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.

This PhD thesis presents a comprehensive study of superconducting and superfluid states in a variety of physical systems.The research is based on 12 papers that explore different phenomena in-depth.

One of the significant contributions of the thesis is the discovery of boundary states in superconductors.The Bardeen-Cooper-Schrieffer (BCS) theory describes the superconducting transition as a single critical point where the gap function or order parameter vanishes uniformly in the entire system. However, in papers $4$ and $8$ we have shown that in a superconductor described by the BCS model, the superconducting gap is enhanced and can survive at surfaces at higher temperatures than in the bulk. This result suggests that conventional superconductors have multiple critical points associated with separate phase transitions at surfaces and bulk. We demonstrate this by finding inhomogeneous solutions of the BCS gap equation near boundaries, which asymptotically decay in the bulk.We revise the microscopic derivation of the superconductor-insulator boundary conditions in paper $8$ for the Ginzburg-Landau (GL) model and obtain a negative contribution to free energy in the form of the surface integral. We show that the boundary conditions for the conventional superconductor have the form $\textbf{n} \cdot \nabla \psi = \text{const} \psi$, which follows from considering the order parameter reflected in the boundary. Additionally, we demonstrate that in the case of an applied external field, the third critical magnetic-field value $H_{c3}$ is higher than what follows from the de Gennes boundary conditions. For two-band superconductors, we show in paper $7$ that boundaries induce variations of the gaps with the presence of multiple length scales.We demonstrate in papers $1,\ 3$, and $8$ that spin-imbalanced superconductors have boundary states with enhanced gap and critical temperature near the surface. We generalize the GL formalism for these systems and show that a cubic superconductor at a mean-field level has a sequence of phase transitions. These transitions occur as the temperature increases, and superconductivity disappears first in the bulk, then at surfaces, next at edges, and finally in the vertices.We also study boundary states for systems on honeycomb lattices in paper $12$.Next, we considered the interface between a BCS superconductor and a non-superconducting band insulator in paper $10$. We showed that such interfaces can have an elevated superconducting critical temperature (higher than at the superconductor-vacuum interface) without increasing the strength of pairing interaction at the interface.

Another significant contribution of the thesis is the study of spiral magnetic fields, unconventional magnetic response, and vortex states in noncentrosymmetric superconductors.Paper $6$ discusses the microscopic derivation of the Ginzburg-Landau free energy, which shows that the system's magnetic response and ratio of coherence and magnetic field penetration lengths can change significantly with temperature due to spin-orbit and Zeeman coupling. The magnetic field in such superconductors decays in spirals, leading to non-monotonic intervortex and vortex-boundary interaction and the formation of bound states with other vortices, antivortices, and boundaries. Paper $11$ shows magnetic dipole or ferromagnetic inclusion in noncentrosymmetric superconductors induces self-knotted magnetic field configurations called "toroflux", which are the superconducting counterparts of the Chandrasekhar-Kendall states in astrophysical and nuclear-fusion plasmas.

Finally, the thesis explores solitons in imbalanced fermionic systems.Paper $5$ presents a study on stable solitons in superfluids with the fermionic imbalance and uniform ground state. The solitons are formed of radial density modulations resulting in nodal rings and can exhibit nontrivial soliton-soliton and soliton-vortex interactions.Paper $2$ shows that in multicomponent imbalanced fermion mixtures, the superfluid states can form three-dimensional lattices of linked vortex loops, which can be interpreted in terms of skyrmions. These solutions are termed "synthetic nuclear Skyrme matter".Paper $9$ proposes a new generalization of crystalline order, called "ground state fractal crystals", which are crystals whose unit cells are fractals. We derive a simple model whose ground state is a fractal crystal.

Denna PhD avhandlingen presenterar en omfattande studie av supraledande och superfluid tillstånd i en mängd olika fysiska system.Forskningen bygger på 12 artiklar som utforskar olika fenomen på djupet.

