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

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

2+1 Dimensional Georgi-Glashow Instantons in Weyl Gauge

Semiclassical instanton solutions in the 3D SU(2) Georgi-Glashow model are transformed into the Weyl gauge. This illustrates the tunneling interpretation of these instantons and provides a smooth regularization of the singular unitary gauge. The 3D Georgi-Glashow model has both instanton and sphaleron solutions, in contrast to 3D Yang-Mills theory which has neither, and 4D Yang-Mills theory which has instantons but no sphaleron, and 4D electroweak theory which has a sphaleron but no instantons. We also discuss the spectral flow picture of fundamental fermions in a Georgi-Glashow instanton background.Comment: 22 pages, 8 figures, revtex4; v2 - references and comments adde

Fredholm Indices and the Phase Diagram of Quantum Hall Systems

The quantized Hall conductance in a plateau is related to the index of a Fredholm operator. In this paper we describe the generic ``phase diagram'' of Fredholm indices associated with bounded and Toeplitz operators. We discuss the possible relevance of our results to the phase diagram of disordered integer quantum Hall systems.Comment: 25 pages, including 7 embedded figures. The mathematical content of this paper is similar to our previous paper math-ph/0003003, but the physical analysis is ne

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 geometry of entanglement: metrics, connections and the geometric phase

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.Comment: 42 page

Topological Phases near a Triple Degeneracy

We study the pattern of three state topological phases that appear in systems with real Hamiltonians and wave functions. We give a simple geometric construction for representing these phases. We then apply our results to understand previous work on three state phases. We point out that the ``mirror symmetry'' of wave functions noticed in microwave experiments can be simply understood in our framework.Comment: 4 pages, 1 figure, to appear in Phys. Rev. Let

Tiling groupoids and Bratteli diagrams

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths dB is homeomorphic to X. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an etale equivalence relation R_B on dB which is homeomorphic to R_X, and contains the AF-relation of "tail equivalence".Comment: 34 pages, 4 figure

Cellular and humoral immune responses and protection against schistosomes induced by a radiation-attenuated vaccine in chimpanzees

The radiation-attenuated Schistosoma mansoni vaccine is highly effective in rodents and primates but has never been tested in humans, primarily for safety reasons. To strengthen its status as a paradigm for a human recombinant antigen vaccine, we have undertaken a small-scale vaccination and challenge experiment in chimpanzees (Pan troglodytes). Immunological, clinical, and parasitological parameters were measured in three animals after multiple vaccinations, together with three controls, during the acute and chronic stages of challenge infection up to chemotherapeutic cure. Vaccination induced a strong in vitro proliferative response and early gamma interferon production, but type 2 cytokines were dominant by the time of challenge. The controls showed little response to challenge infection before the acute stage of the disease, initiated by egg deposition. In contrast, the responses of vaccinated animals were muted throughout the challenge period. Vaccination also induced parasite-specific immunoglobulin M (IgM) and IgG, which reached high levels at the time of challenge, while in control animals levels did not rise markedly before egg deposition. The protective effects of vaccination were manifested as an amelioration of acute disease and overall morbidity, revealed by differences in gamma-glutamyl transferase level, leukocytosis, eosinophilia, and hematocrit. Moreover, vaccinated chimpanzees had a 46% lower level of circulating cathodic antigen and a 38% reduction in fecal egg output, compared to controls, during the chronic phase of infection

Limit theorems for self-similar tilings

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.Comment: 36 pages; some corrections and improved exposition, especially in Section 4; references adde