16 research outputs found
Pattern-Equivariant Homology of Finite Local Complexity Patterns
This thesis establishes a generalised setting with which to unify the study
of finite local complexity (FLC) patterns. The abstract notion of a "pattern"
is introduced, which may be seen as an analogue of the space group of
isometries preserving a tiling but where, instead, one considers partial
isometries preserving portions of it. These inverse semigroups of partial
transformations are the suitable analogue of the space group for patterns with
FLC but few global symmetries. In a similar vein we introduce the notion of a
\emph{collage}, a system of equivalence relations on the ambient space of a
pattern, which we show is capable of generalising many constructions applicable
to the study of FLC tilings and Delone sets, such as the expression of the
tiling space as an inverse limit of approximants.
An invariant is constructed for our abstract patterns, the so called
pattern-equivariant (PE) homology. These homology groups are defined using
infinite singular chains on the ambient space of the pattern, although we show
that one may define cellular versions which are isomorphic under suitable
conditions. For FLC tilings these cellular PE chains are analogous to the PE
cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are
shown to be related through Poincar\'{e} duality.
An efficient and highly geometric method for the computation of the PE
homology groups for hierarchical tilings is presented. The rotationally
invariant PE homology groups are shown not to be a topological invariant for
the associated tiling space and seem to retain extra information about global
symmetries of tilings in the tiling space. We show how the PE homology groups
may be incorporated into a spectral sequence converging to the \v{C}ech
cohomology of the rigid hull of a tiling. These methods allow for a simple
computation of the \v{C}ech cohomology of the rigid hull of the Penrose
tilings.Comment: 159 pages, 8 figures, PhD thesi
Branes, graphs and singularities
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 341-354).In this thesis, we study various aspects of string theory on geometric and nongeometric backgrounds in the presence of branes. In the first part of the thesis, we study non-compact geometries. We introduce "brane tilings" which efficiently encode the gauge group, matter content and superpotential of various quiver gauge theories that arise as low-energy effective theories for D-branes probing singular non-compact Calabi-Yau spaces with toric symmetries. Brane tilings also offer a generalization of the AdS/CFT correspondence. A technique is developed which enables one to quickly compute the toric vacuum moduli space of the quiver gauge theory. The equivalence of this procedure and the earlier approach that used gauged linear sigma models is explicitly shown. As an application of brane tilings, four dimensional quiver gauge theories are constructed that are AdS/CFT dual to infinite families of Sasaki-Einstein spaces. Various checks of the correspondence are performed. We then develop a procedure that constructs the brane tiling for an arbitrary toric Calabi-Yau threefold. This solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. A different approach to the low-energy theory of D-branes uses exceptional collections of sheaves associated to the base of the threefold. We provide a dictionary that translates between the language of brane tilings and that of exceptional collections. Geometric compactifications represent only a very small subclass of the landscape: the generic vacua are non-geometric. In the second part of the thesis, we study perturbative compactifications of string theory that rely on a fibration structure of the extra dimensions. Non-geometric spaces preserving .A = 1 supersymmetry in four dimensions are obtained by using T-dualities as monodromies. Several examples are discussed, some of which admit an asymmetric orbifold description. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber.by David Vegh.Ph.D
Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases
Conformal field theory (CFT) has been extremely successful in describing large-scale universal effects in one-dimensional (1D) systems at quantum critical points. Unfortunately, its applicability in condensed matter physics has been limited to situations in which the bulk is uniform because CFT describes low-energy excitations around some energy scale, taken to be constant throughout the system. However, in many experimental contexts, such as quantum gases in trapping potentials and in several out-of-equilibrium situations, systems are strongly inhomogeneous. We show here that the powerful CFT methods can be extended to deal with such 1D situations, providing a few concrete examples for non-interacting Fermi gases. The system's inhomogeneity enters the field theory action through parameters that vary with position; in particular, the metric itself varies, resulting in a CFT in curved space. This approach allows us to derive exact formulas for entanglement entropies which were not known by other means
Duality and dynamics of supersymmetric field theories from D-branes on singularities
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.Includes bibliographical references (p. 359-373).We carry out various investigations regarding gauge theories on the worldvolume of D-branes probing toric singularities. We first study the connection that arises in Toric Duality between different dual gauge theory phases and the multiplicity of fields in the gauged linear sigma models associated with the probed geometries. We introduce a straightforward procedure for the determination of toric dual theories and partial resolutions based on the (p, q) web description of toric singularities. We study the non-conformal theories that arise in the presence of fractional branes. We introduce a systematic procedure to study the resulting cascading RG flows, including the effect of anomalous dimensions on beta functions. Supergravity solutions dual to logarithmic RG flows are constructed, validating the field theory analysis of the cascades. We systematically study the IR dynamics of cascading gauge theories. We show how the deformation in the dual geometries is encoded in a quantum modification of the moduli space. We construct an infinite family of superconformal quiver gauge theories which are AdS/CFT dual to Sasaki-Einstein horizons with explicit metrics. The gauge theory and geometric computations of R-charges and central charges are shown to agree. We introduce new Type IIB brane constructions denoted brane tilings which are dual to D3-branes probing arbitrary toric singularities. Brane tilings encode both the quiver and superpotential of the gauge theories on the D-brane probes. They give a connection with the statistical model of dimers.(cont.) They provide the simplest known method for computing toric moduli spaces of gauge theories, which reduces to finding the determinant of the Kasteleyn matrix of a bipartite graph.by Sebastián Federico Franco.Ph.D
