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1. A first proof of knot localization for polymers in a nanochannel

   https://doi.org/10.1088/1751-8121/ad6c01  

   Beaton, Nicholas R; Ishihara, Kai; Atapour, Mahshid; Eng, Jeremy W; Vazquez, Mariel; Shimokawa, Koya; Soteros, Christine E (August 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-typeKlattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition ofK. This Knot Entropy (KE) conjecture is consistent with the idea that for unconfined polymers, knots occur in a localized way (the knotted part is relatively small compared to polymer length). For full confinement (to a sphere or box), numerical evidence suggests that knots are much less localized. Numerical evidence for nanochannel or tube confinement is mixed, depending on how the size of a knot is measured. Here we outline the proof that the KE conjecture holds for polygons in the $\mathrm{\infty }\times 2\times 1$lattice tube and show that knotting is localized when a connected-sum measure of knot size is used. Similar results are established for linked polygons. This is the first model for which the knot entropy conjecture has been proved. 

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2. Low-energy magnetic states of Tb adatom on graphene

   https://doi.org/10.1088/1361-648X/ad8fe9  

   Shaikh, Monirul; Klein, Alison; Wysocki, Aleksander L (November 2024, Journal of Physics: Condensed Matter)

   Abstract Electronic structure and magnetic interactions of a Tb adatom on graphene are investigated from first principles using combination of density functional theory and multiconfigurational quantum chemistry techniques including spin–orbit coupling (SOC) . We determine that the six-fold symmetry hollow site is the preferred adsorption site and investigate electronic spectrum for different adatom oxidation states including Tb³⁺, Tb²⁺, Tb¹⁺, and Tb⁰. For all charge states, the Tb $4f^8$configuration is retained with other adatom valence electrons being distributed over $5d_{xy}$, $5d_{x2+y2}$, and $6s/5d_0$single-electron orbitals. We find strong intra-site adatom exchange coupling that ensures that the $5d6s$spins are parallel to the4fspin. For Tb³⁺, the energy levels can be described by theJ = 6 multiplet split by the graphene crystal field (CF). For other oxidation states, the interaction of4felectrons with spin and orbital degrees of freedom of $6s5d$electrons in the presence of SOC results in the low-energy spectrum composed closely lying effective multiplets that are split by the graphene CF. Stable magnetic moment is predicted for Tb³⁺and Tb²⁺adatoms due to uniaxial magnetic anisotropy and effective anisotropy barrier around 440 cm⁻¹controlled by the temperature assisted quantum tunneling of magnetization through the third excited doublet. On the other hand, in-plane magnetic anisotropy is found for Tb¹⁺and Tb⁰adatoms. Our results indicate that the occupation of the $6s5d$orbitals can dramatically affect the magnetic anisotropy and magnetic moment stability of rare earth adatoms. 

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3. A tile model of circuit topology for self-entangled biopolymers

   https://doi.org/10.1038/s41598-023-35771-8  

   Flapan, Erica; Mashaghi, Alireza; Wong, Helen (June 2023, Scientific Reports)

   Abstract Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are$$3_1$$ $3_1$,$$4_1$$ $4_1$,$$5_1$$ $5_1$,$$5_2$$ $5_2$,$$6_1$$ $6_1$,$$6_2$$ $6_2$,$$6_3$$ $6_3$,$$7_7$$ $7_7$,$$8_{12}$$ $8_{12}$and connected sums of these knots. This includes all proteins knots that have thus far been identified. 

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4. Designing multicomponent hydrides with potential high T _c superconductivity

   https://doi.org/10.1073/pnas.2413096121  

   Denchfield, Adam; Park, Hyowon; Hemley, Russell J (November 2024, Proceedings of the National Academy of Sciences)

   While hydrogen-rich materials have been demonstrated to exhibit high T_csuperconductivity at high pressures, there is an ongoing search for ternary, quaternary, and more chemically complex hydrides that achieve such high critical temperatures at much lower pressures. First-principles searches are impeded by the computational complexity of solving the Eliashberg equations for large, complex crystal structures. Here, we adopt a simplified approach using electronic indicators previously established to be correlated with superconductivity in hydrides. This is used to study complex hydride structures, which are predicted to exhibit promisingly high critical temperatures for superconductivity. In particular, we propose three classes of hydrides inspired by the Fm $\overline{3}$m RH ${}_3$structures that exhibit strong hydrogen network connectivity, as defined through the electron localization function. The first class [RH ${}_{11}$X ${}_3$Y] is based on a Pm $\overline{3}$m structure showing moderately high T_c, where the T_cestimate from electronic properties is compared with direct Eliashberg calculations and found to be surprisingly accurate. The second class of structures [(RH ${}_{11}$) ${}_2$X ${}_6$YZ] improves on this with promisingly high density of states with dominant hydrogen character at the Fermi energy, typically enhancing T_c. The third class [(R ${}^1$H ${}_{11}$)(R ${}^2$H ${}_{11}$)X ${}_6$YZ] improves the strong hydrogen network connectivity by introducing anisotropy in the hydrogen network through a specific doping pattern. These design principles and associated model structures provide flexibility to optimize both T_cand the structural stability of complex hydrides. 

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5. Quantum group intertwiner space from quantum curved tetrahedron

   https://doi.org/10.1088/1361-6382/ad5f71  

   Han, Muxin; Hsiao, Chen-Hung; Pan, Qiaoyin (July 2024, Classical and Quantum Gravity)

   Abstract In this paper, we develop a quantum theory of homogeneously curved tetrahedron geometry, by applying the combinatorial quantization to the phase space of tetrahedron shapes defined in Haggardet al(2016Ann. Henri Poincaré172001–48). Our method is based on the relation between this phase space and the moduli space of SU(2) flat connections on a 4-punctured sphere. The quantization results in the physical Hilbert space as the solution of the quantum closure constraint, which quantizes the classical closure condition $M_4{M}_3{M}_2{M}_1=1$, $M_{\nu }\in \mathrm{SU}\left(2\right)$, for the homogeneously curved tetrahedron. The quantum group $U_q\left(\mathfrak{su}\right(2\left)\right)$emerges as the gauge symmetry of a quantum tetrahedron. The physical Hilbert space of the quantum tetrahedron coincides with the Hilbert space of 4-valent intertwiners of $U_q\left(\mathfrak{su}\right(2\left)\right)$. In addition, we define the area operators quantizing the face areas of the tetrahedron and compute the spectrum. The resulting spectrum is consistent with the usual Loop-Quantum-Gravity area spectrum in the large spin regime but is different for small spins. This work closely relates to 3+1 dimensional Loop Quantum Gravity in presence of cosmological constant and provides a justification for the emergence of quantum group in the theory. 

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