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Topological protection breakdown: A route to frustrated ferroelectricity

Ludovica Falsi1,2, Pablo Villegas2,3,*, Tommaso Gili4, A. J. Agranat5, and E. DelRe1,6

  • *Contact author: pablo.villegas@cref.it
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Phys. Rev. Research 7, 043038 – Published 10 October, 2025

DOI: https://doi.org/10.1103/h6j7-cgwz

Abstract

Phases manifesting topological patterns in functional systems, like ferroelectric and ferromagnetic vortex superlattices, can manifest intricate and apparently ungovernable behavior, typical of frustrated nonergodic states with high-dimensional energy landscapes. This is also the case for potassium-tantalate-niobate (KTN) crystals. These transparent ferroelectrics manifest remarkable but little understood metastable domain patterns at optical (micrometer and above) scales near the cubic-to-tetragonal structural phase transition. Here, we formulate the topological breakdown model based on the competition between intrinsic scales of domain-domain collinear and noncollinear interactions associated with polarization-charge screening. The model is able to explain observed KTN mesoscopic domain patterns and phase diagram as a function of temperature and external electric field. Findings include a precise set of sharp and broad percolative transitions that are experimentally verified, validating our model. Our study identifies the central role played by competing topologically protected states, identifying a fundamental link between topological protection and frustration that supports a hitherto unexplored functional nonergodic arena.

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References (91)

