Topological defects in matter behave collectively to form highly non-trivial structures called topological textures that are characterised by conserved quantities such as the winding number. Here we show that an epitaxial ferroelectric square nanoplate of bismuth ferrite subjected to a large strain gradient (as much as 105 m−1) associated with misfit strain relaxation enables five discrete levels for the ferroelectric topological invariant of the entire system because of its peculiar radial quadrant domain texture and its inherent domain wall chirality. The total winding number of the topological texture can be configured from − 1 to 3 by selective non-local electric switching of the quadrant domains. By using angle-resolved piezoresponse force microscopy in conjunction with local winding number analysis, we directly identify the existence of vortices and anti-vortices, observe pair creation and annihilation and manipulate the net number of vortices. Our findings offer a useful concept for multi-level topological defect memory.

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We investigate the ferroelectric state of a tetragonal BiFeO3 thin film grown on a LaAlO3 (001) substrate using an optical second harmonic generation (SHG) microscope. Whereas the ferroelectric state of this material hosts nanometer-sized domains which again form micrometer-sized domains of four different configurations, we could figure out the characteristic features of each domain from the SHG mapping with various sizes of the probe beam, i.e., from 0.7 to 3.9 μm in its diameter. In particular, we demonstrate that a single micrometer-sized domain contributes to the SHG as a coherent summation of the constituent nanometer-sized domains, and multi-micrometer-sized domains contribute to the SHG as an incoherent summation of each micro-domain.