Nature Nanotechnology 11, 444–448 (2016); published online 18 January 2016; corrected online 23 March 2016; corrected after print 20 July 2016

In the version of this Article originally published, the equation 'W = (1/4π) ∫ s · (∂xs × ∂ys) dxdy' was missing the symbol for the vector product and the first erratum did not correct the problem. Further, unrelated, changes were also required: in the sentence beginning 'We present next the micromagnetic simulations...', 'ten' should have been 'eleven', and in the sentence beginning 'With the smallest exchange constant...', '1.6 pJ m−1' should have read '1.6 mJ m−2'. All of these errors have now been corrected in the online versions of the Article.