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1. Realization of unpinned two-dimensional dirac states in antimony atomic layers

   https://doi.org/10.1038/s41467-022-32327-8  

   Lu, Qiangsheng ; Cook, Jacob ; Zhang, Xiaoqian ; Chen, Kyle Y. ; Snyder, Matthew ; Nguyen, Duy Tung ; Reddy, P. V. Sreenivasa ; Qin, Bingchao ; Zhan, Shaoping ; Zhao, Li-Dong ; et al ( August 2022 , Nature Communications)

   Two-dimensional (2D) Dirac states with linear dispersion have been observed in graphene and on the surface of topological insulators. 2D Dirac states discovered so far are exclusively pinned at high-symmetry points of the Brillouin zone, for example, surface Dirac states at$$\overline{{{\Gamma }}}$$$\overline{\Gamma }$in topological insulators Bi₂Se(Te)₃and Dirac cones atKand$$K^{\prime}$$$K^{\prime }$points in graphene. The low-energy dispersion of those Dirac states are isotropic due to the constraints of crystal symmetries. In this work, we report the observation of novel 2D Dirac states in antimony atomic layers with phosphorene structure. The Dirac states in the antimony films are located at generic momentum points. This unpinned nature enables versatile ways such as lattice strains to control the locations of the Dirac points in momentum space. In addition, dispersions around the unpinned Dirac points are highly anisotropic due to the reduced symmetry of generic momentum points. The exotic properties of unpinned Dirac states make antimony atomic layers a new type of 2D Dirac semimetals that are distinct from graphene.

    

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2. Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

   https://doi.org/10.1007/s40993-022-00388-9  

   Adžaga, Nikola ; Chidambaram, Shiva ; Keller, Timo ; Padurariu, Oana ( October 2022 , Research in Number Theory)

   We complete the computation of all$$\mathbb {Q}$$$Q$-rational points on all the 64 maximal Atkin-Lehner quotients$$X_0(N)^*$$$X_0{\left(N\right)}^{\ast }$such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all$$\mathbb {Q}$$$Q$-rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the$$\mathbb {Q}$$$Q$-rational points on all of their modular coverings.

    

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3. Stochastic quantisation of Yang–Mills–Higgs in 3D

   https://doi.org/10.1007/s00222-024-01264-2  

   Chandra, Ajay ; Chevyrev, Ilya ; Hairer, Martin ; Shen, Hao ( August 2024 , Inventiones mathematicae)

   We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space$\mathcal{S}$$S$is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to$\mathcal{S}$$S$and thus define a quotient space of “gauge orbits”$\mathfrak {O}$$O$. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on$\mathfrak {O}$$O$(up to a potential finite time blow-up) associated to the stochastic YMH flow.

    

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4. Trilateration Using Unlabeled Path or Loop Lengths

   https://doi.org/10.1007/s00454-023-00605-x  

   Gkioulekas, Ioannis ; Gortler, Steven J. ; Theran, Louis ; Zickler, Todd ( November 2023 , Discrete & Computational Geometry)

   Let$$\textbf{p}$$$p$be a configuration ofnpoints in$$\mathbb R^d$$$R^d$for somenand some$$d \ge 2$$$d\ge 2$. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing$$\textbf{p}$$$p$given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when$$\textbf{p}$$$p$will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that$$\textbf{p}$$$p$is generic.

    

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5. Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves

   https://doi.org/10.1007/s12220-024-01652-3  

   Bauer, Martin ; Heslin, Patrick ; Maor, Cy ( May 2024 , The Journal of Geometric Analysis)

   We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order$$q\in [0,\infty )$$$q\in [0,\infty )$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if$$q>1/2$$$q>1/2$. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if$$q>3/2$$$q>3/2$, whereas if$$q<3/2$$$q<3/2$then finite-time blowup may occur. The geodesic completeness for$$q>3/2$$$q>3/2$is obtained by proving metric completeness of the space of$$H^q$$$H^q$-immersed curves with the distance induced by the Riemannian metric.

    

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