Topological States in the Kuramoto Model

The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning to each steady state a tuple of integers which records how the state twists around the cycles in the network. We then use this new classification of steady states to obtain a "Weyl" type of asymptotic estimate for the number of steady states as the number of oscillators becomes arbitrarily large while preserving the cycle structure. We further show how this asymptotic estimate can be maximized, and as a special case we obtain an asymptotic lower bound for the number of stable steady states of the model.Comment: 10 pages, 7 figure

Inside the Muchnik Degrees II: The Degree Structures induced by the Arithmetical Hierarchy of Countably Continuous Functions

It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial $\Pi^0_1$ subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty $\Pi^0_1$ subsets of Cantor space, we show the existence of a finite-$\Delta^0_2$-piecewise degree containing infinitely many finite-$(\Pi^0_1)_2$-piecewise degrees, and a finite-$(\Pi^0_2)_2$-piecewise degree containing infinitely many finite-$\Delta^0_2$-piecewise degrees (where $(\Pi^0_n)_2$ denotes the difference of two $\Pi^0_n$ sets), whereas the greatest degrees in these three "finite-$\Gamma$-piecewise" degree structures coincide. Moreover, as for nonempty $\Pi^0_1$ subsets of Cantor space, we also show that every nonzero finite-$(\Pi^0_1)_2$-piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable-$\Delta^0_2$-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-$(\Pi^0_2)_2$-countable-$\Delta^0_2$-piecewise degree includes infinitely many countable-$\Delta^0_2$-piecewise degrees, and every nonzero Muchnik (i.e., countable-$\Pi^0_2$-piecewise) degree includes infinitely many finite-$(\Pi^0_2)_2$-countable-$\Delta^0_2$-piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-$\Delta^0_2$-piecewise degree of a nonempty $\Pi^0_1$ subset of Cantor space have the strong anticupping properties. Finally, we obtain an elementary difference between the Medvedev (Muchnik) degree structure and the finite-$\Gamma$-piecewise degree structure of all subsets of Baire space by showing that none of the finite-$\Gamma$-piecewise structures are Brouwerian, where $\Gamma$ is any of the Wadge classes mentioned above

Stochastic stability of continuous time consensus protocols

A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and those whose topology and strength of connections may vary in time. The graph theoretic interpretation of the analytical results is emphasized. We show how the spectral properties, such as algebraic connectivity and total effective resistance, as well as the geometric properties, such the dimension and the structure of the cycle subspace of the underlying graph, shape stability of the corresponding CPs. In addition, we explore certain implications of the spectral graph theory to CP design. In particular, we point out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly. Similarly, we highlight the benefits of using random versus regular network topologies for CP design. We illustrate these observations with numerical examples and refer to the relevant graph-theoretic results. Keywords: consensus protocol, dynamical network, synchronization, robustness to noise, algebraic connectivity, effective resistance, expander, random graphComment: SIAM Journal on Control and Optimization, to appea

Quantum criticality at the Chern-to-normal insulator transition

Using the non-commutative Kubo formula for aperiodic solids [1-3] and a recently developed numerical implementation [4], we study the conductivity $\sigma$ and resistivity $\rho$ tensors as functions of Fermi level and temperature, for models of strongly disordered Chern insulators. The formalism enabled us to converge the transport coefficients at temperatures low enough to enter the quantum critical regime at the Chern-to-trivial insulator transition. We find that the $\rho_{xx}$-curves at different temperatures intersect each other at one single critical point, and that they obey a single-parameter scaling law with an exponent close to the universally accepted value for the unitary symmetry class. However, when compared with the established experimental facts on the plateau-insulator transition in the Integer Quantum Hall Effect, we find a universal critical conductance $\sigma_{xx}^c$ twice as large, an ellipse rather than a semi-circle law, and absence of the quantized Hall insulator phase.Comment: 7 figures; minor typos eliminated; references adde

