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In this thesis, we study some features of the quantum chromodynamics (QCD) phase diagram at purely imaginary chemical potential using lattice techniques. This is one of the possible methodologies to get insights about the situation at finite density, where the sign problem prevents direct investigations from first principles.
We focus, in particular, on the Roberge-Weiss plane, where the phase structure with two degenerate flavours is studied both in the light and in the heavy quark mass limit. On the lattice, any result is affected by cut-off effects and so are the positions of the two tricritical points m_{tric}^{1,2} separating the second-order intermediate mass region from the first-order triple light and heavy mass regions. Therefore, changing the lattice spacing 'a', the values of m_{tric}^1 and m_{tric}^2 will change. In order to find their position in the continuum limit – i.e. for 'a' going to 0 – they have to be located on finer and finer lattices. Typically, in lattice QCD (LQCD) simulations, the temperature T is tuned through the bare coupling β, on which 'a' depends, while keeping Nt fixed. Hence, it is common to implicitly refer to how fine the lattice is just mentioning its temporal extent.
Using both Wilson and staggered fermions, we simulate Nf=2 QCD on Nt=6 lattices, varying the quark bare mass from the chiral (m_{u,d} going to 0) to the quenched (m_{u,d} going to infinity) limit. For each quark mass, a thorough finite scaling analysis is carried out, taking advantage of two different but consistent methods. In this way we identify the order of the phase transition locating, then, the position of the tricritical points. In order to convert our measurements to physical units we fix the scale measuring the lattice spacing as well as the pion mass corresponding to the quark bare mass used. This allows a comparison between different discretisation, getting a first idea of how serious are cut-off effects.
To be able to make a comparison between two different discretisations, we added an RHMC algorithm with staggered fermions to the CL2QCD software, a GPU code based on OpenCL, which we released in 2014. A considerable part of our work has been invested in ameliorating and optimising CL2QCD, as well as in developing new analysis tools regularly used next to it. Just to mention one, the multiple histogram method has been implemented in a completely general way and we took advantage of it in order to obtain more precise results. Finally, in order to efficiently handle and monitor the hundreds of simulations that are typically concurrently run in finite temperature LQCD, a completely new Bash library of tools has been developed. We plan to release it as a byproduct of CL2QCD in the near future.
The term superconductivity describes the phenomenon of vanishing electrical resistivity in a certain material, then called a superconductor, below a critical typically very low temperature. Since the discovery of superconductivity in mercury in 1911 many other superconductors have been found and the critical temperature below which superconductivity occurs could recently be raised to the temperatures encountered in a cold antarctic winter.
Superconductors are promising materials for applications. They can serve as nearly loss-free cables for energy transmission, in coils for the generation of high magnetic fields or in various electronic devices, such as detectors for magnetic fields. Despite their obvious advantages, the cost for using superconductors, however, depends a lot on the cooling effort needed to realize the superconducting state. Therefore, the search for a superconductor with critical temperature above room-temperature, which would avoid the need for any specialized cooling system, is one of the main projects of contemporary research in condensed matter physics.
While a theory of superconductivity in simple metals has already been developed in the 1950s, it has meanwhile been recognized that many superconductors are unconventional in the sense that their behavior does not follow the aforementioned theory. Unconventional superconductors differ from conventional superconductors mainly by the momentum- and real-space symmetry of the order parameter, which is associated with the superconducting state. While conventional superconductors have a uniform order parameter, unconventional superconductors can have an order parameter that bears structure. Of course, alternative theoretical descriptions have been suggested, but the discussion on the right theory for unconventional superconductivity has not yet been settled. Ultimately, this lack of a general theory of superconductivity prevents a targeted search for the room-temperature superconductor. Any new theoretical approach must, however, prove its value by correctly predicting the structure of the superconducting order parameter and further material properties.
