Browsing by discipline "Teoreettinen fysiikka"

In recent years statistical physics and computational complexity have found mutually interesting subjects of research. The theory of spin glasses from statistical physics has been successfully applied to the boolean satisfiability problem, which is the canonical topic of computational complexity. The study of spin glasses originated from experimental studies of the magnetic properties of impure metallic alloys, but soon the study of the theoretical models outshone the interest in the experimental systems. The model studied in this thesis is that of Ising spins with random interactions. In this thesis we discuss two analytical derivations on spin glasses: the famous replica trick on the Sherrington-Kirkpatrick model and the cavity method on a Bethe lattice spin glass. Computational complexity theory is a branch of theoretical computer science that studies how the running time of algorithms scales with the size of the input. Two important classes of algorithms or problems are P and NP, or colloquially easy and hard problems. The first problem to be proven to belong to the class of NP-complete problems is that of boolean satisfiability, i.e., the study of whether there is an assignment of variables for a random boolean formula so that the formula is satisfied. The boolean satisfiability problem can be tackled with spin glass theory; the cavity method can be applied to it. Boolean satisfiability exhibits a phase transition. As one increases the ratio of constraints to variables the probability of a random formula being satisfiable drops from unity to zero. This transition of random formulas from satisfiable to unsatisfiable is continuous for small formulas. It grows sharper with increasing problem size and becomes discrete at the limit of an infinite number of variables. The cavity method gives a value for the location of the phase transition that is in agreement with the numerical value. The cavity method is an analytical tool for studying average values over a distribution, but it introduces so called surveys that can also be calculated numerically for a single instance. These surveys inspire the survey propagation algorithm that is implemented as a numerical program to efficiently solve large instances of random boolean satisfiability problems. In this thesis I present a parallel version of survey propagation that achieves a speedup by a factor of 3 with 4 processors. With the improved version we are able to gain further knowledge on the detailed workings of survey propagation. It is found, firstly, that the number of iterations needed for one convergence of survey propagation depends on the number of variables, seemingly as ln(N). Secondly, it is found that the constraint to variable ratio for which survey propagation succeeds is dependent on the number of variables.

In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.

Sähkömagneettisen aallon käyttäytymisen simuloimisesta on tullut tärkeä työkalu nykyaikana. Laskentaresurssien kasvu on mahdollistanut Maxwellin yhtälöiden ratkaisemisen mitä monimutkaisemmille systeemeille ilman, että systeemiä tarvittaisiin fyysisesti rakentaa. Tästä on hyötyä tähtitieteelle, sillä kaukaisten kohteiden ominaisuuksia voidaan tutkia vertaamalla kohteesta sironnutta valoa laskennalliseen dataan. Ongelmana laskennallisen datan tuottamisessa on, että sähkömagneettisen aallon käyttäytymistä kuvaavia Maxwellin yhtälöitä ei pystytä ratkaisemaan nykyisillä menetelmillä eksaktisti, kun mallinnettava kohde on suuri. Tästä syystä ongelmaa pitää yksinkertaistaa, kuten tehdään säteilynkuljetusohjelmissa. Säteilynkuljetusohjelmissa seurataan sähkömagneettisen aallon kulkua aineessa ja näin pyritään ratkaisemaan, kuinka aine sirottaa säteilyä. Säteilynkuljetusohjelmat onnistuvat ratkaisemaan harvojen aineiden sirontaongelmia, koska sirottajien väliset vuorovaikutusten oletetaan tapahtuvan kaukokenttien kautta, mikä on pätevä approksimaatio harvojen aineiden tapauksessa. Tiheiden aineiden sirontaongelmissa lähikenttien vaikutusta ei voi kuitenkaan sivuuttaa ja siksi säteilynkuljetuksen tulokset tiheille aineille eroavat eksaktista ratkaisusta huomattavasti. Tässä pro gradu -tutkielmassa on esitetty uudenlaisen säteilynkuljetusohjelma R²T² (Radiative Transfer with Reciprocal Transactions) kehitystyötä. Oletamme, että käyttämällä niin sanottua vapaan avaruuden sirontakenttää kaukokenttien sijaan, pystymme laskemaan säteilynkuljetusohjelma R²T²:lla myös tiheiden aineiden sirontaongelmia. Sähkökentät esitetään palloharmonisilla funktioilla ja säteilynkuljetuksessa yksittäiset sirottajat korvataan palloklustereilla, joiden väliset vuorovaikutukset lasketaan käyttämällä superpositio T-matriisimenetelmää. Tutkielmassa perehdytään säteilynkuljetusyhtälön johtoon, sähkömagneettisen kentän esittämiseen palloharmonisilla funktioilla ja superpositio T-matriisimenetelmään. Tutkielmassa käydään läpi R²T²:n algoritmia, jatkosuunnitelmia ja näytetään, että R²T² onnistuu kelvollisesti ratkaisemaan absorboimattomien aineiden ongelmia. Absorption huomioiminen vaatii kuitenkin vielä menetelmän jatkokehitystä.

