Novel Phenomena in Superconductors and Superfluids: Boundary States,
Spiral Magnetic Fields, and Solitons.

Abstract

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.