440 research outputs found

### Casimir effect with a helix torus boundary condition

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
($D+1$)-dimensional spacetime.
We obtain the exact results of the Casimir energy density and pressure for
any $D$ 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 $r_{crit}$ of
the ratio $r$ of the lengths at which the pressure vanishes. The pressure
changes from negative to positive as the ratio $r$ passes through $r_{crit}$
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 $r_{crit}^{\prime}$ and the pressure is
independent of the mass at $r=r_{crit}^{\prime}$ 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

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

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

We introduce partially defined dynamical systems defined on a topological
space. To each such system we associate a functor $s$ from a category $G$ to
\Top^{\op} and show that it defines what we call a skew category algebra $A
\rtimes^{\sigma} G$. We study the connection between topological freeness of
$s$ and, on the one hand, ideal properties of $A \rtimes^{\sigma} G$ and, on
the other hand, maximal commutativity of $A$ in $A \rtimes^{\sigma} G$. In
particular, we show that if $G$ is a groupoid and for each e \in \ob(G) the
group of all morphisms $e \rightarrow e$ is countable and the topological space
$s(e)$ is Tychonoff and Baire, then the following assertions are equivalent:
(i) $s$ is topologically free; (ii) $A$ has the ideal intersection property,
that is if $I$ is a nonzero ideal of $A \rtimes^{\sigma} G$, then $I \cap A
\neq \{0\}$; (iii) the ring $A$ is a maximal abelian complex subalgebra of $A
\rtimes^{\sigma} G$. 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

Lattice gauge theory with gauge group $Z_{P}$ is reconsidered in four
dimensions on a simplicial complex $K$. One finds that the dual theory,
formulated on the dual block complex $\hat{K}$, contains topological modes
which are in correspondence with the cohomology group $H^{2}(\hat{K},Z_{P})$,
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
$H^{1}(\hat{K},Z_{2})$.Comment: 10 pages, DIAS 94-3

### Inverse monoids and immersions of 2-complexes

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

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

We are interested in the uniqueness of solutions to Maxwell's equations when
the magnetic permeability $\mu$ and the permittivity $\varepsilon$ are
symmetric positive definite matrix-valued functions in $\mathbb{R}^{3}$. We
show that a unique continuation result for globally $W^{1,\infty}$ coefficients
in a smooth, bounded domain, allows one to prove that the solution is unique in
the case of coefficients which are piecewise $W^{1,\infty}$ with respect to a
suitable countable collection of sub-domains with $C^{0}$ 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 $L^\infty$ in sets of small measure.Comment: 9 pages, 4 figure

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

A complex manifold $X$ 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 $S$, where the spaces of holomorphic and continuous maps from $S$ to
$X$ are given the compact-open topology. Gromov's Oka principle states that if
$X$ 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 $X$ 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

We explicitly construct a series of lattice models based upon the gauge group
$Z_{p}$ 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-$p$ flatness condition. The simplest model of this type yields the
Dijkgraaf-Witten invariant of a $3$-manifold and is based upon a single link,
or $1$-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

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