Frustrierter Quantenmagnetismus und korrelierte Kondo Systeme
This work investigates in detail the physical properties of the frustrated quantum Heisenberg model on the square lattice with spatial anisotropy, together with the correlated Kondo lattice model. We have used the linear spin-wave theory, to determine the classical ground-state as well as the effects of quantum fluctuations. The phase diagram of the system has been obtained, and the spin-wave spectra as well as the behavior of the magnetization and the ordered moment in the presence of magnetic field is studied. In addition, numerical exact diagonalization technique has been used to obtain the ground-state as well as the finite-temperature properties. Ground-state energy, as well as spin correlation functions and static structure factors are calculated using the Lanczos algorithm. A detailed finite-size scaling analysis of the ground-state properties is carried out, and a new method for selecting most compatible tilings of the infinite lattice, and the way to construct those is described thoroughly. We have shown that with choosing the most square tiles, we can successfully construct a stable finite-size scaling analysis in the ordered regions of the phase diagram. Besides, we have demonstrated the strong effect of frustration over field dependence of the ordered moment, and used this behavior to propose a new method to determine the frustration ratio in this and similar compounds. Moreover, we investigated the correlated Kondo lattice model, which is a paradigm for the competition of singlet formation and magnetic order. The model is introduced and its ground-state as well as finite-temperature properties are obtained. The dependence of the Kondo temperature scale over the Coulomb repulsion U is examined. We report a new nonmonotonic dependence of the local moment on the correlation strength U. We also show that the Kondo temperature scale increases with U, resolving an existing controversy on this subject.Die vorliegende Doktorarbeit untersucht ausführlich die physikalischen Eigenschaften des frustrierten Heisenberg Modells für Quantenspins auf dem quadratischen Gitter einschliesslich der räumlichen Anisotropie, sowie das korrelierte Kondogitter Modell. Der klassische Grundzustand und die Effekte der Quantenfluktuationen im Heisenberg Modell wurden mit Hilfe der analytischen linearen Spinwellentheorie untersucht. Das Phasendiagramm des Systems wurde bestimmt und die Spinwellenspektren sowie das Verhalten der Magnetisierung und des geordneten magnetischen Moments in Anwesenheit eines Magnetfeldes untersucht. Darüber hinaus wurde die numerische Methode der exakten Diagonalisierung verwendet, um den Grundzustand sowie die Eigenschaften bei endlichen Temperaturen zu erhalten. Grundzustandsenergie sowohl als Spin- Korrelationsfunktionen und der statische Strukturfaktor werden mit Hilfe des Lanczos- Algorithmus berechnet. Eine detaillierte Analyse der Eigenschaften des Grundzustands mit der Finite- Size- Scaling Methode wird durchgeführt. Ausserdem wird eine neue Methode zur Auswahl der am besten geeigneten Parkettierungen des unendlichen Gitters sowie seine Konstruktion gründlich beschrieben. Wir haben gezeigt, dass mit der Auswahl der maximal quadratischen Parkettierung eine stabile Finite-Size-Scaling-Analyse für die geordneten Bereiche des Phasendiagramms durchgeführt werden kann. Ausserdem haben wir einen spürbar starken Einfluss der Frustration auf die Magnetfeldabhängigkeit des geordneten magnetischen Momentes demonstriert. Dieses Verhalten wird verwendet, um eine neue Methode vorzuschlagen die geeignet ist, den Frustrationsgrad in dieser und ähnlichen Verbindungen zu bestimmen. Weiterhin haben wir das korrelierte Kondogittermodell untersucht. Das Modell wird eingeführt und die Eigenschaften seines Grundzustands sowie bei endlichen Temperaturen berechnet
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Quasicrystalline optical lattices for ultracold atoms
Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth
of intriguing physical phenomena, such as the inheritance of topological properties from higher
dimensions, self-similarity, and the presence of non-trivial structure on all scales. The concept of
aperiodic order has been extensively studied in mathematics and geometry, exemplified by the
celebrated Penrose tiling. However, the understanding of physical quasicrystals (the vast majority
of them are intermetallic compounds) is still incomplete owing to their complexity, regarding
both growth processes and stability.
Ultracold atoms in optical lattices offer an ideal, yet untested environment for investigating
quasicrystals. Optical lattices, i.e. standing waves of light, allow the defect-free formation of a
large variety of potential landscapes, including quasiperiodic geometries. In recent years, optical
lattices have become one of the most successful tools in the large-scale quantum simulation of
condensed-matter problems.
This study presents the first experimental realisation of a two-dimensional quasicrystalline potential
for ultracold atoms, based on an eightfold symmetric optical lattice. It is aimed at bringing
together the fields of ultracold atoms and quasicrystals – and the more general concept of
aperiodic order. The first part of this thesis introduces the theoretical aspects of aperiodic order
and quasicrystalline structure. The second part comprises a detailed account of the newly designed
apparatus that has been used to produce quantum-degenerate gases in quasicrystalline
lattices. The third and final part summarises the matter-wave diffraction experiments that have
been performed in various lattice geometries. These include one- and two-dimensional simple
cubic lattices, one-dimensional quasiperiodic lattices, as well as two-dimensional quasicrystalline
lattices. The striking self-similarity of this quasicrystalline structure has been directly
observed, in close analogy to Shechtman’s very first discovery of quasicrystals using electron
diffraction. In addition, an in-depth study of the diffraction dynamics reveals the fundamental
differences between periodic and quasicrystalline lattices, in excellent agreement with ab initio
theory. The diffraction dynamics on short timescales constitutes a continuous-time quantum
walk on a homogeneous four-dimensional tight-binding lattice.
On the one hand, these measurements establish a novel experimental platform for investigating
quasicrystals proper. On the other hand, ultracold atoms in quasicrystalline optical lattices are
worth studying in their own right: Possible avenues include the observation many-body localisation
and Bose glasses, as well as the creation of topologically non-trivial systems in higher
dimensions.Funded in part by a Vice-Chancellor's Award from the Cambridge Commonwealth, European, and International Trus