In this work we participate in the search for a theory of unconventional superconductivity. We discuss the theory of superconductivity mediated by electron-electron interactions, which has been popular in the last few decades due to its success in explaining various properties of the copper-based superconductors that emerged in the 1980s. We give a detailed derivation of the so-called random phase approximation for the Hubbard model in terms of a diagrammatic many-body theory and apply it in conjunction with low-energy kinetic Hamiltonians, which we construct from first principles calculations in the framework of density functional theory. Density functional theory is an established technique for calculating the electronic and magnetic properties of materials solely based on their crystal structure. Its practical implementations in computer codes, however, do for example not describe complicated many-electron phenomena like the superconducting state that we are interested in here. Nevertheless, it can provide important information about the properties of the normal state of the material, which superconductivity emerges from. In our theory we use these information and approach the superconducting state from the normal state.
Such an interfacing of different calculational techniques requires a lot of implementation work in the form of computer code. Inclusion of the computer code into this work would consume by far too much space, but since some of the decisions on approximations in the calculational formalism are guided by the feasibility of the associated computer calculations, we discuss the numerical implementation in great detail.
We apply the developed methods to quasi-two-dimensional organic charge transfer salts and iron-based superconductors. Finally, we discuss implications of our findings for the interpretation of various experiments.
In den vergangen Jahren wurde erkannt, dass eine Quantenfeldtheorie (QFT) namens Quantenchromodynamik (QCD) die richtige Theorie der starken Wechselwirkungen ist. QCD beschreibt erfolgreich die starken Wechselwirkungen, die Quarks zu Nukleonen und Nukleonen zu Atomkernen zusammenbinden. Jedoch ist die theoretische Beschreibung vieler Phänomene der starken Wechselwirkung aufgrund des starken Kopplungsverhaltens bei niedrigen Energien schwierig. Stoßexperimente mit Schwerionen sind ein möglicher Weg, um die charakteristischen Phänomene und Eigenschaften der QCD-Materie zu untersuchen. In Stoßexperimenten mit Schwerionen werden schwere (d.h. große) Atomkerne aufeinander geschossen, beispielsweise Gold (am RHIC) oder Blei (am CERN, LHC), mit einer ultrarelativistischen Energie √s im Schwerpunktsystem. Auf diese Art ist es möglich, eine große Menge von Materie mit hoher Energiedichte hervorzubringen. Das Ziel von Schwerionenkollisionen ist die Erzeugung und Charakterisierung einer makroskopischen Phase von freien Quarks und Gluonen im lokalen thermischen Gleichgewicht. Ein solcher Aggregatzustand kann neue Informationen über das QCD-Phasendiagramm und den QCD-Phasenübergang liefern. Man nimmt an, dass ein solcher Übergang stattfand, als sich die Materie des frühen Universums von einem Plasma aus Quarks und Gluonen (QGP) in ein Gas von Hadronen umwandelte...
The elliptic flow of heavy-flavour decay electrons is measured at midrapidity |eta| < 0.8 in three centrality classes (0-10%, 10-20% and 20-40%) of Pb-Pb collisions at sqrt(sNN) = 2.76TeV with ALICE at LHC. The collective motion of the particles inside the medium which is created in the heavy-ion collisions can be analyzed by a Fourier decomposition of the azimuthal anisotropic particle distribution with respect to the event plane. Elliptic flow is the component of the collective motion characterized by the second harmonic moment of this decomposition. It is a direct consequence of the initial geometry of the collision which is translated to a particle number anisotropy due to the strong interactions inside the medium. The amount of elliptic flow of low-momentum heavy quarks is related to their thermalization with the medium, while high-momentum heavy quarks provide a way to assess the path-length dependence of the energy loss induced by the interaction with the medium.
The heavy-quark elliptic flow is measured using a three-step procedure.
First the v2 coefficient of the inclusive electrons is measured using the event-plane and scalar-product methods. The electron background from light flavours and direct photons is then simulated, calculating the decay kinematics of the electron sources which are initialised by their respective measured spectra. The final result of this work emerges by subtracting the background from the inclusive measurement. A significant elliptic flow is observed after this subtraction. Its value is decreasing from low to intermediate pT and from semi-central to central collisions.
