Haag Duality for 2D Quantum Spin Systems

• URL: http://arxiv.org/abs/2509.23734v1
• Date: Sun, 28 Sep 2025 08:27:49 GMT
• Title: Haag Duality for 2D Quantum Spin Systems
• Abstract summary: Haag duality is a strong notion of locality for two-dimensional lattice quantum spin systems.<n>We prove that two-dimensional tensor network states based on biconnected $C*$-weak Hopf algebras satisfy Haag duality.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Haag duality is a strong notion of locality for two-dimensional lattice quantum spin systems, requiring that the commutant of the algebra of observables supported in a cone-like region coincides with the algebra of observables in its complement. Originally introduced within algebraic quantum field theory, Haag duality has recently become pivotal in the operator-algebraic analysis of quantum many-body systems. In particular, it plays a key role in the description of anyonic excitations, which are widely believed to classify two-dimensional non-chiral gapped quantum phases of matter. Prior to this work, Haag duality had only been rigorously established for Kitaev's quantum double models with abelian input groups. In this paper, we establish that two-dimensional tensor network states based on biconnected $C^*$-weak Hopf algebras satisfy Haag duality. These states include as particular cases Kitaev quantum double and Levin-Wen string-net models, and we expect them to encompass representatives of all non-chiral phases. Our proof relies on deriving an operator-algebraic sufficient condition for Haag duality in finite systems, which we verify using tensor-network methods. As part of our analysis, we extend the tensor-network bulk-boundary correspondence, construct explicit commuting parent Hamiltonians for these models, and prove that in the biconnected case they satisfy the local topological quantum order condition.

Related papers

• Quantum non-local games: Quantum relations, projection lattices and rule operators [51.56484100374058]
  (mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.<n>(mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.<n>(mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.<n>(mathcal R)-projection-test quantum games with finite-dimensional input and output von Neumann algebras.
  arXiv  Detail & Related papers  (2026-08-31T09:41:14Z)
• Quantum Fuzzy Sets Revisited: Density Matrices, Decoherence, and the Q-Matrix Framework [0.0]
  In 2006 we proposed Quantum Fuzzy Sets, observing that states of a quantum register could serve as characteristic functions of fuzzy subsets.<n>In the two decades since, the idea has been taken up by researchers working on quantum annealers, intuitionistic fuzzy connectives, and quantum machine learning.
  arXiv  Detail & Related papers  (2026-03-22T19:51:08Z)
• Hysteretic squashed entanglement in many-body quantum systems [42.085941481155295]
  Entanglement in many-body quantum systems is distributed across spatial regions.<n>We propose hysteretic entanglement $T_sq$, a conditional entanglement that measures the genuine quantum correlations between two subregions.<n>We show that $T_sq$ can detect genuine quantum correlations across both adjacent and long-range subsystems.
  arXiv  Detail & Related papers  (2026-03-10T17:00:49Z)
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains [0.0]
  We introduce the notion of inequality-depth duality transformations, which relates two sets of operator algebras through a conjugation by a poly-depth quantum circuit.<n>We make use of this to construct efficient Gibbs samplers for a variety of interesting quantum Hamiltonians as they are poly-depth dual to classical Hamiltonians.
  arXiv  Detail & Related papers  (2025-07-31T19:25:13Z)
• On Infinite Tensor Networks, Complementary Recovery and Type II Factors [39.58317527488534]
  We study local operator algebras at the boundary of infinite tensor networks.<n>We decompose the limiting Hilbert space and the algebras of observables in a way that keeps track of the entanglement in the network.
  arXiv  Detail & Related papers  (2025-03-31T18:00:09Z)
• Theory of the correlated quantum Zeno effect in a monitored qubit dimer [41.94295877935867]
  We show how the competition between two measurement processes give rise to two distinct Quantum Zeno (QZ) regimes.<n>We develop a theory based on a Gutzwiller ansatz for the wavefunction that is able to capture the structure of the Hilbert phase diagram.<n>We show how the two QZ regimes are intimately connected to the topology of the flow of the underlying non-Hermitian Hamiltonian governing the no-click evolution.
  arXiv  Detail & Related papers  (2025-03-28T19:44:48Z)
• The category of anyon sectors for non-abelian quantum double models [0.0]
  We study Kitaev's quantum double model for arbitrary finite gauge group in infinite volume.<n>We organize endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons.<n>This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.
  arXiv  Detail & Related papers  (2025-03-19T18:00:30Z)
• Quantum cellular automata and categorical dualities of spin chains [0.0]
  We study categorical dualities, which are bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain.<n>A fundamental question about dualities is whether they can be extended to quantum cellular automata.<n>We present a solution to the extension problem using the machinery of Doplicher-Haag-Roberts bimodules.
  arXiv  Detail & Related papers  (2024-10-11T15:00:50Z)
• A Universal Kinematical Group for Quantum Mechanics [0.0]
  In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms.
  arXiv  Detail & Related papers  (2024-04-28T18:46:24Z)
• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension [44.41126861546141]
  We provide a classification of Lie algebras generated by translation-invariant 2-local spin chain Hamiltonians. We consider chains with open and periodic boundary conditions and find 17 unique dynamical Lie algebras. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches.
  arXiv  Detail & Related papers  (2023-09-11T17:59:41Z)
• A vertical gate-defined double quantum dot in a strained germanium double quantum well [48.7576911714538]
  Gate-defined quantum dots in silicon-germanium heterostructures have become a compelling platform for quantum computation and simulation. We demonstrate the operation of a gate-defined vertical double quantum dot in a strained germanium double quantum well. We discuss challenges and opportunities and outline potential applications in quantum computing and quantum simulation.
  arXiv  Detail & Related papers  (2023-05-23T13:42:36Z)
• The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario [1.9798034349981157]
  We show that the quantum inflation hierarchy is complete for the bilocal scenario in the commuting observables model of locality. We also give a bilocal version of an observation by Tsirelson, namely that in finite dimensions, the commuting observables model and the tensor product model of locality coincide.
  arXiv  Detail & Related papers  (2022-12-21T19:00:09Z)
• Self-adjoint extension schemes and modern applications to quantum Hamiltonians [55.2480439325792]
  monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly.
  arXiv  Detail & Related papers  (2022-01-25T09:45:16Z)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)
• A quantum system with a non-Hermitian Hamiltonian [0.0]
  relevance in Physics of non-Hermitian operators with real eigenvalues is being widely recognized. In this note, a quantum system described by a non-Hermitian Hamiltonian is investigated.
  arXiv  Detail & Related papers  (2020-04-15T17:03:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.