Haag Duality for 2D Quantum Spin Systems
Yoshiko Ogata, David Pérez-García, Alberto Ruiz-de-Alarcón
TL;DR
Haag duality is a central locality principle in algebraic quantum field theory, and its rigorous realization in 2D quantum spin systems supports the anyon-based classification of non-chiral topological phases. The authors develop a general finite-volume condition, show that it suffices to check finite systems, and prove it for renormalization-fixed-point MPO-injective PEPS arising from $C^*$-weak Hopf algebras, including Kitaev quantum doubles and Levin–Wen string-nets. By merging operator-algebraic techniques with tensor-network constructions, they derive a bulk–boundary correspondence and local topological quantum order, yielding approximate Haag duality and exact duality on coarse-grained lattices. This work provides a phase-stable foundation for the anyon framework across a broad class of 2D topological phases, enabling rigorous connections between local algebras, superselection sectors, and braided tensor categories in realistic lattice models.
Abstract
Haag duality is a fundamental locality property introduced in the pioneering formulation of algebraic quantum field theory by Haag and Kastler in the 1960s. Since then, it has played a central role, most notably in the classification of superselection sectors by Doplicher, Haag, and Roberts in the 1970s. Over the past two decades, this concept has migrated from its relativistic origins to quantum spin systems, becoming a cornerstone of the operator-algebraic approach to the long-standing problem of classifying two-dimensional topological quantum phases of matter. In physics, it is widely conjectured that such phases are classified by their emergent anyons, a view supported by exactly solvable models exemplifying all known non-chiral phases: Kitaev's quantum double models, Levin-Wen string-net models, and their slight generalizations. In these models, elementary excitations behave as quasi-particles, namely anyons, whose fusion and braiding properties form a tensor category expected to characterize the phase of matter. A major open problem was to derive the emergence of anyons and the stability of their fusion and braiding beyond these solvable models. Recently, it has been shown that a weaker, phase-stable form of Haag duality resolves these questions. However, rigorous proofs of Haag duality in two dimensions were previously restricted to systems exhibiting abelian anyons. In this work, we establish Haag duality for a broad class of tensor network models based on $C^*$-weak Hopf algebras, encompassing all Kitaev quantum double and Levin-Wen string-net models, and expected to include all non-chiral topological quantum phases of matter.