The results are described by model calculations with significant elastic interactions of the heavy quarks with the expanding strongly-interacting medium.
At sufficiently high temperatures and baryon densities, nuclear matter is expected to undergo a transition into the Quark-Gluon-Plasma (QGP) consisting of deconfined quarks and gluons and accompanied by chiral symmetry restoration. Signals of these two fundamental characteristics of Quantum-Chromo-Dynamics (QCD) can be studied in ultra-relativistic heavy-ion collisions producing a relatively large volume of high energy and nucleon densities as existent in the early universe. Dileptons are unique bulk-penetrating sources for this purpose since they penetrate through the surrounding medium with negligible interaction and are created throughout the entire evolution of the initially created fireball. A multitude of experiments at SIS18, SPS and RHIC have taken on the challenging task to measure these rare probes in a heavy-ion environment. NA60's results from high-quality dimuon measurements have identified the broadened ρ spectral function as favorable scenario to explain the low-mass dilepton excess, and partonic sources as dominant at intermediate dilepton masses.
Enabled by the addition of a TOF detector system in 2010, the first phase of the Beam Energy Scan (BES-I) at RHIC allows STAR to conduct an unprecedented energy-dependent study of dielectron production within a homogeneous experimental environment, and hence close the wide gap in the QCD phase diagram between SPS and top RHIC energies. This thesis concentrates on the understanding of the LMR enhancement regarding its invariant mass, transverse momentum and energy dependence. It studies dielectron production in Au+Au collisions at beam energies of 19.6, 27, 39, and 62.4 GeV with sufficient statistics. In conjunction with the published STAR results at top RHIC energy, this thesis presents results on the first comprehensive energy-dependent study of dielectron production.
This includes invariant mass- and transverse momenta-spectra for the four beam energies measured in 0-80% minimum-bias Au+Au collisions with high statistics up to 3.5 GeV/c² and 2.2 GeV/c, respectively. Their comparison with cocktail simulations of hadronic sources reveals a sizeable and steadily increasing excess yield in the LMR at all beam energies. The scenario of broadened in-medium ρ spectral functions proves to not only serve well as dominating underlying source but also to be universal in nature since it quantitatively and qualitatively explains the LMR enhancements measured over the wide range from SPS to top RHIC energies. It shows that most of the enhancement is governed by interactions of the ρ meson with thermal resonance excitations in the late(r)-stage hot and dense hadronic phase. This conclusion is supported by the energy-dependent measurement of integrated LMR excess yields and enhancement factors. The former do not exhibit a strong dependence on beam energy as expected from the approximately constant total baryon density above 20 GeV, and the latter show agreement with the CERES measurement at SPS energy. The consistency in excess yields and agreement with model calculations over the wide RHIC energy regime makes a strong case for LMR enhancements on the order of a factor 2-3.
The extent of the results presented here enables a more solid discussion of its relation to chiral symmetry restoration from a theoretical point of view. High-statistics measurements at BES-II hold the promise to confirm these conclusions along with the LMR enhancment's relation to total baryon density with decreasing beam energy.
Lepton pairs emerging from decays of virtual photons represent promising probes of nuclear matter under extreme conditions of temperature and density. These etreme conditions can be reached in heavy-ion collisions in various facilities around the world. Hereby the collision energy in the center-of-mass system (√SNN) varies from few GeV (SIS) to the TeV (LHC). In the energy domain of 1 - 2 GeV per nucleon (GeV/u), the HADES experiment at GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt studies dielectrons and strangeness production.
