Crystallization of the superconducting phase in unconventional superconductors
Doctoral thesis, 2019

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 Poincaré-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.

pattern formation

spontaneous symmetry breaking

Andreev bound states

mesoscopic superconductivity

phase crystals

thin superconducting films

phase transitions

non-local Ginzburg-Landau theory

time-reversal symmetry

flat bands

unconventional superconductivity

quasiclassical theory

translational symmetry


inhomogeneous superconductivity

Kollektorn (A423), 4th floor
Opponent: Professor Manfred Sigrist, ETH Zürich, Switzerland


Patric Holmvall

Chalmers, Microtechnology and Nanoscience (MC2), Applied Quantum Physics

P. Holmvall, M. Fogelström, T. Löfwander and A. B. Vorontsov, "Phase Crystals", arXiv:1906.04793 (2019)

Broken translational symmetry at edges of high-temperature superconductors

Nature Communications,; Vol. 9(2018)

Journal article

Spontaneous generation of fractional vortex-Antivortex pairs at single edges of high-Tc superconductors

Journal of Physics: Conference Series,; Vol. 969(2018)

Paper in proceedings

Nanoscience and nanotechnology deal with materials on the atomic scale, where interface effects and quantum phenomena become increasingly relevant. It constitutes one of the most active areas of research, with bold promises of new groundbreaking technologies. A cornerstone in this line of research are superconducting devices, due to their extraordinary properties that couple the microscopic and macroscopic regimes. Superconductivity is typically characterized by zero electrical resistance and a perfect expulsion of magnetic fields. It would therefore seem counterintuitive that superconductors could develop and host such magnetic fields spontaneously. In this thesis, it is studied how currents and magnetic fields might appear in the absence of external perturbations at low temperatures, and furthermore, arranged in peculiar periodic patterns. The necessary conditions for this to occur are derived, and are shown to be satisfied in certain nano-scale systems due to interface and quantum effects. The influence that this might have on superconducting devices and nanotechnology is discussed.

Areas of Advance

Nanoscience and Nanotechnology (2010-2017)


Basic sciences


C3SE (Chalmers Centre for Computational Science and Engineering)

Subject Categories

Other Physics Topics

Condensed Matter Physics



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4684


Chalmers University of Technology

Kollektorn (A423), 4th floor

Opponent: Professor Manfred Sigrist, ETH Zürich, Switzerland

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