A cubic lattice with random parameters is reduced to a linear chain by the means of the projection technique. The continued fraction expansion (c.f.e.) approach is herein applied to the density of states. Coefficients of the c.f.e. are obtained numerically by the recursion procedure. Properties of the non-stationary second moments (correlations and dispersions) of their distribution are studied in a connection with the other evidences of transport in a one-dimensional Mori chain. The second moments and the spectral density are computed for the various degrees of disorder in the prototype lattice. The possible directions of the further development are outlined. The physical problem that is addressed in the dissertation is the possibility of the existence of a non-Anderson disorder of a specific type. More precisely, this type of a disorder in the one-dimensional case would result in a positive localization threshold. A specific type of such non-Anderson disorder was obtained by adopting a transformation procedure which assigns to the matrix expressing the physics of the multidimensional crystal a tridiagonal Hamiltonian. This Hamiltonian is then assigned to an equivalent one-dimensional tight-binding model. One of the benefits of this approach is that we are guaranteed to obtain a linear crystal with a positive localization threshold. The reason for this is the existence of a threshold in a prototype sample. The resulting linear model is found to be characterized by a correlated and a nonstationary disorder. The existence of such special disorder is associated with the absence of Anderson localization in specially constructed one-dimensional lattices, when the noise intensity is below the non-zero critical value. This work is an important step towards isolating the general properties of a non-Anderson noise. This gives a basis for understanding of the insulator to metal transition in a linear crystal with a subcritical noise.

This cumulative dissertation is based on the publications [P1–P6], covering various aspects in theoretical studies of isolated quantum many-body systems. The transport and relaxation dynamics in quantum lattice models are studied with a particular focus on (i) the effect of a mass imbalance between different particles on their relaxation dynamics as well as (ii) the influence of generic perturbations on different reference dynamics. As for (i), the dynamics of two mutually interacting fermionic particle species on a lattice are investigated for different mass ratios between the two species [P4]. Numerical studies of density dynamics show that diffusive transport which is expected for small mass imbalances persists also for moderate imbalances and becomes anomalous for stronger imbalances. On the other hand, while transport is suppressed in the limit of infinite imbalance, i.e., if one particle species is immobile, this effective localization is shown to give way to anomalous diffusion as soon as the heavy particle species gains a finite mobility. Regarding (ii), the effect of perturbations on dynamics is investigated from the perspective of projection-operator techniques [P6]. As a main result, it is demonstrated that simple exponential damping, which is expected in the overwhelming majority of cases, may only occur for the density matrix in the interaction picture. Within this approach, this simple damping carries over to the time dependence of standard correlation functions only in certain cases. In particular, the possibility of nontrivial damping in physically relevant perturbation scenarios is discussed. A considerable portion of this work is concerned with the implementation of powerful numerical and (semi-)analytical tools to overcome the enhanced computational complexity in numerical studies of quantum many-body systems. This includes the concept of dynamical quantum typicality [P2, P3], numerical linked-cluster expansions [P5], and projection-operator techniques, as well as the combined use of available symmetries [P1].

>Magister Scientiae - MSc / Tin dioxide (SnO2) thin films are a worthy candidate for an electron transport layer (ETL) in perovskite solar cells, due to its suitable energy level, high electron mobility of 240 cm2 v-1 s- 1, desirable band gap of 3.6 - 4.0 eV, and ultimately proves to be suited for a low temperature thermal oxidation technique for ETL production. A variety of methods are available to prepare SnO2 thin films such as spin and dip coating and chemical bath deposition. However, the customary solid-state method, which incorporates thermal decomposition and oxidation of a metallic Sn precursor compound in an oxygen abundant atmosphere prevails to be low in cost, is repeatable and allows for large-scale processing.

