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1. Anomalous Landau quantization in intrinsic magnetic topological insulators

   https://doi.org/10.1038/s41467-023-40383-x  

   Chong, Su Kong; Lei, Chao; Lee, Seng Huat; Jaroszynski, Jan; Mao, Zhiqiang; MacDonald, Allan H.; Wang, Kang L. (August 2023, Nature Communications)

   Abstract The intrinsic magnetic topological insulator, Mn(Bi_1−xSb_x)₂Te₄, has been identified as a Weyl semimetal with a single pair of Weyl nodes in its spin-aligned strong-field configuration. A direct consequence of the Weyl state is the layer dependent Chern number,$$C$$ $C$. Previous reports in MnBi₂Te₄thin films have shown higher$$C$$ $C$states either by increasing the film thickness or controlling the chemical potential. A clear picture of the higher Chern states is still lacking as data interpretation is further complicated by the emergence of surface-band Landau levels under magnetic fields. Here, we report a tunable layer-dependent$$C$$ $C$ = 1 state with Sb substitution by performing a detailed analysis of the quantization states in Mn(Bi_1−xSb_x)₂Te₄dual-gated devices—consistent with calculations of the bulk Weyl point separation in the doped thin films. The observed Hall quantization plateaus for our thicker Mn(Bi_1−xSb_x)₂Te₄films under strong magnetic fields can be interpreted by a theory of surface and bulk spin-polarised Landau level spectra in thin film magnetic topological insulators. 

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2. Spin structure and dynamics of the topological semimetal Co3Sn2-xInxS2

   https://doi.org/10.1038/s41535-022-00523-w  

   Neubauer, Kelly J.; Ye, Feng; Shi, Yue; Malinowski, Paul; Gao, Bin; Taddei, Keith M.; Bourges, Philippe; Ivanov, Alexandre; Chu, Jiun-Haw; Dai, Pengcheng (December 2022, npj Quantum Materials)

   Abstract The anomalous Hall effect (AHE), typically observed in ferromagnetic (FM) metals with broken time-reversal symmetry, depends on electronic and magnetic properties. In Co₃Sn_2-xIn_xS₂, a giant AHE has been attributed to Berry curvature associated with the FM Weyl semimetal phase, yet recent studies report complicated magnetism. We use neutron scattering to determine the spin dynamics and structures as a function ofxand provide a microscopic understanding of the AHE and magnetism interplay. Spin gap and stiffness indicate a contribution from Weyl fermions consistent with the AHE. The magnetic structure evolves fromc-axis ferromagnetism at$$x = 0$$ $x=0$to a canted antiferromagnetic (AFM) structure with reducedc-axis moment and in-plane AFM order at$$x = 0.12$$ $x=0.12$and further reducedc-axis FM moment at$$x = 0.3$$ $x=0.3$. Since noncollinear spins can induce non-zero Berry curvature in real space acting as a fictitious magnetic field, our results revealed another AHE contribution, establishing the impact of magnetism on transport. 

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3. Higher-group symmetry of (3+1)D fermionic $$\mathbb{Z}_2$$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry

   https://doi.org/10.21468/SciPostPhys.16.5.122  

   Barkeshli, Maissam; Hsin, Po-Shen; Kobayashi, Ryohei (January 2024, SciPost Physics)

   It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D\mathbb{Z}_2 ${\mathbb{Z}}_2$gauge theory with an emergent fermion, and point out that pumping chiralp+ip $p+ip$topological states gives rise to a\mathbb{Z}_{8} ${\mathbb{Z}}_8$0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization ofT^3 $T^3$(3-torus) andT^2 \rtimes_{C_2} S^1 $T^2{⋊}_{C_2}{S}^1$(2-torus bundle over the circle) respectively, and pumpingp+ip $p+ip$states. Our considerations also imply the possibility of a logicalT $T$gate by placing the code on\mathbb{RP}^3 ${ℝ\mathbb{P}}^3$and pumping ap+ip $p+ip$topological state. 

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4. The half-volume spectrum of a manifold

   https://doi.org/10.1007/s00526-025-02949-z  

   Mazurowski, Liam; Zhou, Xin (May 2025, Calculus of Variations and Partial Differential Equations)

   Abstract We define the half-volume spectrum$$\{{\tilde{\omega }_p\}_{p\in \mathbb {N}}}$$ $\{{\tilde{\omega }}_p{\}}_{p\in N}$of a closed manifold$$(M^{n+1},g)$$ $(M^{n+1},g)$. This is analogous to the usual volume spectrum ofM, except that we restrict top-sweepouts whose slices each enclose half the volume ofM. We prove that the Weyl law continues to hold for the half-volume spectrum. We define an analogous half-volume spectrum$$\tilde{c}(p)$$ $\tilde{c}\left(p\right)$in the phase transition setting. Moreover, for$$3 \le n+1 \le 7$$ $3\le n+1\le 7$, we use the Allen–Cahn min-max theory to show that each$$\tilde{c}(p)$$ $\tilde{c}\left(p\right)$is achieved by a constant mean curvature surface enclosing half the volume ofMplus a (possibly empty) collection of minimal surfaces with even multiplicities. 

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5. High pressure raman spectroscopy and X-ray diffraction of K2Ca(CO3)2 bütschliite: multiple pressure-induced phase transitions in a double carbonate

   https://doi.org/10.1007/s00269-023-01262-5  

   Zeff, G.; Kalkan, B.; Armstrong, K.; Kunz, M.; Williams, Q. (January 2024, Physics and Chemistry of Minerals)

   Abstract The crystal structure and bonding environment of K₂Ca(CO₃)₂bütschliite were probed under isothermal compression via Raman spectroscopy to 95 GPa and single crystal and powder X-ray diffraction to 12 and 68 GPa, respectively. A second order Birch-Murnaghan equation of state fit to the X-ray data yields a bulk modulus,$${K}_{0}=46.9$$ $K_0=46.9$GPa with an imposed value of$${K}_{0}^{\prime}= 4$$ $K_0^{\prime }=4$for the ambient pressure phase. Compression of bütschliite is highly anisotropic, with contraction along thec-axis accounting for most of the volume change. Bütschliite undergoes a phase transition to a monoclinicC2/mstructure at around 6 GPa, mirroring polymorphism within isostructural borates. A fit to the compression data of the monoclinic phase yields$${V}_{0}=322.2$$ $V_0=322.2$ Å³$$,$$ $,$$${K}_{0}=24.8$$ $K_0=24.8$GPa and$${K}_{0}^{\prime}=4.0$$ $K_0^{\prime }=4.0$using a third order fit; the ability to access different compression mechanisms gives rise to a more compressible material than the low-pressure phase. In particular, compression of theC2/mphase involves interlayer displacement and twisting of the [CO₃] units, and an increase in coordination number of the K⁺ion. Three more phase transitions, at ~ 28, 34, and 37 GPa occur based on the Raman spectra and powder diffraction data: these give rise to new [CO₃] bonding environments within the structure. 

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