Various reactions, for example collisions of pions, protons, deuterons and heavy-ions with nuclei have been studied since its installation in the year 2001. Hereby the so called DLS Puzzle was solved experimentally, with remeasuring C+C at 1 and 2 GeV/u and by careful studies of inclusive pp and pn reactions at 1.25 GeV. With these measurements the so-called reference spectrum was established. Measurements of e+ e− production Ar+KCl showed an enhancement on the dilepton spectrum above the trivial NN back-
ground. Theory predicts a strong enhancement of medium radiation with the system size, due to large production of fast decaying baryonic resonances like ∆ and N∗ . The heaviest system measured so far was Au+Au at a kinetic beam energy of 1.23 GeV/u. The precise determination of the medium radiation depends
on a precise knowledge of the underlying hadronic cocktail composed of various sources contributing to the measured dilepton spectrum. In general the medium radiation needs to be separated from contributions coming from long-lived particles, that decay after the freeze out of the system. For a more model independent
understanding of the dilepton cocktail the production cross sections of these particles need to measured independently. In the related energy regime the main contributers are π0 and η Dalitz decays. Both mesons have a dominant decay into two real photons and have been reconstructed successfully in this channel. Since HADES has no electromagnetic calorimeter the mesons can not be identified in this decay channel directly. In this thesis the capability of HADES to detect e+ e− pairs from conversions of real photons is demonstrated.
Therefore not only the conversion probability but also the resulting efficiencies are shown. Furthermore, the reconstruction method for neutral mesons will be explained and the resulting spectra are interpreted. The measurement of neutral pions is compared to the independent measured charged pion distribution, and
extrapolated to full phase space. An integrated approach is used to determine the η yield. Both measurement are compared to the world data and to theory model claculations. Finally, the measurements will be used together with the reconstructed dilepton spectra to determine the amount and the properties of in medium radiation in the Au+Au system.
Great interest has emerged recently in the search for Kitaev spin liquid states in real materials. Such states rely on strongly anisotropic magnetic interactions, which have been suggested to exist in a number of candidate materials based on Ir and Ru. This thesis concentrates on two priority purposes. The first is the investigation of electronic and magnetic properties of candidate materials Na2IrO3, α-Li2IrO3, α-RuCl3, γ-Li2IrO3, and Ba3YIr2O9 for Kitaev physics where both spin-orbit coupling and correlation effects are important. The second is the method development for the microscopic description of correlated materials combining many-body methods and density functional theory (DFT). ...
In this thesis we explore the characteristics of strongly interacting matter, described by Quantum Chromodynamics (QCD). In particular, we investigate the properties of QCD at extreme densities, a region yet to be explored by first principle methods. We base the study on lattice gauge theory with Wilson fermions in the strong coupling, heavy quark regime. We expand the lattice action around this limit, and carry out analytic integrals over the gauge links to obtain an effective, dimensionally reduced, theory of Polyakov loop interactions.
The 3D effective theory suffers only from a mild sign problem, and we briefly outline how it can be simulated using either Monte Carlo techniques with reweighting, or the Complex Langevin flow. We then continue to the main topic of the thesis, namely the analytic treatment of the effective theory. We introduce the linked cluster expansion, a method ideal for studying thermodynamic expansions. The complex nature of the effective theory action requires the development of a generalisation of the linked cluster expansion. We find a mapping between generalised linked cluster expansion and our effective theory, and use this to compute the thermodynamic quantities.
Lastly, various resummation techniques are explored, and a chain resummation is implemented on the level of the effective theory itself. The resummed effective theory describes not only nearest neighbour, next to nearest neighbour, and so on, interactions, but couplings at all distances, making it well suited for describing macroscopic effects. We compute the equation of state for cold and dense heavy QCD, and find a correspondence with that of non-relativistic free fermions, indicating a shift of the dynamics in the continuum.
We conclude this thesis by presenting two possible extensions to new physics using the techniques outlined within. First is the application of the effective theory in the large-$N_c$ limit, of particular interest to the study of conformal field theory. Second is the computation of analytic Yang Lee zeros, which can be applied in the search for real phase transitions.