In this thesis I present contributions to the development, analysis and application of tensor network state methods for numerically exact quantum dynamics in one and two-dimensional lattice systems. The setting of numerically exact quantum dynamics is introduced in Chapter 2. This includes a discussion of exact diagonalization approaches and massively parallel implementations thereof as well as a brief introduction of tensor network states. In Chapter 3, I perform a detailed analysis of the performance of n-ary tree tensor network states for simulating the dynamics of two-dimensional lattices. This constitutes the first application of this class of tensor network to dynamics in two spatial dimensions, a long-standing challenge, and the method is found to perform on par with existing state-of-the-art approaches. Chapter 4 showcases the efficacy of a novel tensor network format I developed, tailored to electron-phonon coupled problems in their single-electron sector, through an application to the Holstein model. The applicability of the approach to a broad range of parameters of the model allows to reveal the strong influence of the spread of the electron distribution on the initial state of the phonons at the site where the electron is introduced, for which a simple physical picture is offered. I depart from method development in Chapter 5 and analyse the prospects of using tensor network states evolved using the time-dependent variational principle as an approximate approach to determine asymptotic transport properties with a finite, moderate computational effort. The method is shown to not yield the correct asymptotics in a clean, non-integrable system and can thus not be expected to work in generic systems, outside of finely tuned parameter regimes of certain models. Chapters 6 and 7 are concerned with studies of spin transport in long-range interacting systems using tensor network state methods. For the clean case, discussed in Chapter 6, we find that for sufficiently short-ranged interactions, the spreading of the bulk of the excitation is diffusive and thus dominated by the local part of the interaction, while the tail of the excitation decays with a powerlaw that is twice as large as the powerlaw of the interaction. Similarly, in the disordered case, analysed in Chapter 7, we find subdiffusive transport of spin and sub-linear growth of entanglement entropy. This behaviour is in agreement with the behaviour of systems with local interactions at intermediate disorder strength, but provides evidence against the phenomelogical Griffith picture of rare, strongly disordered insulating regions. We generalize the latter to long-ranged interactions and show that it predicts to diffusion, in contrast to the local case where it results in subdiffusive behaviour.

The two neutral kaon states in nature, the 𝘒_𝐿 (long-lived) and 𝘒_s (short-lived) mesons, are the two time-evolution eigenstates of the 𝘒⁰ - 𝘒̅⁰̅ mixing system. The prediction of their mass difference 𝚫m_𝘒 based on the standard model is an important goal of lattice QCD. Non-perturbative formalism has been developed to calculate 𝚫 m_𝘒 and the calculation has been extended from the first exploratory calculation with only connected diagrams to full calculations on near-physical[1] and physical ensembles[2]. In this work, we extend the calculation described in Reference [2] from 59 to 152 configurations and present a new analysis method employed to calculate 𝚫 m_𝘒 with better reduction of statistical error on this larger set of configurations. By using a free-field calculation, we will show that the four-point contractions in our calculation method yields results consistent with the Inami-Lim calculation[3] in the local limit. We also report a series of scaling tests performed on 24³ × 64 and 32³ × 64 lattice ensembles to estimate the size of the finite lattice spacing error in our 𝚫 m_K$ calculation. We will present the 𝚫 m_𝘒 calculation on the ensemble of 64³ × 128 gauge configurations with inverse lattice spacing of 2.36 GeV and physical quark masses obtaining results coming from 2.5 times the Monte Carlo statistics used for the result in [2]. With the new analysis method and estimated finite lattice spacing error, we obtain 𝚫 m_𝘒 = 5.8(0.6)_stat(2.3)_sys × 10¯¹²MeV. Here the first error is statistical and the second is an estimate of largest systematic error due to the finite lattice spacing effects. The new results also imply the validity of the OZI rule for the case of physical kinematics in contrast to the previous calculation of 𝚫 m_𝘒 with unphysical kinematics[1], where contributions from diagrams with disconnected parts are almost half the size of the contributions from fully connected diagrams but with the opposite sign.

A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, restricted multisets, and lattice paths. For subsets, we show that a u-cycle exists for the κ-subsets of an n-set if we let κ vary in a non zero length interval. We use this result to construct a "covering" of length (1+o(1))(n/κ) for all subsets of [n] of size exactly κ with a specific formula for the o(1) term. We also show that u-cycles exist for all n-length words over some alphabet ∑, which contain all characters from R ⊂ ∑. Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.

