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The flexoelectric-type coupling between the gradients of electric polarization, described by a true polar vector $\vec{P}$, and the antiphase rotation of structural groups, described by an axial pseudovector $\vec{\Phi }$, can lead to the appearance of interfacial polarization at the twin walls, antiphase boundaries and surfaces of antiferrodistortive (AFD) ferroelastics [1]. The linear-quadratic flexo-AFD coupling may contribute significantly to observable interfacial polarization induced by oxygen octahedral rotations at the antiphase boundaries and/or twin walls in YMnO3, Ca3Mn2O7, CaTiO3 and SrTiO3, as well as lead to the appearance of versatile spatially modulated structures in multiferroics [1].
The quadratic dependence of the linear-quadratic flexo-AFD coupling energy on $\vec{\Phi}$ appeared principally important for the emergence of interfacial polarization, because the odd powers of anti-phase (i.e., sign-alternating in neighboring sublattices) pseudovector $\vec{\Phi }$ cannot induce the continuous polarization of structural domain walls, surfaces or interfaces in AFD ferroelastics. Therefore, it seemed that it made little sense to consider odd powers of $\vec{\Phi }$ when constructing the sublattice-insensitive flexo-AFD coupling energy. However, recently Visser et al. [2] and Fang et al. [3] proposed the concept of alterelectricity, an electrical analogue of altermagnetism, in which two switchable states possess alternating band structures.
Using the Landau-Ginsburg-Devonshire approach we show that the linear gradient-type coupling between the electric polarization vector $\vec{P}$ and antiferrodistortive long-range order parameter pseudovector $\vec{\Phi }$, that has the form of Lifshitz invariant $\left(\vec{P}\bullet \mathrm{\nabla }\times \vec{\Phi }\; -\vec{\Phi }\bullet \mathrm{\nabla }\times \vec{P}\right)/2$ and named “bilinear flexo-antiferrodistortive coupling”, can emerge in all antiferrodistortive ferroelastics, since it is symmetry-allowed [4]. Using the four sublattices model we reveal that the bilinear flexo-antiferrodistortive coupling can induce the sublattice-sensitive polarization at the twin walls and antiphase boundaries in antiferrodistortive ferroelastics without any ferroelectric or antiferroelectric ordering. Since the induced polarization $\vec{\delta P}$ is perpendicular to $\vec{\Phi}$ and counter-directed in neighboring sublattices with checkerboard-type direction of $\vec{\Phi}$, such structure of $\vec{\delta P}$ may correspond to the alterelectric-type quadrupolar electric order. However, physical manifestations of the bilinear flexo-antiferrodistortive coupling are invisible in most nanostructured antiferrodistortive ferroelectrics and antiferroelectrics due to the domination of piezoelectric and/or omnipresent linear flexoelectric couplings. We have shown that the bilinear flexo-antiferrodistortive coupling can induce alterelectric-type polarization near antiferrodistortive domain boundaries in ferroelastics without any ferroelectric or antiferroelectric long-range ordering [1].
Acknowledgements. The work is primary supported as part of the Computational Materials Sciences Program funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DE-SC0020145. The part of A.N.M. and E.A.E. efforts is also supported by National Academy of Sciences of Ukraine (grants No. 5.8/26-П, 1.4.B/222, III-6-26).
[1]. A.N. Morozovska, E.A. Eliseev, M.D. Glinchuk, L.-Q. Chen, V. Gopalan. Interfacial Polarization and Pyroelectricity in Antiferrodistortive Structures Induced by a Flexoelectric Effect and Rotostriction. Phys.Rev. B. 85, 094107 (2012); https://doi.org/10.1103/PhysRevB.85.094107
[2]. A. Visser, V. Könye, O. Janson, J. van den Brink, C. Coulais, and J. van Wezel. "Multipolar Piezoelectricity and Anisotropic Surface Transport in Alterelectrics." arXiv preprint arXiv:2604.18324 (2026); https://doi.org/10.48550/arXiv.2604.18324
[3]. S. Fang, J. Wang, Z. Guo, J. Gong, H. Meng, W. Wang, Z. Cheng, X. Wang, and Y. S. Ang. "Alterelectricity: Electrical Analogue of Altermagnetism." arXiv preprint arXiv:2604.07112 (2026); https://doi.org/10.48550/arXiv.2604.07112
[4]. E. A. Eliseev, A. N. Morozovska, A. Saha and V. Gopalan. Bilinear Flexo-Antiferrodistortive Coupling in Ferroelastics: Polar Twins, Antiphase Boundaries and Fingerprints of Alterelectricity. ArXiv Preprint (2026); https://doi.org/10.48550/arXiv.2606.29456
