Abstract:
The Hadron Resonance Gas (HRG) model is considered to study the QCD equation of state for the case of nonzero external magnetic fields. Thermodynamic observables including the pressure, energy density, entropy density, magnetization and the speed of sound are presented as functions of the temperature and the magnetic field. The magnetization is determined to be positive, indicating that the hadronic phase of QCD is paramagnetic. The behavior of the speed of sound suggests that the deconfinement transition temperature is lowered as the magnetic field grows. Moreover, a simple correspondence is derived, which relates the magnetic catalysis of the quark condensate to the positivity of the beta-function in scalar QED.

Abstract:
Lattice simulations have demonstrated that a background (electro)magnetic field reduces the chiral/deconfinement transition temperature of quantum chromodynamics for eB < 1 GeV^2. On the level of observables, this reduction manifests itself in an enhancement of the Polyakov loop and in a suppression of the light quark condensates (inverse magnetic catalysis) in the transition region. In this paper, we report on lattice simulations of 1+1+1-flavor QCD at an unprecedentedly high value of the magnetic field eB = 3.25 GeV^2. Based on the behavior of various observables, it is shown that even at this extremely strong field, inverse magnetic catalysis prevails and the transition, albeit becoming sharper, remains an analytic crossover. In addition, we develop an algorithm to directly simulate the asymptotically strong magnetic field limit of QCD. We find strong evidence for a first-order deconfinement phase transition in this limiting theory, implying the presence of a critical point in the QCD phase diagram. Based on the available lattice data, we estimate the location of the critical point.

Abstract:
We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing the deviation between its derivatives and the measured gradient. Unlike a multidimensional integration along some path, the present method results in a continuous, smooth surface, furthermore, it also applies to input data that are non-equidistant and not aligned on a rectangular grid. Function values, first and second derivatives and integrals are easy to calculate. The proper estimation of the statistical and systematical errors is also incorporated in the method.

Abstract:
The Hadron Resonance Gas (HRG) model is considered to study the QCD equation of state for the case of nonzero external magnetic fields. Thermodynamic observables including the pressure, energy density, entropy density, magnetization and the speed of sound are presented as functions of the temperature and the magnetic field. The magnetization is determined to be positive, indicating that the hadronic phase of QCD is paramagnetic. The behavior of the speed of sound suggests that the deconfinement transition temperature is lowered as the magnetic field grows. Moreover, a simple correspondence is derived, which relates the magnetic catalysis of the quark condensate to the positivity of the beta-function in scalar QED.

Abstract:
In this thesis the finite temperature transition between confined and deconfined matter is studied at zero and nonzero quark densities. The findings are relevant for the understanding of the evolution of the early Universe and contemporary and upcoming heavy ion experiments. The results were obtained using large-scale simulations on lattices with various physical extents and lattice spacings, at physical values of the parameters of the theory. First the phase diagram of QCD in the chemical potential-temperature plane is determined for small chemical potentials via a Taylor-expansion technique. Second the equation of state is studied with 2+1 flavors of dynamical quarks. Finally, the equation of state is calculated in pure gauge theory up to extremely high temperatures, where a comparison to perturbation theory is carried out.

Abstract:
We introduce dressed Wilson loops as a novel confinement observable. It consists of closed planar loops of arbitrary geometry but fixed area and its expectation values decay with the latter. The construction of dressed Wilson loops is based on chiral condensates in response to magnetic (and electric) fields, thus linking different physical concepts. We present results for generalized condensates and dressed Wilson loops on dynamical lattice configurations and confirm the agreement with conventional Wilson loops in the limit of large probe mass. We comment on the renormalization of dressed Wilson loops.

Abstract:
We present and test a new method to compute the hadronic vacuum polarization function in lattice simulations. This can then be used, e.g., to determine the leading hadronic contribution to the anomalous magnetic moment of the muon. The method is based on computing susceptibilities with respect to external electromagnetic plane wave fields and allows for a precision determination of both the connected and the disconnected contributions to the vacuum polarization. We demonstrate that the statistical errors obtained with our method are much smaller than those quoted in previous lattice studies, primarily due to a very effective suppression of the errors of the disconnected terms. These turn out to vanish within small errors, enabling us to quote an upper limit. We also comment on the accuracy of the vacuum polarization function determined from present experimental R-ratio data.

Abstract:
We introduce dressed Wilson loops as a novel confinement observable. It consists of closed planar loops of arbitrary geometry but fixed area, and its expectation values decay with the latter. The construction of dressed Wilson loops is based on chiral condensates in response to magnetic and electric fields, thus linking different physical concepts. We present results for generalized condensates and dressed Wilson loops on dynamical lattice configurations and confirm the agreement with conventional Wilson loops in the limit of large probe mass. We comment on the renormalization of dressed Wilson loops.

Abstract:
We propose a physical mechanism for inverse magnetic catalysis, the suppression of the chiral condensate by an external magnetic field in QCD around the critical temperature. We show that this effect, seen in lattice simulations, is a result of how the sea quarks react to the magnetic field. We find that the suppression of the condensate happens because the quark determinant can suppress low quark modes by ordering the Polyakov loop. This mechanism is particularly efficient around $T_c$ where the Polyakov loop effective potential is flat and the determinant can have a significant ordering effect. Our picture suggests that for the description of QCD in large magnetic fields it is crucial to properly capture the interaction between the Polyakov loop and the sea quarks, both in low-energy effective models and on the lattice.

Abstract:
Using finite temperature SU(3) lattice gauge theory in the fixed scale approach we analyze center properties of the local Polyakov loop L(x). We construct spatial clusters of points x where the phase of L(x) is near the same center element and study their properties as a function of temperature. We find that below the deconfinement transition the clusters form objects with a fractal dimension D < 3. As the temperature is increased, the largest cluster starts to percolate and its dimensionality approaches D=3. The fractal structure of the clusters in the transition region may have implications regarding both the small shear viscosity and the large opacity of the Quark Gluon Plasma observed in heavy-ion collision experiments.