with a type $\mathrm{III}_1$ factor. The counterexample was discovered with AI assistance.

quant-pharXiv:2610.24154•Oct 20, 2026•0 citations
We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.
Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region AAA if and only if the mutual information of the inner region and the exterior, conditioned on the part of AAA inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for abstract quantum spin systems which allow one to access topological order via a boundary algebra construction. Using the LTO axioms, we produced a canonical pure state on the quasi-local algebra, which gives a net of von Neumann algebras associated to a poset of cones in Rn\mathbb{R}^nRn. In this article, motivated by [arXiv:2509.23734], we introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory. We prove this axiom is satisfied for all known topologically ordered commuting projector models. We thus get an independent proof of Haag duality for the Levin-Wen string net models originally proved in [arXiv:2509.23734]. We also give a reflection positivity axiom for LTOs, connecting to the recent article [arXiv:2510.20662]. We again prove this axiom is satisfied for all known topologically ordered commuting projector models about some Z/2\mathbb{Z}/2Z/2-reflection symmetry.
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 $\clb$.
We construct translation-invariant pure quasifree fermionic states on the one-dimensional lattice whose half-chain von Neumann algebras are hyperfinite factors of type II∞\mathrm{II}_\inftyII∞​ or IIIλ\mathrm{III}_λIIIλ​ for any 0<λ≤10<λ\leq 10<λ≤1. Via the Jordan-Wigner-Araki transformation, these yield translation-invariant pure spin-chain states with the same half-chain factors, demonstrating that such factors need not be of type I or III. Each state is the unique ground state of a gapless, translation-invariant parent Hamiltonian with superpolynomially decaying interactions. For any fixed factor type, the interval entanglement entropy can exhibit widely varying growth rates, from arbitrarily slow divergence to nearly power-law growth, showing that entropy scaling does not determine the factor type. In contrast, we prove that gauge-invariant quadratic Hamiltonians with finitely many flavours and exponentially decaying hopping exhibit a dichotomy: their ground states either satisfy an area law with a type I half-chain factor, or have logarithmic entropy growth with a type III1\mathrm{III}_1III1​ factor. The counterexample was discovered with AI assistance.