Symbolic computation of exact solutions for fractional differential-difference equation models

The aim of the present study is to extend the G'/G-expansion method to fractional differential-difference equations of rational type. Particular time-fractional models are considered to show the strength of the method. Three types of exact solutions are observed: hyperbolic, trigonometric and rational. Exact solutions in terms of topological solitons and singular periodic functions are also obtained. As far as we are aware, our results have not been published elsewhere previously

The Ablowitz-Ladik lattice system by means of the extended (G′ / G)-expansion method

We analyzed the Ablowitz-Ladik lattice system by using the extended (G′ / G)-expansion method. Further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained. We observed that some previously known results can be recovered by assigning special values to the arbitrary parameters. © 2010 Elsevier Inc. All rights reserved

Thermodynamics and Universality in Anisotropic Higher Curvature Spacetimes

In my thesis, I describe new results in the thermodynamics of black holes in two gravitational scenarios: spacetime anisotropy and higher curvature gravity. I focus on classifying the critical point of "Large Black Hole / Small Black Hole" phase transitions in higher curvature gravity in various dimensions, for both numerical and analytic black hole solutions. Special emphasis will be placed on five-dimensional cubic and quartic quasitopological gravity. I cover the motivation and document a number of higher curvature black hole solutions as well as the thermodynamic behaviour of these black holes when they are asymptotically Lifshitz symmetric (a form of anisotropy). I describe the methodology used to construct the set of thermodynamic potentials for black holes with general asymptotics from a collection of well-justified conjectures, followed by the development of procedures to numerically and analytically determine unknown quantities such as mass and thermodynamic volume from these conjectures. I will complete this thesis by extracting the critical exponents and thereby finding the universality class of the critical behaviour for a number of black hole solutions. This work has implications for the study of the gauge/gravity duality as well as for the dynamical behaviour of black holes

Numerical Studies of Superconductivity and Charge-Density-Waves: Progress on the 2D Holstein Model and a Superconductor-Metal Bilayer

The problem of superconductivity has been central in many areas of condensed matter physics for over 100 years. Despite this long history, there is still no theory capable of describing both conventional and unconventional superconductors. Recent experimental observations such as the dilute superconductivity in SrTiO3 and near room-temperature superconductivity in hydride compounds under extreme pressure have renewed interest in electron-phonon systems. Adding to this is evidence that electron-phonon coupling may play a supporting role in unconventional systems like the cuprates and monolayer FeSe on SrTiO3. One way to make sense of these observations is to construct simple models that capture the essential physics. Among the models with electron-phonon interactions, the simplest and most studied is the two-dimensional Holstein model. It describes a single band of electrons that hop between sites on a square lattice and interact with atomic oscillators by coupling linearly to their displacements. This model gives rise to superconductivity and charge-density-wave order spanning different regions of doping. Surprisingly, even this model is not entirely understood. First, we present a comprehensive study of the Holstein model phase diagram using self-consistent many-body perturbation theory. We then discuss one potential avenue for accelerating non-perturbative quantum Monte Carlo simulations of electron-phonon models using artificial neural networks. Following these topics, we wrap up the electron-phonon-related part by discussing the importance of nonlinear interaction terms and moving beyond the Holstein model. The last problem of this dissertation revisits a proposal by Steve Kivelson. He hypothesized and later showed that coupling a superconductor with a large pairing scale but low phase stiffness to a metal raises the transition temperature (Tc). Expanding on previous work, we studied a more general case with a 2D negative-U Hubbard model coupled with a metallic layer via single-particle tunneling. Here, we use the dynamical cluster approximation to estimate Tc, finding it is maximal for finite tunneling values, thereby confirming Kivelson’s hypothesis in the general case. Collectively, the results in this dissertation shed new light on superconductivity in conventional systems and demonstrate a need to incorporate more aspects of real materials into models

