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The image shows some of the phase- and current-patterns predicted by the researchers' theory, which are the same patterns found in previous numerical simulations. Illustration: Patric Holmvall

Quantum mechanical phase forms a crystal

Researchers at Chalmers University of Technology in Sweden and Montana State University in the US have developed a theory that derives a so-called "phase crystal", that elicits spontaneous magnetic fields and circulating currents. The theory predicts when a phase crystal can arise, explaining previous numerical results, and is presented in an article recently published in the scientific journal Physical Review Research.
Quantum mechanical states are described by a complex-valued wave function, which similar to a wave has both an amplitude and a phase. In contrast to a classical wave, the amplitude and phase of the wave function are related to purely quantum mechanical phenomena which lack an analogue in classical physics.
“A perfect example is superconductivity, which is a quantum-mechanical state that arises in certain materials due to electron pairing. The pairs have a quantum-mechanical wave function with an amplitude corresponding to the pair density, and a phase which is related to the pair momentum. The pairs move like an inviscid fluid through the material, with zero electrical resistance”, explains Patric Holmvall (below to the left), researcher at the Applied Quantum Physics Laboratory at MC2, and the lead author of the article.
Picture of Patric Holmvall.The researchers’ study shows that in certain superconductors with pathological edges that destroy superconductivity, the kinetic energy can change sign and become favorable as it “heals” the destroyed superconductivity.
“We find that the phase crystallizes and form a periodic pattern, which in turn creates a checker-board pattern of circulating currents and spontaneous magnetic fields”, says Patric Holmvall.
Currents and magnetic fields usually only enter superconductors under external influence and perturbations, but now arise spontaneously. This is an example of spontaneous pattern-formation, where inhomogeneities which usually cost energy instead heal a destroyed system.
“We have derived the conditions for phase crystallization and use a microscopic theory to show that these conditions are satisfied in for example the material YBCO. Our theory combines and explains a number of theoretical studies reaching all the way back to the 1990s, in particular our previous numerical results, which were recently published in Nature Physics and Nature Communications”, says Patric Holmvall.
The researchers' studies show that phase crystals represent a unique class of inhomogeneous ground states.
“To derive the conditions for phase crystallization, we had to generalize the commonly used Ginzburg-Landau theory, to take into account non-local interactions. Since this theory is used not just to study superconductivity, but also in, for instance, biological physics and liquid crystals, we think that new interesting phenomena might be discovered within these disciplines through a similar generalization”, says Patric Holmvall.
The new study has several connections to previous research at Chalmers. Patric Holmvall gives examples of the beautiful patterns found in liquid crystals, the organization of cells and bacteria in thin films, or structural coloration and iridescence in plants and animals, the latter caused by so-called photonic structures. These exemplify how surface interactions can trigger spontaneous pattern formation.
In addition to Patric Holmvall, the Chalmers professors Mikael Fogelström and Tomas Löfwander, as well as Anton Vorontsov at Montana State University in the US, have co-authored the article “Phase crystals”. It was highlighted as Editor's Suggestion, where extra interesting and well-written articles are selected.
Text: Michael Nystås
Illustration: Patric Holmvall
Photo of Patric Holmvall: Kevin Marc Seja
Patric Holmvall, Applied Quantum Physics Laboratory, Department of Microtechnology and Nanoscience – MC2, Chalmers University of Technology,


Phase crystals differ from other inhomogeneous superconducting states (e.g. Abrikosov-vortices and the Fulde-Ferell-Larkin-Ovchinnikov state), as they appear mainly at low temperatures even in the absence of external magnetic fields. Furthermore, an analysis of the free energy shows that it is mainly the phase rather than the amplitude which drives and characterizes the phase transition, in contrast to the traditional picture in superconductivity.


The superconducting ground state is characterized by the pair wave function, with an amplitude proportional to the pair density, and where variations in the phase are proportional to both the pair momentum and the electromagnetic potential. For a given system, the wave function assumes the amplitude and phase with the lowest free energy. Since pairs with a finite momentum (i.e. finite variations in the phase) lead to a kinetic energy, the ground state normally assumes a uniform phase without variations. The ground state is thereby mainly characterized by the amplitude and has zero kinetic energy.


Associate Professor Per Rudquist's Liquid Crystals Research:
Photonic Structures - research at the Photonics Laboratory, among others:
The conditions for the formation of phase crystals are fulfilled in, for example, superconductors of the material YBCO:
Gustafsson, D., Golubev, D., Fogelström, M. et al. Fully gapped superconductivity in a nanometre-size YBa2Cu3O7–δ island enhanced by a magnetic field. Nature Nanotech 8, 25–30 (2013).
The theory combines and explains a number of theoretical studies since the 1990s, especially previous numerical results:
High temperature superconductors can fulfill the hairy ball theorem
The hairy ball theorem in mathematics says that one cannot comb a hairy ball smoothly without forming a vortex. One consequence of this is that there must always be at least one cyclone somewhere on earth. In 2018, researchers at Chalmers conducted a theoretical study of high-temperature superconductors and concluded that there is a low-temperature phase at the edges of the material described by an order parameter, a two-dimensional vector field, which must also fulfill a variant of the hairy ball theorem.
Holmvall, P., Vorontsov, A.B., Fogelström, M., and Löfwander, T., Broken translational symmetry at edges of high-temperature superconductors, Nature Communications 9, 2190 (2018).
A necklace of fractional vortices
Researchers at Chalmers have arrived at how what is known as time-reversal symmetry can break in a class of superconducting materials. Small circulating currents and magnetic fields are created at their edges. Adjacent circulating currents have opposite circulation, which generates magnetic fields of opposite sign. This effect causes the material to appear to have been dressed with a necklace of small magnetic fluxes.
Håkansson, M., Löfwander, T. and Fogelström, M. (2015) Spontaneously broken time-reversal symmetry in high-temperature superconductors, Nature Physics (1745-2473), Vol. 11 (2015), 9, pp. 755-760.

Published: Mon 11 May 2020.