Ett av avhandlingens betydelsefulla bidrag är upptäckten av gränstillstånd i supraledare.Bardeen-Cooper-Schrieffer (BCS) teorin beskriver den supraledande övergången som en enda kritisk punkt där gapfunktionen eller ordningsparametern försvinner enhetligt i hela systemet. Men i tidningar $4$ och $8$ har vi visat att i en supraledare som beskrivs av BCS-modellen, är det supraledande gapet förstärkt och kan överleva vid ytor vid högre temperaturer än i bulken. Detta resultat tyder på att konventionella supraledare har flera kritiska punkter associerade med separata fasövergångar vid ytor och bulk. Vi demonstrerar detta genom att hitta inhomogena lösningar av BCS gap-ekvation nära gränser, som asymptotiskt sönderfaller i bulken.Vi reviderar den mikroskopiska härledningen av gränsvillkoren för supraledare-isolator i papper $8$ för Ginzburg-Landau (GL) modellen och erhåller ett negativt bidrag till fri energi i form av ytintegralen. Vi visar att randvillkoren för den konventionella supraledaren har formen $\textbf{n} \cdot \nabla \psi = \text{const} \psi$, vilket följer av att betrakta ordningsparametern som återspeglas i gränsen. Dessutom visar vi att i fallet med ett applicerat externt fält är det tredje kritiska magnetfältsvärdet $H_{c3}$ högre än vad som följer av de Gennes gränsvillkor. För tvåbandssupraledare visar vi i papper $7$ att gränser inducerar variationer av gapen med närvaron av multipla längdskalor.Vi visar i uppsatser $1,\ 3$ och $8$ att spinn-obalanserade supraledare har gränstillstånd med ökat gap och kritisk temperatur nära ytan. Vi generaliserar GL-formalismen för dessa system och visar att en kubisk supraledare på medelfältsnivå har en sekvens av fasövergångar. Dessa övergångar sker när temperaturen ökar, och supraledning försvinner först i bulken, sedan vid ytorna, därefter vid kanterna och slutligen i hörnen.Vi studerar också gränstillstånd för system på bikakegaller i papper $12$.Därefter övervägde vi gränssnittet mellan en BCS-supraledare och en icke-supraledande bandisolator i papper $10$. Vi visade att sådana gränssnitt kan ha en förhöjd supraledande kritisk temperatur (högre än vid supraledare-vakuumgränssnittet) utan att öka styrkan av parningsinteraktion vid gränssnittet.

Ett annat betydande bidrag från avhandlingen är studiet av spiralmagnetiska fält, okonventionell magnetisk respons och virveltillstånd i icke-centrosymmetriska supraledare.Papper $6$ diskuterar den mikroskopiska härledningen av den fria energin från Ginzburg-Landau, som visar att systemets magnetiska respons och förhållandet mellan koherens och magnetfältets penetrationslängder kan förändras avsevärt med temperaturen på grund av spin-orbit och Zeeman-koppling. Magnetfältet i sådana supraledare sönderfaller i spiraler, vilket leder till icke-monotonisk intervirvel och virvelgränsinteraktion och bildandet av bundna tillstånd med andra virvlar, antivirvlar och gränser. Papper $11$ visar magnetisk dipol eller ferromagnetisk inneslutning i icke-centrosymmetriska supraledare inducerar självknutna magnetfältskonfigurationer som kallas ``toroflux", som är supraledande motsvarigheter till Chandrasekhar-Kendall-tillstånden i astrofysiska och nukleära fusionsplasma.

Slutligen, utforskar avhandlingen solitoner i obalanserade fermioniska system.Paper $5$ presenterar en studie om stabila solitoner i supervätskor med fermionisk obalans och enhetligt grundtillstånd. Solitonerna är bildade av radiella densitetsmoduleringar som resulterar i nodalringar och kan uppvisa icke-triviala soliton-soliton- och soliton-virvelinteraktioner.Papper $2$ visar att i flerkomponents obalanserade fermionblandningar kan de superfluidiska tillstånden bilda tredimensionella gitter av länkade virvelslingor, vilket kan tolkas i termer av skyrmioner. Dessa lösningar kallas ``syntetisk nukleär Skyrme-materia".Paper $9$ föreslår en ny generalisering av kristallin ordning, kallad ``grundtillståndsfraktalkristaller", som är kristaller vars enhetsceller är fraktaler. Vi härleder en enkel modell vars grundtillstånd är en fraktal kristall.

This thesis focuses on superconductivity, a field within condensed matter physics which since its experimental discovery roughly a century ago, not only has lead to significant contributions revealing the fundamental theories of physics, but also to practical applications. This includes for example quantum vortices, which play paramount roles both in other condensed matter settings, but also in high-energy physics. The dissipationless currents in superconductors are essential to achieve the strong magnetic fields necessary when performing Magnetic Resonant Imaging (MRI). 