In this thesis we give self-sufficient introduction to the complex Langevin method, which is a promising simulation method for the quantum field theories with finite chemical potential. We begin by revising the the stochastic methods needed for mathematical understanding of Langevin equation. This revision is done trough the examples of a Brownian motion. After the stochastical preliminaries are deled with, we also give a short introduction to quantum field theories on the lattice and with both finite temperature and chemical potential. We show that introduction of chemical potential can cause a sign problem, which can yield traditional Monte Carlo simulations non usable for this problem. We notice a similarity of form between the stochastic path integrals and the Feynman path integrals. We use this similarity to define a stochastic quantization method, that lets us to use stochastic evolution equations to find the correlation function in quantum field theory. This method is then expanded for complex actions that arise from the use of the finite chemical potential in the quantum field theory. We discuss the possible problems with this expansion and deduct that there is no complete mathematical understanding of complex Langevin equation. After all these theoretical considerations we do our own simulations to inspect if this model works. We find out, that for the U(1) one link model and for the SU(3) spin model, the method works as long as we are not using imaginary noise. Unfortunately we also find out that three dimensional XY-model converges to wrong results. We find no reason for this behavior.

Coronal mass ejections (CMEs) are ejections of plasma and magnetic flux from the Sun. These eruptions are large in size and they propagate over large distances in the heliosphere. If a CME is directed towards the Earth it can cause major space weather disturbances. Thus it is important to know whether a CME will hit the magnetosphere of the Earth and what kind of disturbances it could cause. CMEs contain large-scale twisted magnetic fields called flux ropes. Information about the orientation and the structure of the flux rope is crucial in determining its space weather effects. This thesis is focused on the determination of the orientation and the direction of flux rope CMEs from the corona all the way to the orbit of the Earth. An overview of a number of methods to obtain the necessary flux rope parameters both from remote sensing and in situ measurements is given. In particular, a technique combining the remote and in situ observations with a Sun-to-Earth propagation model for the CME is described in detail. In this thesis, the technique is used to study the propagation of a slow and a fast CME and the results are compared with each other and with earlier studies on the subject. The direction and amount of deflections and rotation were found to be consistent with the earlier results. In addition, differences in the proportional deflections before and after 20 Solar radii with respect to the total deflections were found between the fast and the slow case.

After the finding of a Higgs boson at the Large Hadron Collider of CERN, the next milestone in experimental particle physics is the detection of a new particle beyond the Standard Model. Many of the popular extensions of the Standard Model are so called Two-Higgs-doublet models, that include five physical Higgs bosons in total. Two of the bosons are neutral scalars, two are charged scalars and one is a pseudoscalar. The first part of this thesis presents the Standard Model of particle physics as a gauge field theory where the gauge principle is used to introduce interactions between the particles. Quantum electrodynamics, weak interactions and quantum chromodynamics with their properties are discussed. The electroweak symmetry breaking through the Higgs mechanism and its implications for the masses of the particles are shown. The considerations on the Standard Model are concluded by discussing some of the problems in the Standard Model. Supersymmetric extensions to the Standard Model are introduced, motivated by their potential to solve some of the problems and by the framework for new physics the supersymmetry provides. The Higgs sector in the more constrained minimal supersymmetry scenario is discussed and some of the properties for these new Higgs bosons are given. The latter part of the thesis focuses on the experimental aspects of high energy particle physics. The search for the charged Higgs boson in the H+ → τ+ ντ decay channel, with the tau lepton decaying into hadronic decay products is presented in detail. Vetoing collision events with more than one tau particle is shown to enhance the transverse mass resolution of the analysis, improving the signal detection. Methods for performing a tau veto are discussed and the problems in performing the tau veto are studied using 13 TeV collision event simulations and data from 2015 from the CMS detector at the LHC. No viable way of performing the tau veto in such a way that it improves the overall analysis is found.

Majorana quasiparticles are zero-energy modes theorised to exist at the boundaries of topological superconductors. Due to their topological protection and non-Abelian exchange statistics, Majorana quasiparticles have in recent years garnered much interest within the condensed matter community as a promising platform for fault tolerant quantum computing. To this end, theorists constantly attempt to come up with new experimentally feasible models that can host these quasiparticles. In this thesis, we investigate a recent proposal for realising an effective topological superconductor with Majorana quasiparticles, namely the helical Shiba chain. The system consists of a one-dimensional array of magnetic impurities deposited on a conventional superconductor, such that the overall magnetic texture is helical. A single impurity hosts a bound electron state with an energy within the superconducting gap. For many impurities, these so called Shiba states hybridise and form energy bands, that for a certain range of parameters support topological superconductivity and Majorana bound states at the ends of the chain. Prior to this work, the Shiba chain has only been studied in the deep-dilute limit where the energies of the individual Shiba states are assumed to be very close to the centre of the gap, and the magnetic impurities are placed sufficiently far away from each other as to keep the resulting bands deep within the gap. By re-expressing the problem in terms of a non-linear eigenvalue equation, we go beyond this limit and extend the study of the topological properties of the Shiba chain to a novel domain of experimental interest — a domain which has until now remained unexplored. We compare our results with previous work and observe an agreement between our more general theory and the deep-dilute limit in the expected parameter regime. However, qualitative differences emerge when the coherence length of the underlying superconductor becomes very large.

