Browsing by master's degree program "Master 's Programme in Theoretical and Computational Methods"

The nature of dense matter is one of the greatest mysteries in high energy physics. For example, we do not know how QCD matter behaves in neutron star densities as there the matter is strongly coupled. Thus auxiliary methods have to be applied. One of these methods is the AdS/CFT-correspondence. This maps the strongly coupled field theory to weakly coupled gravity theory. The most well known example of this correspondence is the duality between N = 4 Super Yang-Mills and type IIB supergravity in AdS 5 × S 5 . This duality at finite temperature and chemical potential is the one we invoke in our study. It has been hypothesized that the dense matter would be in a color superconducting phase, where pairs of quarks form a condensate. This has natural interpretation in the gravity theory. The AdS 5 × S 5 geometry is sourced by stack of N coincident D3-branes. This N corresponds to the gauge group SU (N ) of N = 4 SYM. Then to study spontaneous breaking of this gauge group, one studies systems where D3-branes have separated from the stack. In this work we present two methods of studying the possibility of separating these branes from the stack. First we present an effective potential for a probe brane, which covers the dynamics of a single D3-brane in the bulk. We do this by using the action principle. Then we construct an effective potential for a shell constructed from multiple branes. We do this by using the Israel junction conditions. Single brane in the bulk corresponds to SU (N ) → SU (N − 1) × U (1) symmetry breaking and a shell of k-branes corresponds to SU (N ) → SU (N − k) × U (1) k symmetry breaking. Similar spontaneous breaking of the gauge group happens in QCD when we transition to a CSC-phase and hence these phases are called color superconducting. We find that for sufficiently high chemical potential the system is susceptible to single brane nucleation. The phase with higher breaking of the gauge group, which corresponds to having shell made out of branes in the bulk, is metastable. This implies that we were able to construct CSC-phases of N = 4 SYM, however, the exact details of the phase diagram structure is left for future research.

In quantum field theory the objects of interest are the n-point vacuum expectations which can be calculated from the path integral. The path integral usually used in physics is not well-defined and the main motivation for this thesis is to give axioms that a well-defined path integral candidate has to at least satisfy for it to be physically relevant - we want the path integral to have properties which allow us to reconstruct the physically interesting objects from it. The axioms given in this thesis are called the Osterwalder-Schrader axioms and the reconstruction of the physical objects from the path integral satisfying the axioms is called the Osterwalder-Schrader reconstruction. The Osterwalder-Schrader axioms are special in the sense that they are stated in terms of the Euclidean spacetime instead of the physically relevant Minkowski spacetime. As the physical objects live in Minkowski spacetime this means that when reconstructing the physically relevant objects we have to go back to Minkowski spacetime at some point. This thesis has three parts (and an introduction). In the first part we give a brief introduction to parts of functional analysis which we will need later - theory about distributions and about generators of families of operators. The second part is about the Osterwalder-Schrader axioms and about the reconstruction of the physically relevant objects from the path integral. In the last part we check that the path integral for the free field of mass m satisfies the Osterwalder-Schrader axioms.

Numerical techniques have become powerful tools for studying quantum systems. Eventually, quantum computers may enable novel ways to perform numerical simulations and conquer problems that arise in classical simulations of highly entangled matter. Simple one dimensional systems of low entanglement are efficiently simulatable on a classical computer using tensor networks. This kind of toy simulations also give us the opportunity to study the methods of quantum simulations, such as different transformation techniques and optimization algorithms that could be beneficial for the near term quantum technologies. In this thesis, we study a theoretical framework for a fermionic quantum simulation and simulate the real-time evolution of particles governed by the Gross-Neveu model in one-dimension. To simulate the Gross-Neveu model classically, we use the Matrix Product State (MPS) method. Starting from the continuum case, we discretise the model by putting it on a lattice and encode the time evolution operator with the help of fermion-to-qubit transformations, Jordan-Wigner and Bravyi-Kitaev. The simulation results are visualised as plots of probability density. The results indicate the expected flavour and spatial symmetry of the system. The comparison of the two transformations show better performance of the Jordan-Wigner transformation before and after the gate reduction.