My research on superconductors spans across three topics: superconducting surfaces, multiband superconductivity and inhomogeneous states formed in imbalanced superfluids. A brief introduction and summary of the scientific contribution of this thesis to each of these topics is given below. 

Bardeen-Cooper-Schrieffer (BCS) theory tells us that from a microscopic perspective, superconductivity is the phenomenon of condensation of bound electron pairs, so-called Cooper pairs. The superconducting state is described by a complex-valued field known as the superconducting gap parameter. In the most simple superconducting materials, where there is only one electronic band, only one complex field is necessary to describe the superconducting state, which spontaneously breaks U(1)-symmetry. In other superconducting materials, such as the iron-based superconductor Ba_1−xK_xFe₂As₂, the band structure is more complicated and multiple electronic bands are present. Such multiband superconductors may require multiple complex fields to describe the superconducting state, which can spontaneously break other symmetries, such as time-reversal symmetry, in addition to U(1)-symmetry. 

Two proposed pairing symmetries for spontaneous time-reversal symmetry breaking (TRSB) spin-singlet superconductors are s+is and s+id. In Paper IV, we demonstrate how magnetic features of pinned domain walls in anisotropic TRSB superconductors can be used to distinguish between s+is and s+id pairing. 

Classifying topological excitations in superconductors is crucial to understand the superconducting state. For example, quantum vortices are key in understanding the magnetic response of type-II superconductors, and the thermal fluctuations-induced phase transitions in superconductors and superfluids. It has been hypothesized that multiband superconductors, which are described by multiple complex fields, can host topological excitations which are different from the ordinary quantum vortices. Understanding the properties of these new topological excitations carries similar importance to that of ordinary quantum vortices. In Paper VII and Paper VIII, we provide the first microscopic demonstration of multiband fractional vortices and CP²-skyrmions using fully self-consistent Bogoliubov-de Gennes (BdG) theory. Previous demonstrations of such topological excitations have been done using classical field theory approaches, such as Ginzburg-Landau (GL) theory. Our BdG calculations maintain microscopic degrees of freedom which are neglected using GL and quasiclassical theories of superconductivity. 

The most well-known inhomogeneous superconducting phase is the Abrikosov vortex lattice, which forms in the presence of an external magnetic field in type-II superconductors. Fulde, Ferrell, Larkin and Ovchinnikov (FFLO) proposed another type of inhomogeneous superconducting state, which may form in the presence of a sufficiently large population imbalance between spin up and spin down electrons. The origin of this supersolid state is the formation of Cooper pairs with non-zero net momentum due to spin-dependent Fermi surfaces. In Paper V, we show that spin-imbalanced superfluids can host a unique type of solitons, even before the FFLO regime is entered. These solitons are not present in ordinary uniform superconducting states, and can therefore act as identifiable traces of the FFLO state. 

The Fulde-Ferrell state and the Larkin-Ovchinnikov state are characterized respectively by modulation in the phase and the density of the superconducting gap parameter. In Paper II, we explored the possibility of other types of inhomogeneous states caused by imbalance in multiband superconductors. Using GL theory, we demonstrated two new types of inhomogeneous states, characterized by spatially alternating chirality and nematicity. 

Understanding the superconducting properties of surfaces and boundaries is important, both fundamentally to the theory of superconductivity and practically in the construction of superconducting devices. In Paper I and Paper III we demonstrate using both GL and BdG theory that pair-density-wave superconductors support superconducting surface states with critical temperatures larger than the bulk critical temperature. In Paper VI we show increased critical temperatures of superconductor-insulator interfaces. The increase in critical temperature occurs without locally increasing the superconducting pairing strength near the boundaries, or without the introduction of modified surface phonons. 

Den här avhandlingen fokuserar på supraledning, ett forskningsområde inom kondenserade materiens fysik som sedan dess upptäckt för cirka ett sekel sedan har haft stor betydelse både för grundläggande fysik och praktiska tillämpningar. Kvantvirvlar spelar till exempel en viktig roll både inom den kondenserade materiens fysik och högenergifysik. De starka magnetfälten som krävs i magnetkameror genereras tack vare resistansfria supraledare. 

