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1. Ground state degeneracy on torus in a family of ZN toric code

   https://doi.org/10.1063/5.0134010  

   Watanabe, Haruki; Cheng, Meng; Fuji, Yohei (May 2023, Journal of Mathematical Physics)

   Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species. 

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2. Conceptual Neighborhood Graphs of Topological Relations in Z2

   https://doi.org/10.3390/ijgi14040150  

   Hall, Brendan Patrick; Dube, Matthew Paul (April 2025, ISPRS International Journal of Geo-Information)

   Topological relations form the backbone of qualitative spatial reasoning and, as such, play a paramount role in geographic information systems. Three decades of research have provided a proliferation of sets of qualitative topological relations in both continuous and discretized spaces, but only in continuous spaces has the concept of organizing these relations into a larger framework (called a conceptual neighborhood graph) been considered. Previous work leveraged matrix differences to derive the anisotropic scaling neighborhood for these relations. In this paper, a simulation protocol is used to derive conceptual neighborhood graphs of qualitative topological relations in Z2 for the operations of translation and isotropic scaling. It is further shown that, when aggregating raster relations into their continuous counterparts and collapsing neighborhood connections within these groups, the familiar conceptual neighborhood structures for continuous regions appear. 

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3. Catalog of topological phonon materials

   https://doi.org/10.1126/science.adf8458  

   Xu, Yuanfeng; Vergniory, M G; Ma, Da-Shuai; Mañes, Juan L; Song, Zhi-Da; Bernevig, B Andrei; Regnault, Nicolas; Elcoro, Luis (May 2024, Science)

   Phonons play a crucial role in many properties of solid-state systems, and it is expected that topological phonons may lead to rich and unconventional physics. On the basis of the existing phonon materials databases, we have compiled a catalog of topological phonon bands for more than 10,000 three-dimensional crystalline materials. Using topological quantum chemistry, we calculated the band representations, compatibility relations, and band topologies of each isolated set of phonon bands for the materials in the phonon databases. Additionally, we calculated the real-space invariants for all the topologically trivial bands and classified them as atomic or obstructed atomic bands. We have selected more than 1000 “ideal” nontrivial phonon materials to motivate future experiments. The datasets were used to build the Topological Phonon Database. 

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4. Emergence of analogy from relation learning

   https://doi.org/10.1073/pnas.1814779116  

   Lu, Hongjing; Wu, Ying Nian; Holyoak, Keith J. (February 2019, Proceedings of the National Academy of Sciences)

   By middle childhood, humans are able to learn abstract semantic relations (e.g., antonym, synonym, category membership) and use them to reason by analogy. A deep theoretical challenge is to show how such abstract relations can arise from nonrelational inputs, thereby providing key elements of a protosymbolic representation system. We have developed a computational model that exploits the potential synergy between deep learning from “big data” (to create semantic features for individual words) and supervised learning from “small data” (to create representations of semantic relations between words). Given as inputs labeled pairs of lexical representations extracted by deep learning, the model creates augmented representations by remapping features according to the rank of differences between values for the two words in each pair. These augmented representations aid in coping with the feature alignment problem (e.g., matching those features that make “love-hate” an antonym with the different features that make “rich-poor” an antonym). The model extracts weight distributions that are used to estimate the probabilities that new word pairs instantiate each relation, capturing the pattern of human typicality judgments for a broad range of abstract semantic relations. A measure of relational similarity can be derived and used to solve simple verbal analogies with human-level accuracy. Because each acquired relation has a modular representation, basic symbolic operations are enabled (notably, the converse of any learned relation can be formed without additional training). Abstract semantic relations can be induced by bootstrapping from nonrelational inputs, thereby enabling relational generalization and analogical reasoning. 

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5. Global aspects of measure preserving equivalence relations and graphs

   https://doi.org/10.53733/96  

   Kechris, Alexander (September 2021, New Zealand Journal of Mathematics)

   This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view. 

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