In this thesis, the production of charged kaons and Φ mesons in Au+Au collisions at sqrt sAuAu = 2.4 GeV is studied. At this energy, all particles carrying open and hidden strangeness are produced below their respective free nucleon-nucleon threshold with the corresponding so-called excess energies: sqrt sK+ exc = -0.15 GeV, sqrt sK- exc = -0.46 GeV, sqrt sΦ exc = -0.49 GeVGeV. As a consequence, the production cross sections are very sensitive to medium effects like momentum distributions, two- or multistep collisions, and modification of the in-medium spectral distribution of the produced states [1]. K+ and K- mesons exhibit different properties in baryon dominated matter, since only K- can be resonantly absorbed by nucleons. Although strangeness exchange reactions have been proposed to be the dominant channel for K- production in the analyzed energy regime, the production yield and kinematic distributions could also be explained in smaller systems based on statistical hadronization model fits to the measured particle yields, including a canonical strangeness suppression radius RC, and taking the Φ feed-down to kaons into account [2, 3]. For the first time in central Au+Au collisions at such low energies, it is possible to reconstruct and do a multi differential analysis of K- and Φ mesons. In principle, this should be the ideal environment for strangeness exchange reactions to occur, as the particles are produced deeply sub-threshold in a large and long-living system. Therefore, it is the ultimate test to differentiate between the different sources for K- production in HIC.
In total 7.3x10exp9 of the 40% most central Au(1.23 GeV per nucleon)+Au collisions are analyzed. The data has been recorded with the High Acceptance DiElectron Spectrometer HADES located at Helmholtzzentrum für Schwerionenforschung GSI in April/May 2012. A substantially improved reconstruction method has been employed to reconstruct the hadrons with high purity in a wide phase space region.
The estimated particle multiplicities follow a clear hierarchy of the excess energy: 41.5 ± 2.1|sys protons at mid-rapidity per unit in rapidity, 11.1 ± 0.6|sys ± 0.4|extrapol π-, (3.01 ± 0.03|stat ± 0.15|sys ± 0.30|extrapól) x10 exp -2 K+, (1.94 ± 0.09|stat ± 0.10|sys ± 0.10|extrapol)x10 exp -4 K- and (0.99 ± 0.24|stat ± 0.10|sys ± 0.05|extrapol)x10 exp -4 Φ per event. The multiplicities of the strange hadrons increase more than linear with the mean number of participating nucleons hAparti, supporting the assumption that the necessary energy to overcome the elementary production threshold is accumulated in multi-particle interactions. Transport models predict such an increase, but are overestimating the measured particle yield and are not able to describe the kinematic distributions of K+ mesons perfectly. However, the best description is given by the IQMD model with a density dependent kaonnucleon potential of 40 MeV at nuclear ground state density.
The K-=K+ multiplicity ratio is constant as a function of centrality and follows with (6.45 ± 0.77)x10 exp -3 the trend of increasing with beam energy indicated from previous experiments [4]. The effective temperature of K- TK+eff = (84 ± 6) MeV is found to be systematically lower than the one of K+ TK+eff = (104 ± 1) MeV, which has also been observed by the other experiments.
The Φ=K- ratio is with a value of 0.52 ± 0.16 higher than the one obtained at higher center-of-mass energies and smaller systems. This behavior is predicted from a tuned version of the UrQMD transport model [5], when including higher mass baryonic resonances which can decay into Φ mesons and from statistical hadronization models when suppressing open strangeness canonically. The found ratio is constant as a function of centrality and results with a branching ratio of 48.9%, that ~ 25% of all measured K- originate from Φ feed-down decays. A two component PLUTO simulation, consisting of a pure thermal and a K- contribution originating from Φ decays, can fully explain the observed lower effective temperature in comparison to K+ and the shape of the measured rapidity distribution of K-. As a result, we find no indication for strangeness exchange reactions being the dominant mechanism for K- production in the SIS18 energy regime, if taking the contribution from Φ feed-down decays into account.
