Twodimensional (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 highsymmetry points of the Brillouin zone, for example, surface Dirac states at
Hard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as
 Award ID(s):
 1839370
 NSFPAR ID:
 10370527
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Granular Matter
 Volume:
 24
 Issue:
 3
 ISSN:
 14345021
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract in topological insulators Bi_{2}Se(Te)_{3}and Dirac cones at$$\overline{{{\Gamma }}}$$ $\overline{\Gamma}$K and points in graphene. The lowenergy 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.$$K^{\prime}$$ ${K}^{\prime}$ 
Abstract We complete the computation of all
rational points on all the 64 maximal AtkinLehner quotients$$\mathbb {Q}$$ $Q$ 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 fourthnamed author) combined with the MordellWeil sieve. Additionally, for squarefree levels$$X_0(N)^*$$ ${X}_{0}{\left(N\right)}^{\ast}$N , we classify all rational points as cusps, CM points (including their CM field and$$\mathbb {Q}$$ $Q$j invariants) and exceptional ones. We further indicate how to use this to compute the rational points on all of their modular coverings.$$\mathbb {Q}$$ $Q$ 
Abstract We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space
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”$\mathcal{S}$ $S$ . 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 blowup) associated to the stochastic YMH flow.$\mathfrak {O}$ $O$ 
Abstract Let
be a configuration of$$\textbf{p}$$ $p$n points in for some$$\mathbb R^d$$ ${R}^{d}$n and some . 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$$d \ge 2$$ $d\ge 2$ 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 edgesetting when given a sufficiently rich set of edge measurements and assuming that$$\textbf{p}$$ $p$ is generic.$$\textbf{p}$$ $p$ 
Abstract We investigate the geometry of the space of immersed closed curves equipped with reparametrizationinvariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractionalorder
. 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$$q\in [0,\infty )$$ $q\in [0,\infty )$ . Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if$$q>1/2$$ $q>1/2$ , whereas if$$q>3/2$$ $q>3/2$ then finitetime blowup may occur. The geodesic completeness for$$q<3/2$$ $q<3/2$ is obtained by proving metric completeness of the space of$$q>3/2$$ $q>3/2$ immersed curves with the distance induced by the Riemannian metric.$$H^q$$ ${H}^{q}$