Min forskning inom supraledning berör tre områden: supraledande ytor, flerbandssupraledning och inhomogena tillstånd i suprafluider med obalans. En kort introduktion och sammanfattning av denna avhandlings vetenskapliga bidrag till dessa områden ges nedan. 

Enligt Bardeen-Cooper-Schrieffer-teori uppstår supraledning genom kondensation av bundna elektronpar, så kallade Cooperpar. Det supraledande tillståndet beskrivs av ett komplexvärt fält som kallas det supraledande gapet. I de enklaste supraledarna med enbart ett elektronband behövs endast ett komplexvärt fält för att beskriva det supraledande tillståndet, som spontant bryter U(1)-symmetri. I andra supraledande material vars bandstruktur är mer komplicerad, som till exempel Ba_1−xK_xFe₂As₂, kan det behövas flera komplexvärda fält för att beskriva det supraledande tillståndet. Sådana flerbandssupraledare kan spontant bryta andra symmetrier, såsom tidsinversionssymmetri, utöver U(1)-symmetrin. 

s+is och s+id är två parningssymmetrier som föreslagits beskriva spin-singlet-supraledare som spontant bryter tidsinversionssymmetri. I artikel IV visar vi hur magnetfälten som genereras av domänväggar i anisotropa sådana supraledare kan användas för att särskilja de två föreslagna parningssymmetrierna. 

Att identifiera tillåtna topologiska excitationer i supraledare är avgörande för att förstå det supraledande tillståndet. Exempelvis så spelar kvantvirvlar en viktig roll då typ-II-supraledare utsätts för externa magnetfält och vid fasövergångar i supraledare och suprafluider orsakade av termiska fluktuationer. Det har föreslagits att flerbandssupraledare som beskrivs av flera komplexa fält kan inneha topologiska excitationer som skiljer sig från vanliga kvantvirvlar. Dessa topologiska excitationers egenskaper i flerbandssupraledare är lika viktiga som kvantvirvlars egenskaper är i vanliga supraledare. I artikel VII och artikel VIII presenterar vi de första mikroskopiska lösningarna av partiella virvlar och CP²-skyrmioner med hjälp av Bogoliubov-de Gennes (BdG)-teori. Sådana tillstånd har tidigare påvisats genom klassiska fältteorier såsom Ginzburg-Landau (GL)-teori. Våra BdG-beräkningar behåller mikroskopiska frihetsgrader som försummas i GL-teori och kvasiklassiska beskrivningar av supraledning. 

Abrikosovs virveltillstånd, som bildas i typ-II-supraledare i ett externt magnetfält, är det mest välkända inhomogena supraledande tillståndet. Fulde, Ferrell, Larkin och Ovchinnikov (FFLO) lade fram idén att ett annorlunda inhomogent tillstånd kan bildas i supraledare med tillräckligt stor spin-obalans. Detta supersolida tillstånd uppstår då Cooperpar med nollskild rörelsemängd bildas på grund av spin-beroende Fermiytor. I artikel V visar vi att suprafluider med spin-obalans kan inneha en unik typ av solitoner, trots att FFLO-tillståndet inte ännu har nåtts. Dessa solitoner är inte tillåtna i vanliga homogena supraledande tillstånd, och är därför identifierbara spår av FFLO-tillståndet. 

Fulde-Ferrell-tillståndet och Larkin-Ovchinnikov-tillståndet karaktäriseras av modulerad fas respektive amplitud av det supraledande gapet. I artikel II undersöker vi ifall det existerar andra typer av inhomogena tillstånd orsakade av obalans i flerbandssupraledare. Med hjälp av GL-teori hittade vi två nya sorters inhomogena tillstånd som karaktäriseras av rumsligt varierande kiralitet respektive nematicitet. 

Det är viktigt att förstå hur supraledning beter sig vid ytor, både ur ett fundamentalt perspektiv och i praktiken vid konstruktion av supraledande enheter. I artikel I och artikel III visar vi med hjälp av både GL- och BdG-teori att partäthetsvågssupraledare kan ha supraledande yttillstånd vars kritiska temperatur överskrider den supraledande bulkens kritiska temperatur. I artikel VI visar vi förhöjda kritiska temperaturer vid supraledare-isolator-ytskikt. Denna ökning i kritisk temperatur sker utan att lokalt öka parningsstyrkan nära ytskiktet, eller genom att introducera modifierade ytfononer.