S.D. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative, under grant GBMF5307. Funding for the synthesis and characterization work (to A.B.M., D.G.S. and R.R.) was also provided by the Army Research Office under grant W911NF-16-1-0315. Y.L.T. and R.R. acknowledge support by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract number DE-AC02-05-CH11231 (Quantum Materials program KC2202) for detailed polarization vector analysis. M.R.M. acknowledges support from the National Science Foundation Graduate Research Fellowship under grant number DGE-1106400. M.A.P.G. and J.Í. are funded by the Luxembourg National Research Fund through the CORE programme (grant FNR/C15/MS/10458889 NEWALLS). Z.H. acknowledges support from the National Science Foundation (DMR-1210588). F.G.-O., P.G.-F. and J.J. acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through grant number FIS2015-64886-C5-2-P, and P.G.-F. acknowledges support from Ramón y Cajal grant number RyC-2013-12515. L.Q.C. is supported by the US DOE, Office of Basic Energy Sciences under award FG02-07ER46417. V.A.S. acknowledges support from the US DOE, Office of Science, Office of Basic Energy Sciences, under award number DE-SC- 0012375. M.R.M. and S.D. acknowledges use of the Advanced Photon Source, which was supported by the US DOE, Office of Science, Office of Basic Energy Science (DE-AC02-06CH11357), for the synchrotron-based reciprocal space map studies of samples at the Sector 33-BM-C and 7-ID-C beamline. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility, under contract number DE-AC02-05CH11231. L.W.M. acknowledges support from the US DOE, Office of Science, Office of Basic Energy Sciences, under award number DE-SC-0012375 for the development of novel ferroic heterostructures. Electron microscopy of superlattice structures was performed at the Molecular Foundry, LBNL, supported by the Office of Science, Office of Basic Energy Sciences, US DOE (DE-AC02-05CH11231). STEM and phase-field analysis and visualization performed by C.T.N. was supported by the US DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. EMPAD-STEM polar mapping at the Cornell Center for Materials Research was funded by the US DOE, grant DE-SC0002334. The Cornell EM Facility is supported by the Cornell Center for Materials Research through the National Science Foundation MRSEC programme, award #DMR DMR-1719875.

Reviewer information

Nature thanks Marty Gregg, Sergey Prosandeev and the other anonymous reviewer(s) for their contribution to the peer review of this work.