This thesis is a compilation of theoretical works focused on simulating and studying open questions regarding single and multiband superconductivity. In the last decades, with the discovery of multiband superconductors, the spectrum of potential applications has greatly widened. Superconductors are not only employed to realize dissipationless current carrying devices, but are used to construct quantum-based measurement instruments, such as single photon detectors as well as superconducting qubits. The properties of superconductors, as critical temperatures and vortex nucleation barriers are of key importance for applications, and still poorly understood. They are strongly affected by the physics of the boundaries, as well as by the sample's geometry and by the presence of impurities. The open questions can be answered with new theoretical methods, which can then guide and optimize the construction process of superconducting devices, which constitutes a crucial challenge today. 

There are several models that can be utilized to describe superconductors, from the microscopic Bardeen Cooper Schrieffer theory, up to the macroscopic Ginzburg Landau models. Each of these theories carries advantages and limitations, making it impossible to rely only on a specific model. In this thesis we utilize microscopic and macroscopic models to answer the following questions:

• How can we determine the free energy barriers to vortex nucleation in single band and multiband superconductors without relying on uncontrolled approximations?
• What are the properties of the superconducting states which spontaneously break time reversal symmetry?
• How do boundaries and interfaces influence the critical temperatures of superconductors?

We answer these questions in eight papers, which we shortly summarize in the following. 

In Paper 1, Magnetic signatures of domain walls in s+is and s+id superconductors: Observability and what that can tell us about the superconducting order parameter, we consider an effective two-band anisotropic Ginzburg Landau model, describing a superconductor breaking time reversal symmetry. There is high interest on spontaneous time reversal symmetry breaking due to recent muon-spin rotation experiments, claiming to measure spontaneous magnetic field in Fe-based superconductors such as Ba_1-xK_xFe₂As₂. However, the symmetry of the superconducting order parameters remains undetermined, and the most promising candidates are s+is and s+id states. In the work, we obtain solutions for domain walls within the Ginzburg Landau model.  By studying the spontaneous magnetic signatures of domain walls, having different orientations with respect to the crystalline axes, for both s+is and s+id order parameters, we demonstrate their differences and propose a procedure to infer the order parameter's symmetry from magnetic field measurements.

In Paper 2, Vortex nucleation barrier in superconductors beyond the Bean-Livingston approximation: A numerical approach for the sphaleron problem in a gauge theory, we address the long standing problem of calculating the energy barriers for the vortex nucleation in a superconductor. The only available tool to do so, was the Bean-Livingston theory, which relies on uncontrollable approximations. This does not allow to take into account the non-linear nature of the Ginzburg Landau model, or the presence of impurities and surface roughness. Therefore, we develop the gauged string method, a gauge invariant numerical framework, based on the simplified string method, which enables us to accurately compute the minimum free energy path for the vortex nucleation. Moreover, we present a study of how the nucleation energy barrier changes in the presence of impurities and surface roughness. 

In Paper 3, Vortex nucleation barriers and stable fractional vortices near boundaries in multicomponent superconductors, we extend the gauged string method to multiband superconductors, where the energy landscape is much broader than in the single band case, and the number of possible processes is higher. In multiband superconductors the topological excitations are fractional vortices, which once bounded, form composite vortices. Fractional vortices are energetically unfavorable, as they are associated to an energy cost which scales logarithmically with the system size. Once they bind and form a composite vortex, the extra energy cost is canceled. However, it was previously shown in the London model that fractional vortices can be stabilized near boundaries. In this paper, we study the energy barriers for the nucleation of fractional vortices, and for the formation composite vortices. Moreover, we show how the presence of anisotropies can influence such barriers. Then we study how the same processes are influenced by the interband Josephson interactions. By using the gauged string method, we demonstrate how the fractionalized nucleation process results in multiple saddle points and intermediate metastable configurations.

In Paper 4, Boundary effects in two-band superconductors, we study microscopically the behavior of the superconducting order parameters near the boundaries of a two-band s-wave superconductor. We describe the system using a tight binding Bardeen Cooper Schrieffer model with interband interaction. We show the existence of surface states, and calculate how the difference between bulk and surface critical temperatures depends on the strength of the interband coupling. Then, we focus the analysis on weak interband interactions to show, at the level of a fully microscopic theory, how the variations of the gaps near the boundaries occur with multiple length scales. 

In Paper 5, Spontaneous edge and corner currents in s+is superconductors and time-reversal-symmetry-breaking surface states, we consider a three band superconductor, described with a microscopic tight binding Bardeen Cooper Schrieffer model with interband interaction. In the current classification scheme, an s+is state is a non-topological and non-chiral state, which does not exhibit topological surface states and therefore no spontaneous surface currents. In the article, we consider a system where the three bands have slightly different intraband pairing potential but the same interband coupling, resulting in slightly asymmetric bands. We show that, as the temperature is increased, the state which spontaneously break time reversal symmetry becomes localized near the sample boundaries, and generate spontaneous magnetic signatures. Finally, we show how, by changing the sample geometry, the magnetic signatures can be enhanced. We underline that, this phenomenon is not a general property of time reversal symmetry breaking states, but can account for the presence of spontaneous magnetic fields in s+is superconductors and cannot be predicted using the macroscopic Ginzburg Landau theory. Moreover, the paper shows that spontaneous surface currents can arise for non-topological reasons.

In Paper 6, Demonstration of CP2 skyrmions in three-band superconductors by self-consistent solutions to a Bogoliubov-de-Gennes model, we continue the study of three component s+is superconductors, described using a microscopic tight binding Bardeen Cooper Schrieffer model. In this work, we consider three symmetric bands, and we study the CP2 skyrmionic topological excitations of the system. We present not only the configurations of the superconducting order parameters, but also the respective magnetic field and density of states. Moreover, the simulation of CP2 skyrmions in superconductors, described a with fully microscopic model, had not been done before. In the context of superconductivity, CP2 skyrmion solutions were previously described only within the phenomenological macroscopic Ginzburg-Landau theory.

In Paper 7, Pair-density-wave superconductivity of faces, edges, and vertices in systems with imbalanced fermions we analyze the boundary effects in superconductors exhibiting Fulde-Ferrell-Larkin-Ovchinnikov states. We do so by employing and comparing Bogoliubov-de-Gennes and Ginzburg Landau formalisms. We show that, within the Ginzburg Landau theory, in a three dimensional superconductor, there is a sequence of phase transitions as the temperature increases. Then, we perform the same sequence of simulations for two dimensional samples described using the Bogoliubov-de-Gennes formalism, showing the same sequence of phase transitions.

In Paper 8, Elevated critical temperature at BCS superconductor-band insulator interfaces, we study the physics of interfaces between a superconductor, described using a tight-binding mean field Hamiltonian, and a band insulator. We limit the study to one-dimensional samples and demonstrate that, within certain parameter ranges, it is indeed possible to enhance the critical temperature in the vicinity of the interface. This occurs without changing the strength of the superconducting-pairing interaction. Then we present the parameters regimes in which the near-interface critical temperature exceeds the critical temperature of a conventional superconductor-vacuum interface.