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1. Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

   https://doi.org/10.1007/s00209-024-03627-0  

   Çineli, Erman; Ginzburg, Viktor L; Gürel, Başak Z (December 2024, Mathematische Zeitschrift)

   We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistent homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy. 

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2. Barcode entropy of geodesic flows

   https://doi.org/10.4171/JEMS/1572  

   Ginzburg, Viktor L; Gürel, Başak Z; Mazzucchelli, Marco (December 2024, Journal of the European Mathematical Society)

   We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional. 

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3. On the barcode entropy of Reeb flows

   https://doi.org/10.1007/s00029-025-01062-5  

   Çineli, Erman; Ginzburg, Viktor L; Gürel, Başak Z; Mazzucchelli, Marco (September 2025, Selecta Mathematica)

   In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows. 

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4. A topological kagome magnet in high entropy form

   https://doi.org/10.1038/s42005-022-00842-1  

   Min, Lujin; Sretenovic, Milos; Heitmann, Thomas W.; Valentine, Tyler W.; Zu, Rui; Gopalan, Venkatraman; Rost, Christina M.; Ke, Xianglin; Mao, Zhiqiang (March 2022, Communications Physics)

   Abstract Topological kagome magnets RMn₆Sn₆(R = rare earth element) attract numerous interests due to their non-trivial band topology and room-temperature magnetism. Here, we report a high entropy version of kagome magnet, (Gd_0.38Tb_0.27Dy_0.20Ho_0.15)Mn₆Sn₆. Such a high entropy material exhibits multiple spin reorientation transitions, which is not seen in all the related parent compounds and can be understood in terms of competing magnetic interactions enabled by high entropy. Furthermore, we also observed an intrinsic anomalous Hall effect, indicating that the high entropy phase preserves the non-trivial band topology. These results suggest that high entropy may provide a route to engineer the magnetic structure and expand the horizon of topological materials. 

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5. 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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