The hadron yields for the 20% most central collisions can be described by a statistical hadronization model fit with the chemical freeze-out temperature of Tchem = (68 ± 2) MeV and baryochemical potential of μB = (883 ± 25) MeV, which is higher than expected from previous parameterizations. The analysis of the transverse mass spectra of protons indicate a kinetic freeze-out temperature of Tkin = (70 ± 4) MeV and radial flow velocity of βr = 0.43 ± 0.01, which is in agreement with the parameters obtained from the linear dependence of the effective temperatures on the particle mass Tkin = (71.5 ± 4.2) MeV and βr = 0.28 ± 0.09.
The Standard Model is one of the greatest successes of modern theoretical physics. Itl describes the physics of elementary particles by means of three forces, the electro-magnetisc, the weak and the strong interactions. The electro-magnetic and the weak interaction are rather well understood in comparison to the strong interaction.
The latest is as fundamental as the others, it is responsible for the formation of all hadrons which are classified into mesons and baryons. Well-known examples of the former is the pion and of the latter is the proton and the neutron, which form the nucleus of every atom. This fundamental force is believed to be described by the Quantum Chromodynamics (QCD) theory. According to this theory, hadrons are not elementary particles but are composed of quarks and gluons. The latter are the vector particles of the force and so are bosons of spin 1 and the former constitute the matter and are fermions with spin 1/2. To describe the interaction a new quantum number had to be introduced: the color charge which exists in three different types (blue, green and red). The name has not been chosen arbitrary as elements created from three quarks of different colors are colorless in the same way that mixing the three primary colors leads to white. However, experimentally no colored structure has ever been observed. The quarks and the gluons seem to be confined in colorless hadrons. This property of QCD is called confinement and results from a large coupling constant at low energy (or large distance). For high energy (or small distance), the perturbative analysis of QCD permits to establish the coupling constant to be small and quarks and gluons are almost free. This property is called asymptotic freedom. The possibility for QCD to describe both behaviors is one of its amazing characteristics. However, both phenomena are not well understood and one needs a method to study both the pertubative and the confining regime.
The only known method which fulfills the above criteria is Lattice QCD and more generally Lattice Quantum Field Theory (LQFT). It consists of a discretization of the spacetime and a formulation of QCD on a four-dimensional Euclidean spacetime grid of spacing a. In this way, the theory is naturally regularized and mathematically well-defined. On the other hand, the path integral formalism allows the theory to be treated as a Statistical Mechanics system which can be evaluated via a Markov chain Monte-Carlo algorithm. This method was first suggested by Wilson in 1974 [1] and shortly after Creutz performed the first numerical simulations of Yang-Mills theory [2] using a heath-bath Monte-Carlo algorithm. It appears that this method is extremely demanding in computational power. In its early days the method was criticized as the only feasible simulations involved non-physical values such as extremely large quark masses, large lattice spacing a and no dynamical quarks. With the progress of the computers and the appearance of the super-computer, the studies have come close to the physical point. But one still needs to deal with discrete space time and finite volume. Several techniques have been developed to estimate the infinite volume limit and the continuum limit. The smaller the lattice spacing and the larger the volume, the better the extrapolation to continuum and infinite volume limits is. The simulations are still very expensive and for the moment a typical length of the box is L ≈ 4fm and a ≈ 0.08fm. However, it has been realized simulating pure Yang-Mills theory and other lower dimensional models that the topology is freezing at small a [3]. It was also observed recently on full QCD simulations [4,5].
The typical lattice spacing for which this problem appears in QCD is a ≈ 0.05fm but this value depends on the quark mass used and on the algorithm. The freezing of topology leads to results which differ from physical results. Solving this issue is important for the future of LQCD [6]. Recently several methods to overcome the problem have been suggested, one of the most popular is the used of open boundary conditions [7] but this promising method has still its own issues, mainly the breaking of translation invariance.