Collective quantum effects in field theory and gravity

Collective quantum effects have traditionally not received much attention in high energy physics. Recently, however, a model for black hole physics was put forward, in which black holes are described as Bose-condensates of gravitons close to a critical point. In a different line of research, estimates of high-energy collisions in the electro-weak theory have hinted that scattering processes with multiple Higgs or vector Bosons in the final state might be in reach for future particle colliders. In both scenarios, collective quantum effects may be crucial for understanding the physics. In the first part of this thesis, we address the black hole condensate picture of Dvali and Gomez. We study a Bosonic many-body system (attractive Lieb-Liniger) which exhibits a quantum phase transition and was proposed as a model for the graviton condensate. We demonstrate that, even for macroscopic particle number, quantum effects are prominent at the critical point. This becomes especially clear in the entanglement of different momentum modes and in the quantum discord between two successive density measurements. We point out that the leading contribution to these phenomena arises from long-wavelength modes and is therefore insensitive to ultra-violet physics. For black holes in the graviton condensate picture, these findings imply a breakdown of the semiclassical description and may be the key to resolving the long-standing information problem. We then turn our attention to the question of information processing in black holes. Inspired by the properties of three-dimensional attractive Bose condensates, we propose a concrete mechanism for fast scrambling in graviton-condensate black holes. To bolster our claims, we perform simulations of the Lieb-Liniger model in an appropriate regime that reveal entanglement-generation in logarithmic time. We also point out that the idea of instability and possibly chaos as the origin of fast quantum breaking and scrambling may also be relevant for other models of black holes. In the second part of this thesis, we use techniques of integrability (Bethe ansatz) to address the phase transition of the attractive Lieb-Liniger model analytically. We derive the continuum limit of the Bethe equations and solve it for the ground state at arbitrary coupling. We establish an exact equivalence between the Bethe-ansatz description in the large-particle-number limit and the large-N saddle point of Euclidean two dimensional U(N) Yang-Mills theory quantized on a sphere. The transition between the homogeneous and solitonic phases of the Lieb-Liniger model is thus dual to the Douglas-Kazakov confinement-deconfinement transition. In the last part, we consider scattering amplitudes involving many particles. In a simple integral-model, we study in detail the breakdown of perturbation theory and emphasize that the pure tree-level approximation fails earlier, parametrically. We then demonstrate, in three different (integral and quantum mechanical) model systems, that the physical high multiplicity amplitudes can be predicted on the basis of leading-order information from (non-perturbative) saddle points. In the non-Borel summable cases, one non-perturbative saddle contribution alone dominates the amplitudes. We highlight that high-multiplicity amplitudes may thus be a fruitful application for the methods of resurgence theory.Kollektive Quanteneffekte haben traditionell keine große Aufmerksamkeit in der Hochenergiephysik erfahren. Vor kurzem ist jedoch ein Modell für die Physik schwarzer Löcher vorgeschlagen worden, in dem diese als Bose-Kondensate von Gravitonen nahe an einem kritischen Punkt beschrieben werden. In einer anderen Forschungsrichtung haben Abschätzungen der Hochenergiekollisionen in der elektroschwachen Theorie Hinweise darauf geliefert, dass Streuprozesse mit mehreren Higgs- oder Vektorbosonen im Endzustand in Reichweite künftiger Teilchenbeschleuniger sein könnten. In beiden Fällen dürften kollektive Quanteneffekte zentral für das Verständnis der Physik sein. Im ersten Teil dieser Arbeit behandeln wir das Gravitonkondensat-Bild für schwarze Löcher von Dvali und Gomez. Wir untersuchen ein Bosonisches Vielteilchensystem (attraktives Lieb-Liniger), das einen Quantenphasenübergang zeigt und als Modell für Gravitonkondensate vorgeschlagen worden ist. Wir zeigen, dass - selbst für makroskopische Teilchenzahlen - Quanteneffekte am kritischen Punkt wichtig sind. Das wird an der Verschränkung unterschiedlicher Impulsmoden und dem Quantenmissklang zwischen zwei aufeinanderfolgenden Dichtemessungen besonders klar. Wir heben hervor, dass der führende Beitrag zu diesen Phänomenen aus langwelligen Moden hervorgeht und daher von der ultravioletten Physik unabhängig ist. Diese Ergebnisse implizieren für schwarze Löcher im Gravitonkondensat-Bild, dass die semiklassische Beschreibung zusammenbricht, und sie könnten der Schlüssel dazu sein, das lange bestehende Informationsproblem zu lösen. Dann wenden wir uns der Frage der Informationsverarbeitung in schwarzen Löchern zu. Inspiriert von den Eigenschaften dreidimensionaler attraktiver Bose-Kondensate schlagen wir einen konkreten Mechanismus für das schnelle Scrambling in Gravitonkondensat-schwarzen-Löchern vor. Um diese Behauptung zu stützen führen wir Simulationen am Lieb-Liniger Modell in einem geeigneten Regime durch, die Verschränkungs-Erzeugung in logarithmischer Zeit offenbaren. Wir weisen auch darauf hin, dass die Idee, Instabilität und gegebenenfalls Chaos als Ursache für schnelles Quantenbrechen und Scrambling zu betrachten, relevant für andere Modelle von schwarzen Löchern sein kann. Im zweiten Teil dieser Arbeit verwenden wir Integrabilitäts-Techniken (Bethe-Ansatz) um den Phasenübergang des attraktiven Lieb-Liniger Modells analytisch zu analysieren. Wir leiten den Kontinuumslimes der Bethe-Gleichungen her und lösen ihn für den Grundzustand bei beliebiger Kopplungsstärke. Wir stellen eine genaue Äquivalenz zwischen der Bethe-Ansatz Beschreibung im Vielteilchen-Limes und dem groß-N Sattelpunkt von Euklidischer zweidimensionaler U(N) Yang-Mills Theorie, auf der Sphäre quantisiert, her. Der Übergang zwischen der homogenen und solitonischen Phase des Lieb-Liniger Modells ist dadurch dual zum Douglas-Kazakov Übergang zwischen Confinement und Deconfinement. Im letzten Teil widmen wir uns Streuamplituden von vielen Teilchen. In einem einfachen Integralmodell untersuchen wir im Detail den Zusammenbruch der Störungstheorie und betonen, dass reine Baum-Näherungen parametrisch noch früher versagen. Wir demonstrieren dann, dass sich die Streuamplituden hoher Multiplizität, in drei verschiedenen (Integral- und quantenmechanischen) Modellsystemen, auf Basis der führenden Ordnung von (nicht-perturbativen) Sattelpunkten, vorhersagen lassen. In den nicht Borel-summierbaren Fällen dominiert allein der Beitrag eines nicht-perturbativen Sattelpunkts. Wir zeigen auf, dass die Amplituden hoher Multiplizität daher wohl eine lohnenswerte Anwendung für die Techniken der Resurgenztheorie sind.Deutsche Übersetzung des Titels: Kollektive Quanteneffekte in Feldtheorie und Gravitatio

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed