Theses supervised by Prof. Dr. Dündar Tekin Dereli
10 theses · Koç University
Generalizations of differential geometric structures on pre-Leibniz algebroids
Algebroids are mathematical structures that generalize the tangent bundle of a manifold. Hence, they can be considered as an appropriate mathematical framework for generalizations of differential geometric structures on a manifold. For example, metric-affine geometry can be constructed on a Lie algebroid of which the tangent bundle constitutes an example. On the other hand, generalized geometry can be written on a Courant algebroid. As differential geometric structures play a fundamental role in classical field theories, these algebroids are also crucial for physical purposes. For instance, general relativity can be expressed in terms of metric-affine geometric structures, whereas generalized geometry is closely related to the double field theory formulation of bosonic string theory. There are other algebroid structures such as metric algebroids, higher-Courant algebroids that are widely used in string and M theories. It is then natural to seek for a comprehensive structure that includes all of these algebroids as special cases. Local pre-Leibniz algebroids, and in particular anti-commutable pre-Leibniz algebroids, are such general ones. Differential geometry on these algebroids is of great interest, as they create a chance to work with all of these individual structures at once. In this thesis, various differential geometric structures on pre-Leibniz algebroids are studied. At the fundamental level, metric-connection geometries are constructed, and their properties are examined. On anti-commutable pre-Leibniz algebroids, for a class of connections that we call admissible, many relations that hold in the usual differential geometry are proven. These include Bianchi and Ricci identities, Cartan structure equations and the decomposition of a connection in terms of its torsion and non-metricity. Moreover, we introduce statistical, Hessian, conformal, projective and Weyl structures on pre-Leibniz algebroids, and prove some results analogous to the manifold setting.
Generalized theories of gravity and supergravity in three dimensions
Some generalizations inspired by the high and low energy modifications of general relativity have been studied in three dimensions. In particular modifications of massive gravity models have been considered. These generalizations have been studied using language of exterior algebra on Riemann-Cartan-Weyl space-times. All generalizations are defined in terms of a Lagrangian and variational field equations have been found using a first order variational formalism. Some exact solutions and properties of these models have been discussed.
Non-associative quantum mechanics and jordan algebras
We present a Jordan algebraic formulation of the non-commutative Landau problem coupled to a harmonic potential. To achieve this, we use a new formulation of the Hilbert space version of quantum mechanics. Using this construction, the Hilbert space corresponding to the non-commutative Landau problem is obtained. For that problem, non-commutative parameters are described in terms of an associator in the Jordan algebraic setting. Pure states and density matrices arising from this problem are characterized. This in turn leads us to the Jordan-Schrödinger time-evolution equation for the state vectors for this specific problem.
Crossing the singularity of a gravitational wave collision
A reformulation of general relativity inspired by the Belinski–Khalatnikov–Lifshitz conjecture had been introduced by Ashtekar, Henderson and Sloan which is based on variables closely related to the basic variables of loop quantum gravity, thereby providing a way of classically analyzing singularities that may be carried over to the quantum theory. It is reasonable to expect that these variables are regular at generic spacelike singularities. This has been shown on various examples—particularly, cosmological spacetimes. In this thesis we extend this analysis to the singularities of gravitational wave collision spacetimes, which are the result of the mutual focusing of the two waves. We focus on two specific examples and explicitly confirm that the said variables are regular at the singularity and can be smoothly continued beyond it.
Üçgen-sel optik ağlar üstünde Z2-topolojik yalıtkanlar ve kuantumlu spin Hall etkisi
In this thesis we study Z2 topological insulators and quantum spin Hall effect on triangular optical lattices. In the first part we examine the effect of a strong bichromatic deformation to the Z2 topological insulator in a fermionic ultracold atomic system on triangular lattices. We modify a system proposed by B. Beri and N. R. Cooper, Phys. Rev. Lett. 107, 145301 (2011) to a bichromatic optical lattice. Large insulating gap of this system allows for examination of strong perturbations. The calculations are done in the nearly free electron limit. We conclude that the Z2 topological character of the system is robust against large global perturbations that break the inversion symmetry but preserve the time-reversal symmetry. In the second part, we study the tight binding regime of the same system and compare the band structures and density of states of nearly free electron and tight binding limits. In the final part, we also propose a tight binding model for the quantum spin Hall system on triangular optical lattices. We show that the proposed Hamiltonian corresponding to 1/4 magnetic field flux unit can be realised by a time-periodic driving-like lattice shaking. We determine the edge state spectrum which contains gap traversing states as the hallmark of a Z2 topological insulator.
Sicim kuramlarından esinlenen bazı kütleçekim kuramlarının kozmolojik çözümleri
Some modified theories of gravity in four and higher dimensions that are motivated by the low energy effective string field theories are considered. The corresponding variational field equations are derived using the method of Lagrange multipliers in the language of exterior differential forms over the space-time manifold. Cosmological solutions are discussed, paying special attention to the anisotropy and the late-time acceleration of the expansion of our three dimensional space. Comments on recent observational data are made.
Büyük birleştirme E6 modelinde yüksüz fermiyonların kütle ve karışımları
The modern theory of elementary particle physics is based on the Standard Model and the symmetry groups in question are Lie groups as far as continous symmetries are considered. The strong, weak and electromagnetic interactions among the elementary particles are associated with the Lie algebras of SU(3)C, SU(2) and U(1) in the Standard Model. There are many reasons not to believe that the Standard Model is a final theory. New physics on which we will concentrate in this thesis is based on the idea of uni fication where three types of known gauge interactions are uni fied at the same very high energy scale by linking the three of the four forces in nature by combining strong and electroweak forces, called the Grand Uni fied Theory (GUT). We here ground on the exceptional E6 group for Grand Uni fication to give the masses and mixing of neutral leptons. We start by giving some algebraic foundations of exceptional groups including division algebras, octonions and quaternions. Then, we focus on E6 model where the fundamental representation 27 include the fermionic content of the model. Then a brief review of the assignment of elementary fermions and bosons to irreducible multiplets in E6 models is followed by a discussion of di fferent, hierarchical symmetry breaking chains from E6 down to SU(3)C x U(1)EM. We fi rst discuss the symmetry breaking pattern for the E6-> SO(10) ->SU(5) x U(2)EW x SU(3)C ->U(1)QED x SU(3)C chain that involves Georgi and Glashow's SU(5) Model. Next we concentrate on the symmetry breaking chain E6 -> SO(10) -> SU(2)L x SU(2)R x SU(4)C -> U(2)EW x SU(3)C -> U(1)EM x SU(3)C with an intermediate Pati-Salam symmetry for which (B - L) is conserved. In particular, the mass/mixing matrix of electrically neutral fermions (i.e.neutrinos) that would be derived from Yukawa couplings is constructed. The pattern of neutrino masses and some bounds on mixing parameters are discussed.
Analog kara delikler ve bose einstein yoğunlaşmasında tek bir girdaptan süper-ışıma ile enerji çıkarımı
Analog models of gravity, in this case, the acoustic black holes, are based on the mathematical correspondence between the propagation of sound waves in a certain media and the propagation of fields in curved spacetime. Since the direct observation of certain phenomena arising from curved space-time, such as the Hawking radiation and the superradiance, is highly impractical, analog models provide an accessible and controllable way to investigate them in the laboratory. Bose-Einstein condensate provides a convenient system for the implementation of this analogy. Indeed, the propagation of acoustic perturbations in the velocity potential of a Bose-Einstein condensate (BEC) behaves like minimally coupled massless scalar fields in a curved (2+1) dimensional Lorentzian space-time, where their propagation is governed by the Klein-Gordon wave equation. For linearized perturbations, this geometric picture can still apply in the presence of a single vortex state of the BEC. Based on this setting, this thesis work investigates the amplified scattering of axisymmetric perturbations from a vortex, as a manifestation of the acoustic superradiance. We first employ the widely used constant background density under the hydrodynamic approximation and conduct a comparative analysis of the time-domain and asymptotic frequency-domain analysis of acoustic superradiance as a function of the rotational speed of the vortex and the frequency of the impinging fluctuations. This shows that the solutions in both domains are in good agreement near the characteristic rotational speed of the single quantized vortex in its lowest energy state. However, for larger rotational speeds time-domain calculations, especially near the event horizon becomes unreliable, shown by the constraint equations introduced by the excision technique. While the simulations in frequency domain do not suffer from the numerical instabilities due to the coordinate transformations, allowing to increase the rotational velocity of the fluid. In addition, the formulation predicts an upper bound of the reflection coefficient against the rotational velocity of the vortex. The second part of the thesis examines the validity of the constant background density approximation by calculating a self-consistent density profile through the Gross-Pitaevskii equation. The resulting radial density profile around the vortex implies a radially varying speed of sound, which modifies the entire propagation dynamics as well as the loci of the event horizon and the ergosphere. The main conclusions of this part are that the self-consistent density profile remedies the overshoot of the superradiance dynamics temporally and that the spectral profile of the superradiance differs significantly in the vortex-scaled low frequency regime between the constant density and self-consistent density formulations. In this thesis, the superradiance phenomena for a single vortex in Bose Einstein condensate under the hydrodynamic approximation is analyzed, showing that the self-consistent density profile of the condensate improves on the constant background density approximation, particularly at the low frequency regime of acoustic perturbations. And the qualitative corrections in the transient behavior of the scattered wave's energy within the ergosphere justifies the radially varying density. The analysis could be extended for external confining potentials which may be more attainable in terms of the experimental aspects of the study. In reality, setting up a BEC vortex with a drain in (2+1)-dimension may be a difficult task to achieve, acoustic black holes in BEC are already observed for a 1D-flow within a steplike potential combined with a harmonic potential.
Hamilton-Jacobi formulation of the thermodynamics of Einstein-Born-Infeld-AdS Black Holes
Termodinamik için, tek boyutlu termodinamik faz uzayının kontakt yapısına dayalı bir Hamilton-Jacobi formalizmi Rajeev tarafından geliştirilmiş (Ann. Phys., 323 (2008) 2265) ve aynı çalışmada y üksüz, d önmeyen kara deliklere uygulanmıştır. Bu tezde aynı formalizm, Born-Infeld kuramını ve negatif bir kozmolojik sabiti kullanarak y üklü, d önmeyen kara deliklere uygulanmaktadır. Öncelikle uzayzamanın nedensel yapısının, kara deliklerin ve özellikle Reissner-Nordstr öm kara deliğinin temelleri g özden geçirilmektedir. Daha sonra y üzey yer çekimi ve korunumlu Komar y ükleri gibi kara deliklerin termodinamiğiyle ilgili kavramlar sunulmakta, kara delik mekaniğinin yasaları tartışılmaktadır. Ardından termodinamiğin kontakt geometrik form ülasyonu ve Rajeev tarafından geliştirilen formalizm g özden geçirilmektedir. Son olarak, Einstein-Born-Infeld kuramındaki y üklü, d önmeyen, asimptotik olarak AdS kara delik çözümü sunulmakta, Hamilton-Jacobi denklemi t üretilmekte ve en genel çözümü bulunmaktadır. Einstein-Born-Infeld-AdS kara deliğinin hal denkleminden farklı olan çözümler olduğu g österilmekte, bu çözümlerin ne anlama geldiği tartışılmaktadır.
Kiral mıknatıslar ve kuantum dolaşmasında iki katmanlı skyrmiyonlar
The antiferromagnetic coupling and entanglement between skyrmion lattices in magnetic bilayer systems are studied in thesis. We first formulate the problem of large bilayer skyrmions using $\mathbb{CP}^{1}\otimes \mathbb{CP}^{1}$ theory. We have considered bilayer skyrmions under the presence of Dzyaloshinskii-Moriya (DMI) and Zeeman interactions confined in a two-dimensional chiral magnet such as Fe$_{0.5}$Co$_{0.5}$Si. We parametrize the bilayer skyrmions using $SU(4)$ representation, and represent each skyrmion and antiskyrmion using Schmidt decomposition. Moreover, we computed the reduced density matrices for skyrmion and antiskyrmion. The conditions for maximal, partial entanglement and separable bilayer skyrmions are presented. We find intimate connection between bilayer skyrmions and $SU(4)$ skyrmions in multicomponent Hall systems and Graphene from entanglement perspective. Our results can be used for generating entanglement in systems with large number of spins.