UNCONVENTIONAL COOPER PAIRING AND CONFINEMENT Studies of lattice superconductivity and superfluid helium-3 enclosed by surfaces

Abstract

Superconductivity and superfluidity --- two related and quintessentially quantum properties of matter --- manifest in a variety of interacting many-particle systems at low temperatures. The conceptual breakthrough for understanding these properties in fermions was provided by Bardeen, Cooper and Schrieffer in 1957. Their theory demonstrated how a coherent macroscopic quantum state can arise from the microscopic pairing of two fermions into a bound state. Although most well-known for inviscid flow, these many-body systems can showcase a wide array of properties depending both on their internal microscopic states and externally imposed conditions. The internal state of the bound pairs sets a length scale known as the coherence length, and confining these systems to the order of the coherence length can give rise to phases which would otherwise be energetically unfavourable in the bulk. This thesis explores how superconductivity and superfluidity are affected by the presence of strongly confining surfaces. For the project on superconductivity, we use three different techniques --- the Bardeen-Cooper-Schrieffer (BCS), the Bogoliubov-de Gennes (BdG) and the Anderson methods --- to calculate the superconducting order parameters on a one-dimensional (1D) and two-dimensional (2D) lattice. For lattices with periodic boundary conditions (PBCs), these methods are means to the same end, producing identical ``bulk'' order parameters. However, using open boundary conditions (OBCs) exposes the underlying assumptions and limitations of each method; the BCS method is inapplicable to systems with OBCs, while the BdG and Anderson methods produce varying results for identical parameters. To contrast these methods, we first map out the bulk phase diagrams for 1D and 2D lattices using the BCS method, showing how these phase diagrams can change dramatically from the onset of superconductivity at T = Tc down to T = 0. We then explore the T = 0 superconducting phases with OBCs, using the results from the BdG method as a benchmark for comparison with the Anderson method. The BdG method allows us to calculate the OBC superconducting phases to a high degree of accuracy, while the Anderson method provides a trade-off of accuracy for a computationally simpler approach. However, the simplicity of the Anderson method stems from approximation, and we show that these approximations lead to its failure for some of the superconducting phases. On the superfluid side, we study superfluid helium-3 (3-He) when it is confined to a quasi-2D slab. Using Ginzburg-Landau (GL) theory, we construct a theory of a new spatially-modulated superfluid phase called the 2D pair density wave (PDW) phase. This phase demonstrates spontaneous symmetry breaking of the spatial translation and rotation symmetries of the superfluid. However, it still retains its zero-viscosity transport properties across the transition, and thus could be described as a superfluid crystal phase. This work was motivated by recent experiments which have found evidence of a new superfluid 3-He phase appearing under quasi-2D confinement. We hope that our theoretical work will spur further investigation of this confined phase and its predicted spatial structure.