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Register a new userTopological aspects of $mathbb{Z}/2mathbb{Z}$ eigenfunctions for the Laplacian on $S^2$
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This paper concerns the behavior of the eigenfunctions and eigenvalues of the round spheres Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of particular interest is how these eigenvalues and eigenvectors change when viewed as functions on the configuration spaces of points.
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A unitary fusion category is called $mathbb{Z}/2mathbb{Z}$-quadratic if it has a $mathbb{Z}/2mathbb{Z}$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $mathbb{Z}/2mathbb{Z}$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostriks results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larsons rank-finiteness bound for $mathbb{Z}/3mathbb{Z}$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information.
Assume a polynomial-time algorithm for factoring integers, Conjecture~ref{conj}, $dgeq 3,$ and $q$ and $p$ are prime numbers, where $pleq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $log(q)$ that lifts every $mathbb{Z}/qmathbb{Z}$ point of $S^{d-2}subset S^{d}$ to a $mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $mathbb{Z}/qmathbb{Z}$ points of $S^{d-2}subset S^d$.
Topological quantum paramagnets are exotic states of matter, whose magnetic excitations have a topological band structure, while the ground state is topologically trivial. Here we show that a simple model of quantum spins on a honeycomb bilayer hosts a time-reversal-symmetry protected $mathbb{Z}_2$ topological quantum paramagnet ({em topological triplon insulator}) in the presence of spin-orbit coupling. The excitation spectrum of this quantum paramagnet consists of three triplon bands, two of which carry a nontrivial $mathbb{Z}_2$ index. As a consequence, there appear two counterpropagating triplon excitation modes at the edge of the system. We compute the triplon edge state spectrum and the $mathbb{Z}_2$ index for various parameter choices. We further show that upon making one of the Heisenberg couplings stronger, the system undergoes a topological quantum phase transition, where the $mathbb{Z}_2$ index vanishes, to a different topological quantum paramagnet. In this case the counterpopagating triplon edge modes are disconnected from the bulk excitations and are protected by a chiral and a unitary symmetry. We discuss possible realizations of our model in real materials, in particular d$^{4}$ Mott insulators, and their potential applications.
Let $Gamma$ be a co-compact Fuchsian group of isometries on the Poincare disk $DD$ and $Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $Delta$, equivariant by $Gamma$ with real eigenvalue $lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution $dd_{f,s}$ (the Helgason distribution) which is equivariant by $Gamma$ and supported at infinity $partialDD=SS^1$. The geodesic flow on the compact surface $DD/Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:SS^1toSS^1$, the so-called Bowen-Series transformation. Let $ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-sln |T|$. M. Pollicott showed that $dd_{f,s}$ is an eigenfunction of the dual operator $ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $psi_{f,s}$ of $ll_s$ for the eigenvalue 1, given by an integral formula [ psi_{f,s} (xi)=int frac{J(xi,eta)}{|xi-eta|^{2s}} dd_{f,s} (deta), ] oindent where $J(xi,eta)$ is a ${0,1}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $DD/Gamma$.
We prove an Herschs type isoperimetric inequality for the third positive eigenvalue on $mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.
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