Crystallization of the superconducting phase in unconventional superconductors

Superconductivity is a macroscopic quantum phenomenon, in the sense that a macroscopic number of electrons form a pair condensate, that occupies a single ground state. The electrons in this state are phase-coherent, breaking global U(1)-symmetry, and spatial variations of the phase imply superflows that usually cost kinetic energy, resulting in a uniform and rigid phase. It would therefore be surprising if a more ordered state with a non-uniform phase existed. This thesis proposes that such a ground state can occur in the absence of external perturbations, deep inside the superconducting state, where a periodic pattern is spontaneously imprinted on the superconducting phase, breaking continuous translational invariance. The resulting phase gradients break time-reversal symmetry, manifested through finite superflows and equilibrium charge currents with peculiar patterns. In analogy to crystallization in solids, the new order parameter is defined as a finite Fourier amplitude at the wavevector corresponding to the phase-periodicity. This ground state is hence referred to as a phase crystal. The thesis employs the quasiclassical theory of superconductivity, combined with a non-local Ginzburg-Landau theory, to derive the inhomogeneous superfluid density tensor and the conditions under which phase crystallization can occur. It is shown how the phase can be realized at certain interfaces of unconventional superconductors, and in conventional superconductor-ferromagnet structures. The instability phase diagram is obtained, and the transition classified as second-order, surviving moderately strong external fields. The phase is tied to critical points in the superflow field, satisfying a generalized Poincare-Hopf theorem. Geometric perturbations and disorder are studied, and characteristic signatures identified, in an attempt to aid experimental efforts in potential realization of the phase. In conclusion, the model based on the non-local superfluid tensor provides a unified approach to studying surface phenomena, e.g. topological states and inhomogeneous superconductivity, and is used to both verify and explain several previous numerical observations. The model directly highlights the role of non-local correlations and phase variations as drivers in phase transitions, motivating a search for new non-local phenomena in various condensed matter systems.