Topological excitations and external charges in Yang-Mills theories and Ising spin glasses at finite temperature

The present thesis is devoted to an investigation of various physical systems in the vicinity of the boundary between statistical physics and quantum field theory using non-perturbative methods. In the first chapter we present a generalization of the so called dressing formalism to non-abelian gauge theories, initially introduced by Dirac in the context of abelian gauge theories in order to construct gauge invariant charges in quantum electrodynamics. First of all, a method is devised which allows us to compare various Ansaetze for the unknown ground state of a gauge theory in the presence of a pair of static charges. We primarily focus on two classes of states, closely related to the choice of axial and Coulomb gauge respectively. Starting from QED which can be treated analytically it is shown that a comparison of the persistence amplitude of different states, describing charges with different dressings, also provides a viable method to probe the ground state in a numerical investigation of the corresponding theory on a lattice by means of Monte Carlo simulations. The problems arising in a perturbative construction of dressings in non-abelian gauge theories are discussed before turning to the investigation of a SU(2)-Yang-Mills theory coupled to a scalar in the fundamental representation using a non-perturbative approach. Finally we present the results of an investigation of pure SU(2)-Yang-Mills theory in the continuum limit. In the second chapter we study the role of centre vortices in the theory of strong and electroweak interactions at finite temperature. The string breaking phenomenon which can be understood as a change of percolation properties of vortices in the framework of the so called random vortex model proved to be observable in a numerical study of a model of QCD after fixing the Direct Maximal Centre Gauge (DMCG) and projecting onto the centre degrees of freedom. An investigation of the electroweak standard model at finite temperature showed that a sensible definition of centre vortices - as opposed to elementary Z-strings and embedded Nambu monopoles - can also be obtained in the dimensionally reduced theory using local operators only, which allows the detection of objects whose density proves to be invariant under renormalization group transformations. The third chapter is concerned with the investigation of frustrated systems. On the basis of the prototypical two dimensional Ising spin glass we point out the possibility of a gauge invariant characterization of frustration by means of Z₂-vortices which allows for a unified classification of various instances of spin glasses sharing the same thermodynamic properties. The relevance of Landau gauge for the calculation of the ground state energy is explained before we present the results of an exact calculation of this energy in terms of the planar density of vortices which serves as a new parameter of complexity. At finite temperature an algorithm is used which is shown to be able to define and update clusters of spins in a gauge invariant way. As a first application of the cluster algorithm in the context of gauge theories, we study QCD in its dimensionally reduced form near the critical temperature by embedding Ising spins into the theory and mapping it to a frustrated spin model in d=3 dimensions with dynamically generated coupling constants. The topic of the last chapter is an investigation of the fundamentals of the standard model of electroweak interactions as well as the quest for an alternative mechanism for liberation of weak isospin which does not rely on the condensation of fundamental Higgs bosons. To this end we study a SU(3) gauge theory in four space time dimensions coupled to a scalar field in the adjoint representation which enables a transition to a phase characterized by a residual symmetry group SU(2) x U(1). Our numerical results concerning both the static potential of external sources carrying weak isospin and hypercharge and the spectrum of the theory suggest that there might be a point in the phase diagram where an effective quantum field theory is realized which can be characterized by the absence of both of a mass gap and the confinement phenomenon although the symmetry group it is based upon includes a non-abelian factor.