Low-frequency $1/$ noise is ubiquitous, found in all electronic devices and other diverse areas such as as music, economics and biological systems. Despite valiant efforts, the source of $1/f$ noise remains one of the oldest unsolved mysteries in modern physics after nearly 100 years since its initial discovery in 1925. In metallic conductors resistance $1/f$ noise is commonly attributed to diffusion of mobile defects that alter the scattering cross section experienced by the charge carriers. Models based on two-level tunneling systems (TLTS) are typically employed. However, a model based on the dynamics of mobile defects forming temporary clusters would naturally offer long-term correlations required by $1/f$ noise via the nearly limitless number of configurations among a group of defects. Resistance $1/f$ noise due to such motion of mobile defects was studied via Monte Carlo simulations of a simple resistor network resembling an atomic lattice. The defects migrate through the lattice via thermally activated hopping motion, causing fluctuations in the resistance due to varying scattering cross section. The power spectral density (PSD) $S(f)$ of the simulated resistance noise was then calculated and first compared to $S(f)=C/f^\alpha$ noise, where $C$ is a constant and $\alpha$ is ideally close to unity. The value of $\alpha$ was estimated via a linear fit of the noise PSD on a log-log scale. The resistor network was simulated with varying values of temperature, system size and the concentration of defects. The noise produced by the simulations did not yield pure $1/f^\alpha$ noise, instead the lowest frequencies displayed a white noise tail, changing to $1/f^\alpha$ noise between $10^{-4}$ to $10^{-2}$~Hz. In this way the spectrum of the simulated noise resembles a Lorentzian. The value of $\alpha$ was found to be the most sensitive to temperature $T$, which directly affects the motion of the defects. At high $T$ the value of $\alpha$ was closer to 1, whereas at low $T$ it was closer to $1,5$. Varying the size of the system was found to have little impact on $\alpha$ when the temperature and concentration of defects were kept fixed. Increasing the number of defects was found to have slightly more effect on $\alpha$ when the temperature and system size were kept fixed. The value of $\alpha$ was closer to unity when the concentration of defects was higher, but the effect was not nearly as pronounced compared to varying the temperature. In addition, the simulated noise was compared to a PSD of the form $S(f)\propto e^{-\sqrt{N}/T}1/f$, where $N$ is the size of the system, according to recent theoretical proceedings. The $1/f^\alpha$ part of the simulated noise was found to roughly follow the above equation, but the results remain inconclusive. Although the simple toy model did not produce pure $1/f^\alpha$ noise, the dynamics of the mobile defects do seem to have an effect on the noise PSD, yielding noise closer to $1/f$ when there are more interactions between the defects due to either higher mobility or higher concentration of defects. However, this is disregarding the white noise tail. Recent experimental research on high quality graphene employing more rigorous kinetic Monte Carlo simulations have displayed more promising results. This indicates that the dynamics of temporary cluster formation of mobile defects is relevant to understand $1/f$ noise in metallic conductors, offering an objective for future work.

This thesis is aimed to explore the topic of surface diffusion on copper and iron surfaces, using an accelerated molecular dynamics (MD) method known as collective variable-driven hyperdynamics (CVHD). The thesis is divided into six main sections: Introduction, Theory, Methods, Simulations, Results and Conclusion. The introduction briefly explains the main interest behind the topic and why diffusion is a difficult subject for classical MD simulations. In the theory section, the physical description of diffusion in metals is explained, as well as the important quantities that can be determined from these types of simulations. The following section dives into the basics concerning the molecular dynamics simulations method. It also gives a description of the theoretical basis of collective variable-driven hyperdynamics and how it is implemented alongside molecular dynamics. The simulations section more technically explains the system building methodology, discusses key parameters and gives reasoning for the chosen values of these parameters. Since, both copper and iron systems have been simulated, both sets of systems are explained independently. The results section displays the results for the copper and iron systems separately. In both sets of systems, the obtained activation energy of the dominant diffusion mechanisms remain the main point of focus. Lastly, the results are dissected and summarized.

Bayesian networks (BN) are models that map the mutual dependencies and independencies between a set of variables. The structure of the model can be represented as a directed acyclic graph (DAG), which is a graph where the nodes represent variables and the directed edges between variables represent a dependency. BNs can be either constructed by using knowledge of the system or derived computationally from observational data. Traditionally, BN structure discovery from observational data has been done through heuristic algorithms, but advances in deep learning have made it possible to train neural networks for this task in a supervised manner. This thesis provides an overview of BN structure discovery and discusses the strengths and weaknesses of the emerging supervised paradigm. One supervised method, the EQ-model, that uses neural networks for structure discovery using equivariant models, is also explored in further detail with empirical tests. Through a process of hyperparameter optimisation and moving to online training, the performance of the EQ-model is increased. The EQ-model is still observed to underperform in comparison to a competing score-based model, NOTEARS, but offers convenient features, such as dramatically faster runtime, that compensate for the reduced performance. Several interesting lines of further study that could be used to further improve the performance of the EQ-model are also identified.