  1. D. Viehland, S. Jang, L. E. Cross, and M. Wuttig, Freezing of the polarization fluctuations in lead magnesium niobate relaxors, J. Appl. Phys. 68, 2916 (1990).
  2. A. A. Bokov, Ferroelectric domains and polar clusters in disordered structures, Ferroelectrics 190, 197 (1997).
  3. A. Bokov and Z.-G. Ye, Recent progress in relaxor ferroelectrics with perovskite structure, J. Mater. Sci. 41, 31 (2006).
  4. N. Choudhury, L. Walizer, S. Lisenkov, and L. Bellaiche, Geometric frustration in compositionally modulated ferroelectrics, Nature (London) 470, 513 (2011).
  5. A. A. Bokov and Z.-G. Ye, Dielectric relaxation in relaxor ferroelectrics, J. Adv. Dielectr. 02, 1241010 (2012).
  6. F. Xin, L. Falsi, Y. Gelkop, D. Pierangeli, G. Zhang, F. Bo, F. Fusella, A. J. Agranat, and E. DelRe, Evidence of 3D topological-domain dynamics in KTN:Li polarization-supercrystal formation, Phys. Rev. Lett. 132, 066603 (2024).
  7. V. Westphal, W. Kleemann, and M. D. Glinchuk, Diffuse phase transitions and random-field-induced domain states of the “relaxor” ferroelectric PbMg1/3Nb2/3O3, Phys. Rev. Lett. 68, 847 (1992).
  8. D. Pierangeli, M. Ferraro, F. Di Mei, G. Di Domenico, C. De Oliveira, A. Agranat, and E. DelRe, Super-crystals in composite ferroelectrics, Nat. Commun. 7, 10674 (2016).
  9. T. Nishinaga, Handbook of Crystal Growth: Fundamentals (Elsevier, Boston, 2014).
  10. A. Agranat, C. de Oliveira, and G. Orr, Dielectric electrooptic gratings in potassium lithium tantalate niobate, J. Non-Cryst. Solids 353, 4405 (2007).
  11. V. Stoica, N. Laanait, C. Dai, Z. Hong, Y. Yuan, Z. Zhang, S. Lei, M. McCarter, A. Yadav, A. Damodaran et al., Optical creation of a supercrystal with three-dimensional nanoscale periodicity, Nat. Mater. 18, 377 (2019).
  12. M. Hadjimichael, Y. Li, E. Zatterin, G. A. Chahine, M. Conroy, K. Moore, E. N. O. Connell, P. Ondrejkovic, P. Marton, J. Hlinka et al., Metal-ferroelectric supercrystals with periodically curved metallic layers, Nat. Mater. 20, 495 (2021).
  13. I. Luk'Yanchuk, Y. Tikhonov, A. Razumnaya, and V. Vinokur, Hopfions emerge in ferroelectrics, Nat. Commun. 11, 2433 (2020).
  14. V. Govinden, S. Prokhorenko, Q. Zhang, S. Rijal, Y. Nahas, L. Bellaiche, and N. Valanoor, Spherical ferroelectric solitons, Nat. Mater. 22, 553 (2023).
  15. J. Junquera, Y. Nahas, S. Prokhorenko, L. Bellaiche, J. Íñiguez, D. G. Schlom, L.-Q. Chen, S. Salahuddin, D. A. Muller, L. W. Martin et al., Topological phases in polar oxide nanostructures, Rev. Mod. Phys. 95, 025001 (2023).
  16. C. Kittel, Physical theory of ferromagnetic domains, Rev. Mod. Phys. 21, 541 (1949).
  17. L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8, 101 (1935).
  18. K. M. Rabe, M. Dawber, C. Lichtensteiger, C. H. Ahn, and J.-M. Triscone, Modern physics of ferroelectrics: Essential background, in Physics of Ferroelectrics: A Modern Perspective, edited by M. Rabe, C. H. Ahn, and J.-M. Triscone (Springer, Berlin, 2007), pp. 1–30.
  19. A. Gumennik, Y. Kurzweil-Segev, and A. J. Agranat, Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment, Opt. Mater. Express 1, 332 (2011).
  20. M. Eremenko, V. Krayzman, A. Bosak, H. Playford, K. Chapman, J. Woicik, B. Ravel, and I. Levin, Local atomic order and hierarchical polar nanoregions in a classical relaxor ferroelectric, Nat. Commun. 10, 2728 (2019).
  21. P. Gehring, H. Hiraka, C. Stock, S.-H. Lee, W. Chen, Z.-G. Ye, S. Vakhrushev, and Z. Chowdhuri, Reassessment of the burns temperature and its relationship to the diffuse scattering, lattice dynamics, and thermal expansion in relaxor Pb(Mg1/3Nb2/3)O3, Phys. Rev. B 79, 224109 (2009).
  22. I. Muench, A. Renuka Balakrishna, and J. Huber, Periodic boundary conditions for the simulation of 3D domain patterns in tetragonal ferroelectric material, Arch. Appl. Mech. 89, 955 (2019).
  23. Y. Yoshimura, K.-J. Kim, T. Taniguchi, T. Tono, K. Ueda, R. Hiramatsu, T. Moriyama, K. Yamada, Y. Nakatani, and T. Ono, Soliton-like magnetic domain wall motion induced by the interfacial Dzyaloshinskii-Moriya interaction, Nat. Phys. 12, 157 (2016).
  24. Y. Ishikuro, M. Kawaguchi, N. Kato, Y.-C. Lau, and M. Hayashi, Dzyaloshinskii-Moriya interaction and spin-orbit torque at the Ir/Co interface, Phys. Rev. B 99, 134421 (2019).
  25. S.-W. C. Shuai Dong, J.-M. Liu, and Z. Ren, Multiferroic materials and magnetoelectric physics: Symmetry, entanglement, excitation, and topology, Adv. Phys. 64, 519 (2015).
  26. Y. Takeuchi, Y. Yamane, J.-Y. Yoon, R. Itoh, B. Jinnai, S. Kanai, J. Ieda, S. Fukami, and H. Ohno, Chiral-spin rotation of non-collinear antiferromagnet by spin-orbit torque, Nat. Mater. 20, 1364 (2021).
  27. I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958).
  28. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960).
  29. H. Yang, J. Liang, and Q. Cui, First-principles calculations for Dzyaloshinskii-Moriya interaction, Nat. Rev. Phys. 5, 43 (2023).
  30. H. J. Zhao, P. Chen, S. Prosandeev, S. Artyukhin, and L. Bellaiche, Dzyaloshinskii–Moriya-like interaction in ferroelectrics and antiferroelectrics, Nat. Mater. 20, 341 (2021).
  31. H. Katsura, N. Nagaosa, and A. V. Balatsky, Spin current and magnetoelectric effect in noncollinear magnets, Phys. Rev. Lett. 95, 057205 (2005).
