The complex interplay of spin, charge, orbital and lattice degrees of freedom provides a plethora of exotic phases and physical phenomena. In recent years, complex spin topologies have emerged as a consequence of the electronic band structure and the interplay between spin and spin–orbit coupling in materials. Here we produce complex topologies of electrical polarization—namely, nanometre-scale vortex–antivortex (that is, clockwise–anticlockwise) arrays that are reminiscent of rotational spin topologies—by making use of the competition between charge, orbital and lattice degrees of freedom in superlattices of alternating lead titanate and strontium titanate layers. Atomic-scale mapping of the polar atomic displacements by scanning transmission electron microscopy reveals the presence of long-range ordered vortex–antivortex arrays that exhibit nearly continuous polarization rotation. Phase-field modelling confirms that the vortex array is the low-energy state for a range of superlattice periods. Within this range, the large gradient energy from the vortex structure is counterbalanced by the corresponding large reduction in overall electrostatic energy (which would otherwise arise from polar discontinuities at the lead titanate/strontium titanate interfaces) and the elastic energy associated with epitaxial constraints and domain formation. These observations have implications for the creation of new states of matter (such as dipolar skyrmions, hedgehog states) and associated phenomena in ferroic materials, such as electrically controllable chirality.

10aElectronic properties and materials10aFerroelectrics and multiferroics10ainterfaces and thin films10asurfaces1 aYadav, Ajay, K.1 aNelson, Christopher, T.1 aHsu, Shang-Lin1 aHong, Zijian1 aClarkson, James, D.1 aSchlepüetz, Christian, M.1 aDamodaran, Anoop, R.1 aShafer, Padraic1 aArenholz, Elke1 aDedon, Liv, R.1 aChen, Deyang1 aVishwanath, Ashvin1 aMinor, Andrew, M.1 aChen, Long-Qing1 aScott, James, F.1 aMartin, Lane, W.1 aRamesh, Ramamoorthy uhttps://rameshlab.lbl.gov/publications/observation-polar-vortices-oxide01991nas a2200349 4500008004100000022001400041245012600055210006900181260001200250490000700262520093100269653002001200653002501220653001301245653000801258653001501266653002201281100001401303700002201317700002001339700002101359700002301380700001701403700001101420700002101431700001701452700002301469700002601492700001801518700002401536856008101560 2016 eng d a2469-995000aStrain-Induced Nonsymmorphic Symmetry Breaking and Removal of Dirac Semimetallic Nodal Line in an Orthoperovskite Iridate0 aStrainInduced Nonsymmorphic Symmetry Breaking and Removal of Dir c02/20160 v933 aBy using a combination of heteroepitaxial growth, structure refinement based on synchrotron x-ray diffraction, and first-principles calculations, we show that the symmetry-protected Dirac line nodes in the topological semimetallic perovskite SrIrO_{3} can be lifted simply by applying epitaxial constraints. In particular, the Dirac gap opens without breaking the *Pbnm* mirror symmetry. In virtue of a symmetry-breaking analysis, we demonstrate that the original symmetry protection is related to the *n*-glide operation, which can be selectively broken by different heteroepitaxial structures. This symmetry protection renders the nodal line a nonsymmorphic Dirac semimetallic state. The results highlight the vital role of crystal symmetry in spin-orbit-coupled correlated oxides and provide a foundation for experimental realization of topological insulators in iridate-based heterostructures.