Topological representation of intuitionistic and distributive abstract logics

We continue work of our earlier paper (Lewitzka and Brunner: Minimally generated abstract logics, Logica Universalis 3(2), 2009), where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the category of intuitionistic abstract logics with stable logic maps as morphisms, and the category of implicative spectral spaces with spectral maps as morphisms. We show the equivalence of these categories and conclude that the larger categories of distributive abstract logics and distributive sober spaces are equivalent, too.Comment: 19 pages. The results of this article were presented in a session at the XVI. Brazilian Logic Conference EBL in Petr\'opolis, Brazil, in 201

Cationic vacancies as defects in honeycomb lattices with modular symmetries

Layered materials tend to exhibit a myriad of intriguing crystalline symmetries and topological characteristics based on their two dimensional (2D) geometries, dislocations, disclinations and defects. In this letter, we consider the diffusion dynamics of positively charged ions (cations) localized in honeycomb lattices within layered materials when an external electric field, non-trivial topologies, curvatures and cationic vacancies are present. The unit (primitive) cell of the honeycomb lattice is characterized by two generators, $J_1, J_2 \in \rm SL_2(\mathbb{Z})$ of modular symmetries in the special linear group with integer entries, corresponding to discrete re-scaling and rotations respectively. Moreover, applying a 2D conformal metric in an idealized model, we can consistently treat cationic vacancies as topological defects in an emergent manifold. The framework can be utilized to elucidate the molecular dynamics of the cations in exemplar honeycomb layered frameworks and the role of quantum geometry and topological defects in the diffusion processes.Comment: 7 pages, 5 figures, V2 corrections: Typo in the entries of the 1st generator ($J_1$) of modular symmetry corrected; proofrea

Strongly enhanced Berry dipole at topological phase transitions in BiTeI

Transitions between topologically distinct electronic states have been predicted in different classes of materials and observed in some. A major goal is the identification of measurable properties that directly expose the topological nature of such transitions. Here we focus on the giant-Rashba material bismuth tellurium iodine (BiTeI) which exhibits a pressure-driven phase transition between topological and trivial insulators in three-dimensions. We demonstrate that this transition, which proceeds through an intermediate Weyl semi-metallic state, is accompanied by a giant enhancement of the Berry curvature dipole which can be probed in transport and optoelectronic experiments. From first-principles calculations, we show that the Berrry-dipole --a vector along the polar axis of this material-- has opposite orientations in the trivial and topological insulating phases and peaks at the insulator-to-Weyl critical points, at which the nonlinear Hall conductivity can increase by over two orders of magnitude.Comment: As accepted in PR

What lattice theorists can do for superstring/M-theory

The gauge/gravity duality provides us with nonperturbative formulation of superstring/M-theory. Although inputs from gauge theory side are crucial for answering many deep questions associated with quantum gravitational aspects of superstring/M-theory, many of the important problems have evaded analytic approaches. For them, lattice gauge theory is the only hope at this moment. In this review I give a list of such problems, putting emphasis on problems within reach in a five-year span, including both Euclidean and real-time simulations.Comment: Contribution to IJMPA special issue "Lattice gauge theory beyond QCD". v2: References added, explanations have been improved in several places. And the title in the front page has been corrected. Sorry

Cell theory for glass-forming materials and jamming matter, combining free volume and cooperative rearranging regions

We investigate the statistical mechanics of glass-forming materials and jamming matter by means of a geometrically driven approach based on a revised cell theory. By considering the system as constituted of jammed blocks of increasing sizes, we obtain a unified picture that describes accurately the whole process from low densities to limit densities at the glass/jamming transition. The approach retrieves many of the aspects of existing theories unifying them into a coherent framework. In particular, at low densities we find a free volume regime, based on local relaxation process, at intermediate densities a cooperative length sets in, where both local and cooperative relaxation process are present. At even higher densities the increasing cooperative length suppresses the local relaxation and only the cooperative relaxation survives characterized by the divergence of the cooperative length, as suggested by the random first order theory. Finally a relation between the cooperative length and the hyperuniform length is also suggested.Comment: 7 pages, 1 figur

Rethinking the notion of oracle: A link between synthetic descriptive set theory and effective topos theory

We present three different perspectives of oracle. First, an oracle is a blackbox; second, an oracle is an endofunctor on the category of represented spaces; and third, an oracle is an operation on the object of truth-values. These three perspectives create a link between the three fields, computability theory, synthetic descriptive set theory, and effective topos theory