440 research outputs found

    Casimir effect with a helix torus boundary condition

    Full text link
    We use the generalized Chowla-Selberg formula to consider the Casimir effect of a scalar field with a helix torus boundary condition in the flat (D+1D+1)-dimensional spacetime. We obtain the exact results of the Casimir energy density and pressure for any DD for both massless and massive scalar fields. The numerical calculation indicates that once the topology of spacetime is fixed, the ratio of the sizes of the helix will be a decisive factor. There is a critical value rcritr_{crit} of the ratio rr of the lengths at which the pressure vanishes. The pressure changes from negative to positive as the ratio rr passes through rcritr_{crit} increasingly. In the massive case, we find the pressure tends to the result of massless field when the mass approaches zero. Furthermore, there is another critical ratio of the lengths rcritr_{crit}^{\prime} and the pressure is independent of the mass at r=rcritr=r_{crit}^{\prime} in the D=3 case.Comment: 11 pages, 3 figures, to be published in Mod. Phys. Lett.

    Betti number signatures of homogeneous Poisson point processes

    Full text link
    The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.Comment: Submitted to PRE. 11 pages, 10 figure

    Topology of the three-qubit space of entanglement types

    Full text link
    The three-qubit space of entanglement types is the orbit space of the local unitary action on the space of three-qubit pure states, and hence describes the types of entanglement that a system of three qubits can achieve. We show that this orbit space is homeomorphic to a certain subspace of R^6, which we describe completely. We give a topologically based classification of three-qubit entanglement types, and we argue that the nontrivial topology of the three-qubit space of entanglement types forbids the existence of standard states with the convenient properties of two-qubit standard states.Comment: 9 pages, 3 figures, v2 adds a referenc

    Skew Category Algebras Associated with Partially Defined Dynamical Systems

    Full text link
    We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor ss from a category GG to \Top^{\op} and show that it defines what we call a skew category algebra AσGA \rtimes^{\sigma} G. We study the connection between topological freeness of ss and, on the one hand, ideal properties of AσGA \rtimes^{\sigma} G and, on the other hand, maximal commutativity of AA in AσGA \rtimes^{\sigma} G. In particular, we show that if GG is a groupoid and for each e \in \ob(G) the group of all morphisms eee \rightarrow e is countable and the topological space s(e)s(e) is Tychonoff and Baire, then the following assertions are equivalent: (i) ss is topologically free; (ii) AA has the ideal intersection property, that is if II is a nonzero ideal of AσGA \rtimes^{\sigma} G, then IA{0}I \cap A \neq \{0\}; (iii) the ring AA is a maximal abelian complex subalgebra of AσGA \rtimes^{\sigma} G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.Comment: 16 pages. This article is an improvement of, and hereby a replacement for, version 1 (arXiv:1006.4776v1) entitled "Category Dynamical Systems and Skew Category Algebras

    Topological Modes in Dual Lattice Models

    Get PDF
    Lattice gauge theory with gauge group ZPZ_{P} is reconsidered in four dimensions on a simplicial complex KK. One finds that the dual theory, formulated on the dual block complex K^\hat{K}, contains topological modes which are in correspondence with the cohomology group H2(K^,ZP)H^{2}(\hat{K},Z_{P}), in addition to the usual dynamical link variables. This is a general phenomenon in all models with single plaquette based actions; the action of the dual theory becomes twisted with a field representing the above cohomology class. A similar observation is made about the dual version of the three dimensional Ising model. The importance of distinct topological sectors is confirmed numerically in the two dimensional Ising model where they are parameterized by H1(K^,Z2)H^{1}(\hat{K},Z_{2}).Comment: 10 pages, DIAS 94-3

    Inverse monoids and immersions of 2-complexes

    Get PDF
    It is well known that under mild conditions on a connected topological space X\mathcal X, connected covers of X\mathcal X may be classified via conjugacy classes of subgroups of the fundamental group of X\mathcal X. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion f:DCf : {\mathcal D} \rightarrow \mathcal C between CW-complexes is a cellular map such that each point yDy \in {\mathcal D} has a neighborhood UU that is mapped homeomorphically onto f(U)f(U) by ff. In order to classify immersions into a 2-dimensional CW-complex C\mathcal C, we need to replace the fundamental group of C\mathcal C by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex

    Restrictions on Transversal Encoded Quantum Gate Sets

    Full text link
    Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.Comment: 4 pages, v2: minor change

    On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients

    Full text link
    We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability μ\mu and the permittivity ε\varepsilon are symmetric positive definite matrix-valued functions in R3\mathbb{R}^{3}. We show that a unique continuation result for globally W1,W^{1,\infty} coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise W1,W^{1,\infty} with respect to a suitable countable collection of sub-domains with C0C^{0} boundaries. Such suitable collections include any bounded finite collection. The proof relies on a general argument, not specific to Maxwell's equations. This result is then extended to the case when within these sub-domains the permeability and permittivity are only LL^\infty in sets of small measure.Comment: 9 pages, 4 figure

    Excision for simplicial sheaves on the Stein site and Gromov's Oka principle

    Full text link
    A complex manifold XX satisfies the Oka-Grauert property if the inclusion \Cal O(S,X) \hookrightarrow \Cal C(S,X) is a weak equivalence for every Stein manifold SS, where the spaces of holomorphic and continuous maps from SS to XX are given the compact-open topology. Gromov's Oka principle states that if XX has a spray, then it has the Oka-Grauert property. The purpose of this paper is to investigate the Oka-Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka-Grauert property is equivalent to XX representing a finite homotopy sheaf on the Stein site. This expresses the Oka-Grauert property in purely holomorphic terms, without reference to continuous maps.Comment: Version 3 contains a few very minor improvement

    On Dijkgraaf-Witten Type Invariants

    Get PDF
    We explicitly construct a series of lattice models based upon the gauge group ZpZ_{p} which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-pp flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a 33-manifold and is based upon a single link, or 11-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher dimensional simplices.Comment: 18 page
    corecore