In this thesis, we examine a fairly novel area of physics that concerns topological materials, and in particular, topological superconductivity. A goal in the research of topological materials is realizing applications in quantum computing, which could be aided by the emergent quasiparticles that exhibit non-Abelian exchange statistics. These are called Majorana bound states, and they are elusive quasiparticles predicted to be found on the boundary of topological superconductors. We first study a one-dimensional chain of potential impurities placed on the surface of a two-dimensional $p$-wave superconductor. As is usually the case, such chains are composed of perfect lattice structures, which is very challenging to achieve in any laboratory setting. Nevertheless, they serve as a good example of systems where an analytical solution can be well established. We investigate the model without employing any deep-dilute approximation, which gives us an accurate description even far away from the gap center. This is done by formulating the problem as a non-linear eigenvalue equation, which complicates it significantly, but also extends the region of applicability of our theory. We use reciprocal space calculations of two topological invariants to obtain the topological phase diagram of the system. The model is shown to host topological quasiparticle excitations at the ends of the chain, with multiple distinct topological phases. The near-perfect localization of the excitations makes them good candidates for probing Majorana bound states in experimental setups. We then move on to study topological superconductivity in random lattices, as opposed to regular structures which assume arbitrary precision. We frame our work starting with the mathematics of random numbers. Our work is thus in stark contrast with previous studies on topological materials that start off with a perfect lattice structure, and investigate some degree of disorder as perturbations to the regular lattice case. Our work establishes a first-ever realistic candidate for realizing topological superconductivity in an amorphous material. This could enable a novel approach to creating topological materials, and drastically aid in the development of fault-tolerant quantum computing.

Hiukkasfysiikan standardimalli on yksi nykypäivän tarkimmista teorioista. Vuonna 2012 tapahtuneen Higgsin bosonin havaitsemisen myötä olemme havainneet kaikki alkeishiukkaset ja niiden väliset vuorovaikutukset, jotka standardimalli ennustaa. Tarkkuudestaan huolimatta hiukkasfysiikassa on kuitenkin edelleen ilmiöitä, joita standardimalli ei kykene selittämään. Standardimallin laajennuksiksi kutsutut teoriat pyrkivät selittämään standardimallin avoimia kysymyksiä ja useat laajennukset ennustavat myös uusia hiukkasia. Tämä opinnäytetyö keskittyy kahden-Higgsin-dubletin malleihin joka kuuluu standardimallin laajennuksiin. Nämä mallit ennustavat yhden sijasta yhteensä viisi Higgsin bosonia, joista kaksi on sähköisesti varattuja. Analyysi, joka on myös esitelty tässä opinnäytetyössä, pyrkii havitsemaan nämä kaksi sähköisesti varattua Higgsin bosonia. Tähän käytetään dataa, joka on kerätty suurella hadronitörmäyttimellä (eng. Large Hadron Collider) käyttäen kompaktia myonisolenoidi-hiukkasilmaisinta (eng. Compact Muon Solenoid). Data kerätään törmäyttämällä protoneita yhteen suurella energialla. Suuri energia mahdollistaa uusien hiukkasten syntymisen ja näitä lopputuotteita tutkimalla voidaan selvittää syntyikö törmäyksessä mahdollisesti eksoottisia, jopa standardimallin ulkopuolisia hiukkasia, kuten sähköisesti varattuja Higgsin bosoneja. Protonisuihkut törmäävät jopa 40 000 000 kertaa sekunnissa, minkä takia dataa syntyy nopeammin kuin sitä ehditään tallentaa. Tästä syystä tapahtumien lukumäärää on leikattava, mikä tapahtuu käyttämällä liipaisua (eng. trigger system). Liipaisu koostuu hiukkasilmaisimen laitteistoon asennetuista komponenteista sekä laitteiston ulkopuolisista tietokoneohjelmistoista, jotka päättävät mitä osia kerätystä datasta on syytä tallentaa. Liipaisu on suunniteltu niin, että valinnan läpäisevät esimerkiksi hyvin energeettiset hiukkaset, jotka saattavat olla lähtöisin mielenkiintoisista kohteista. Liipaisun jälkeen datan määrä on vähentynyt niin että se on mahdollista kirjoittaa levylle tallennusta varten. Tässä opinnäytetyössä esitän uuden menetelmän mitata liipaisun tehokkuutta. Tehokkuus määritellään liipaisun valintaan sisään tulevien hiukkasten ja valinnan läpäisseiden hiukkasten lukumäärien suhteena. Uusi menetelmä sovittaa mitattuun ja simuloituun liipaisun tehokkuuteen funktion ja näin vähentää tehokkuuden mittauksen systemaattista epävarmuutta. Tämä pienentää koko analyysin systemaattisia virhelähteitä ja parantaa lopullisia tuloksia.