In and around Abelian anyon models

• URL: http://arxiv.org/abs/2004.12048v3
• Date: Wed, 21 Oct 2020 02:23:51 GMT
• Title: In and around Abelian anyon models
• Authors: Liang Wang, Zhenghan Wang
• Abstract summary: We provide an explicit algorithm for a $K$-matrix in Chern-Simons theory and a positive definite even one for a lattice conformal field theory. Anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter.
• Score: 6.509665408765348
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern-Simons theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a $K$-matrix in Chern-Simons theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.

Related papers

• Low-dimensional polaritonics: Emergent non-trivial topology on exciton-polariton simulators [0.0]
  Polaritonic lattice configurations in dimensions $D=2$ are used as simulators of topological phases, based on symmetry class A Hamiltonians. We provide a comprehensive mathematical framework, which fully addresses the source and structure of topological phases in coupled polaritonic array systems.
  arXiv  Detail & Related papers  (2023-10-31T04:22:58Z)
• Geometric Neural Diffusion Processes [55.891428654434634]
  We extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We show that with these conditions, the generative functional model admits the same symmetry.
  arXiv  Detail & Related papers  (2023-07-11T16:51:38Z)
• A Study of Neural Collapse Phenomenon: Grassmannian Frame, Symmetry and Generalization [91.95109845914267]
  We extend original Neural Collapse Phenomenon by proving Generalized Neural Collapse hypothesis. We obtain Grassmannian Frame structure from the optimization and generalization of classification. We provide a theorem to explain Symmetric Generalization of permutation.
  arXiv  Detail & Related papers  (2023-04-18T11:35:14Z)
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory [44.99833362998488]
  We introduce quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple quantum numbers. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories.
  arXiv  Detail & Related papers  (2022-12-28T18:56:25Z)
• Pauli topological subsystem codes from Abelian anyon theories [2.410842777583321]
  We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits.
  arXiv  Detail & Related papers  (2022-11-07T19:00:01Z)
• Dualities in one-dimensional quantum lattice models: topological sectors [0.0]
  We construct a general framework for relating the spectra of dual theories to each other. We find that the mapping between its topological sectors and those of the XXZ model is associated with the non-trivial braided auto-equivalence of the Drinfel'd center.
  arXiv  Detail & Related papers  (2022-11-07T18:54:57Z)
• Ultrahyperbolic Knowledge Graph Embeddings [41.85886045379838]
  We present an ultrahyperbolic KG embedding in an ultrahyperbolic (or pseudo-Riemannian) manifold. In particular, we model each relation as a pseudo-orthogonal transformation that preserves the pseudo-Riemannian bilinear form. Experimental results on three standard KGs show that UltraE outperforms previous Euclidean- and hyperbolic-based approaches.
  arXiv  Detail & Related papers  (2022-06-01T12:31:26Z)
• Abelian Neural Networks [48.52497085313911]
  We first construct a neural network architecture for Abelian group operations and derive a universal approximation property. We extend it to Abelian semigroup operations using the characterization of associative symmetrics. We train our models over fixed word embeddings and demonstrate improved performance over the original word2vec.
  arXiv  Detail & Related papers  (2021-02-24T11:52:21Z)
• Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
  We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow. First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric. We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
  arXiv  Detail & Related papers  (2020-10-29T06:15:30Z)
• Microscopic models for Kitaev's sixteenfold way of anyon theories [0.0]
  In two dimensions, the topological order described by $mathbbZ$ gauge theory is classified by $nu ; mathrmmod; 16$ as predicted by Kitaev. Here we provide a systematic and complete construction of microscopic models realizing this so-called sixteenfold way of anyon theories.
  arXiv  Detail & Related papers  (2020-05-27T22:13:37Z)
• Ground Subspaces of Topological Phases of Matter as Error Correcting Codes [0.9306768284179177]
  We prove that a lattice implementation of the disk axiom and annulus axiom in TQFTs is essentially the equivalence of TQO1 and TQO2 conditions. We propose to characterize topological phases of matter via error correcting properties, and refer to gapped fracton models as lax-topological.
  arXiv  Detail & Related papers  (2020-04-24T20:38:10Z)

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.