Tilings, tiling spaces and topology

To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical properties of the space (such as mixing, or the spectrum of the translation operator) are closely related to bulk properties of the individual tilings (such as the diffraction pattern). The topology of the space of tilings, particularly the Cech cohomology, gives information on how the original tiling can be deformed. Tiling spaces can be constructed as inverse limits of branched manifolds.Comment: 8 pages, including 2 figures, talk given at ICQ

Transport and Dissipation in Quantum Pumps

This paper is about adiabatic transport in quantum pumps. The notion of ``energy shift'', a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers necessarily vanish.Comment: 31 pages, 10 figure

Efficient Optimal Minimum Error Discrimination of Symmetric Quantum States

This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator $\rho_0$. It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator $\Pi_0$, and maximizing $\textrm{Tr}(\rho_0 \Pi_0)$. In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a semidefinite positive operator $X$ commuting with the symmetry operator and such that $X >= \rho_0$. The new formulation gives a deeper insight into the optimization problem and allows to obtain closed-form analytical solutions, as shown by a simple but not trivial explanatory example. Besides the theoretical interest, the result leads to semidefinite programming solutions of reduced complexity, allowing to extend the numerical performance evaluation to quantum communication systems modeled in Hilbert spaces of large dimension.Comment: 5 pages, 1 Table, no figure

Angioarchitectural evolution of clival dural arteriovenous fistulas in two patients.

Dural arteriovenous fistulas (dAVFs) may present in a variety of ways, including as carotid-cavernous sinus fistulas. The ophthalmologic sequelae of carotid-cavernous sinus fistulas are known and recognizable, but less commonly seen is the rare clival fistula. Clival dAVFs may have a variety of potential anatomical configurations but are defined by the involvement of the venous plexus just overlying the bony clivus. Here we present two cases of clival dAVFs that most likely evolved from carotid-cavernous sinus fistulas

Chern numbers and adiabatic transport in networks with leads

We study the Chern numbers associated with the adiabatic conductances of mesoscopic systems with leads. We describe the results of exact calculations of all the Chern numbers of several model networks. For a network with one lead we find the integer conductances 0,±1. For a network with three leads we find noninteger conductances bounded by 1

Topological Invariants in Fermi Systems with Time-Reversal Invariance

We discuss topological invariants for Fermi systems that have time-reversal invariance. The TKN^2 integers (first Chern numbers) are replaced by second Chern numbers, and Berry's phase becomes a unit quaternion, or equivalently an element of SU(2). The canonical example playing much the same role as spin ½ in a magnetic field is spin ½ in a quadrupole electric field. In particular, the associated bundles are nontrivial and have ± 1 second Chern number. The connection that governs the adiabatic evolution coincides with the symmetric SU(2) Yang-Mills instanton

The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps

The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. His method employs energy estimates which rely on a key coercivity property of the Hessian of the energy functional of the theory under consideration. In this paper we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact K\"ahler domain and target. We go on to prove a continuity property for our result, and show that, for the CP^1 model on S^2, the Hessian fails to be globally coercive in the degree 1 sector. We present numerical evidence which suggests that the Hessian is globally coercive in a certain equivariance class of the degree n sector for n>1. We also prove that, within the geodesic approximation, a single CP^1 lump moving on S^2 does not generically travel on a great circle.Comment: 29 pages, 1 figure; typos corrected, references added, expanded discussion of the main function spac