In this thesis, the problem of motion planning under uncertainty is explored. Motion planning under uncertainty is important since even with noise during the execution of the plan, it is desirable to keep the collision risk low. However, for the motion planning to be useful it needs to be possible to perform it in a reasonable time. The introduction of state uncertainty leads to a substantial increase in search time due to the additional dimensions it adds to the search space. In order to alleviate this problem, different approaches to pruning of the search space are explored. The initial approach is to prune states based on having strictly worse uncertainty and path cost than other found states. Having performed this initial pruning, an alternate approach to comparing uncertainties is examined in order to explore if it is possible to achieve a lower search time. The approach taken in order to lower the search time further is to discretize the covariance of a state by using a number of buckets. However, this discretization results in giving up the completeness and optimality of the algorithm. Having implemented these different ways of pruning, their performance is tested on a number of different scenarios. This is done by evaluating the planner using the pruning in several different scenarios including uncertainty and one without uncertainty. It is found that all of the pruning approaches reduce the overall search time compared to when no additional pruning based on the uncertainty is done. Additionally, it is indicated that the bucket-based approach reduce the search time to a greater extent than the strict pruning approach. Furthermore, the extensions made results in no increase in cost or a very small increase in cost for the explored scenarios. Based on these results, it is likely that the bucket pruning approach has some potential. However more studies, particularly with additional scenarios, needs to be made before any definitive conclusions can be made.

Includes bibliographical references. / The thesis is organized as follows. Part I is a general introduction to LGT. The theory is discussed from first principles, so that for the interested reader no previous knowledge is required, although it is assumed that he/she will be familiar with the rudiments of relativistic quantum mechanics. Part II is a review of QCD on the lattice at finite temperature and density. Monte Carlo results and analytical methods are discussed. An attempt has been made to include most relevant data up to the end of 1987, and to update some earlier reviews existing on the subject. To facilitate an understanding of the techniques used in LGT, provision has been made in the form of a separate Chapter on Group Theory and Integration, as well as four Appendices, one of which deals with Grassmann variables and integration.

Thermal transport properties are critical for applications ranging from thermal management to energy conversion. Passive thermal management has been an area of study for over a century and has only grown as technology has advanced because it requires no additional energy to remove heat. Changing the nanostructure of the materials involved in passive heat transfer methods, either by geometric changes or stoichiometric changes, can greatly improve the effectiveness of this heat transfer method. In order to explore this further, this work employs LAMMPS molecular dynamics (MD) simulation software to calculate the lattice thermal conductivity (λp) of a nanoparticle (NP) and material used indifferent passive heat transfer methods after either modifying their geometry or stoichiometry. The NPs this work will simulate are single-wall carbon nanotubes (SWCNTs), which have been well known for high λp, and their applications in improving thermal conductivity in matrix materials. The material this work will simulate is magnesium silicide (Mg2Si), a thermoelectric material. Thermoelectric materials, in general, become more efficient in converting heat into electrical power as their λp decreases. λp will be calculated for SWC-NTs of varying lengths, diameters, and at varying equilibration temperatures (Teq). λp will be calculated for samples of pure Mg2Si and Mg2Si with off-stoichiometry over a range of Teq values. Two methods will be used to induce the off-stoichiometry: atomic silicon (Si) substitutionals, and Si NPs. A range of stoichiometric ratios will be applied to the material by both methods, and then λp will be calculated for each of these cases. This is done so as to observe which method of stoichiometric change, given the same stoichiometric ratio, decreases λp greater, and, therefore, causes Mg2Si to be a better thermoelectric material. It is expected that increases in length will increase the λp of the SWCNT, while increases in diameter and Teq will decrease λp. It is expected that increases in atomic percent (a/o) Si and Teq will decrease λp regardless of the method of stoichiometric change, and that the Si NP method will decrease λp more than the atomic Si substitutional method.

Classical methods of analyzing heat capacity data for the characteristic moments of the frequency distribution are applied to iridium and rhodium. Impurity moments are determined from high and low temperature f values. These moments are combined by modern theory to estimate the magnitude of the host-host to host-impurity force constant ratio. Ratios of the various host moments are also examined.