  32. D. Rusu, J. J. Peters, T. P. Hase, J. A. Gott, G. A. Nisbet, J. Strempfer, D. Haskel, S. D. Seddon, R. Beanland, A. M. Sanchez et al., Ferroelectric incommensurate spin crystals, Nature (London) 602, 240 (2022).
  33. P. Chen, H. J. Zhao, S. Prosandeev, S. Artyukhin, and L. Bellaiche, Microscopic origin of the electric Dzyaloshinskii-Moriya interaction, Phys. Rev. B 106, 224101 (2022).
  34. D. Cassi, Phase transitions and random walks on graphs: A generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures, Phys. Rev. Lett. 68, 3631 (1992).
  35. N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966).
  36. C. Dai, V. A. Stoica, S. Das, Z. Hong, L. W. Martin, R. Ramesh, J. W. Freeland, H. Wen, V. Gopalan, and L.-Q. Chen, Tunable nanoscale evolution and topological phase transitions of a polar vortex supercrystal, Adv. Mater. 34, 2106401 (2022).
  37. A. Fernandez, M. Acharya, H.-G. Lee, J. Schimpf, Y. Jiang, D. Lou, Z. Tian, and L. W. Martin, Thin-film ferroelectrics, Adv. Mater. 34, 2108841 (2022).
  38. F. Ferrari, S. Niu, J. Hasik, Y. Iqbal, D. Poilblanc, and F. Becca, Static and dynamical signatures of Dzyaloshinskii-Moriya interactions in the Heisenberg model on the Kagome lattice, SciPost Phys. 14, 139 (2023).
  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/h6j7-cgwz for further technical details supporting the findings in the main text.
  40. P. Villegas, T. Gili, G. Caldarelli, and A. Gabrielli, Laplacian renormalization group for heterogeneous networks, Nat. Phys. 19, 445 (2023).
  41. P. Villegas, A. Gabrielli, F. Santucci, G. Caldarelli, and T. Gili, Laplacian paths in complex networks: Information core emerges from entropic transitions, Phys. Rev. Res. 4, 033196 (2022).
  42. A. Poggialini, P. Villegas, M. A. Muñoz, and A. Gabrielli, Networks with many structural scales: A renormalization group perspective, Phys. Rev. Lett. 134, 057401 (2025).
  43. A. Ortiz-Ambriz, C. Nisoli, C. Reichhardt, C. J. O. Reichhardt, and P. Tierno, Colloquium: Ice rule and emergent frustration in particle ice and beyond, Rev. Mod. Phys. 91, 041003 (2019).
  44. P. Villegas, J. Hidalgo, P. Moretti, and M. A. Muñoz, Complex synchronization patterns in the human connectome network, in Proceedings of ECCS 2014, Springer Proceedings in Complexity, edited by S. Battiston, F. De Pellegrini, G. Caldarelli, and E. Merelli (Springer, Cham, 2016), pp. 69–80.
  45. P. Moretti and M. Muñoz, Griffiths phases and the stretching of criticality in brain networks, Nat. Commun. 4, 2521 (2013).
  46. D. Pierangeli, A. Tavani, F. Di Mei, A. J. Agranat, C. Conti, and E. DelRe, Observation of replica symmetry breaking in disordered nonlinear wave propagation, Nat. Commun. 8, 1501 (2017).
  47. C. Li, X. Wang, Y. Wu, F. Liang, F. Wang, X. Zhao, H. Yu, and H. Zhang, Three-dimensional nonlinear photonic crystal in naturally grown potassium–tantalate–niobate perovskite ferroelectrics, Light Sci. Appl. 9, 193 (2020).
  48. Y. Wu, X. Wang, G. Tian, L. Zheng, F. Liang, S. Zhang, H. Yu, and H. Zhang, Inverse design of ferroelectric-order in perovskite crystal for self-powered ultraviolet photodetection, Adv. Mater. 34, 2105108 (2022).
  49. Y. Wu, Q. Liu, G. Tian, L. Zheng, F. Liang, X. Wang, D. Wang, Y. Sang, H. Liu, J. Wang et al., Periodically poled nonlinear photonic crystal KTa0.51Nb0.49O3 combining wide transparent range and large quadratic coefficient, Adv. Opt. Mater. 10, 2201010 (2022).
  50. Y.-C. Chang, C. Wang, S. Yin, R. C. Hoffman, and A. G. Mott, Giant electro-optic effect in nanodisordered KTN crystals, Opt. Lett. 38, 4574 (2013).
  51. B. Noheda, D. Cox, G. Shirane, J. Gonzalo, L. Cross, and S. Park, A monoclinic ferroelectric phase in the Pb(Zr1- xTix)O3 solid solution, Appl. Phys. Lett. 74, 2059 (1999).
  52. V. V. Shvartsman and D. C. Lupascu, Lead-free relaxor ferroelectrics, J. Am. Ceram. Soc. 95, 1 (2012).
  53. F. Di Mei, L. Falsi, M. Flammini, D. Pierangeli, P. Di Porto, A. Agranat, and E. DelRe, Giant broadband refraction in the visible in a ferroelectric perovskite, Nat. Photonics 12, 734 (2018).
  54. L. Falsi, L. Tartara, F. Di Mei, M. Flammini, J. Parravicini, D. Pierangeli, G. Parravicini, F. Xin, P. DiPorto, A. J. Agranat et al., Constraint-free wavelength conversion supported by giant optical refraction in a 3D perovskite supercrystal, Commun. Mater. 1, 76 (2020).
  55. L. Falsi, S. Macis, Y. Gelkop, L. Tartara, E. Bonaventura, P. Di Pietro, A. Perucchi, Y. Garcia, G. Perepelitsa, E. DelRe et al., Anomalous optical properties of KTN:Li ferroelectric supercrystals, Nanomaterials 13, 899 (2023).
  56. L. Falsi, M. Aversa, F. Di Mei, D. Pierangeli, F. Xin, A. J. Agranat, and E. DelRe, Direct observation of fractal-dimensional percolation in the 3D cluster dynamics of a ferroelectric supercrystal, Phys. Rev. Lett. 126, 037601 (2021).
  57. M. Ferraro, D. Pierangeli, M. Flammini, G. Di Domenico, L. Falsi, F. Di Mei, A. Agranat, and E. DelRe, Observation of polarization-maintaining light propagation in depoled compositionally disordered ferroelectrics, Opt. Lett. 42, 3856 (2017).
  58. F. Colaiori, Exactly solvable model of avalanches dynamics for Barkhausen crackling noise, Adv. Phys. 57, 287 (2008).
  59. M. F. Atiyah, The Geometry and Physics of Knots (Cambridge University Press, Cambridge, 1990).
  60. L. H. Kauffman, Knots and Physics (World Scientific, Singapore, 2001), Vol. 1.
  61. D. Rolfsen, Knots and Links (American Mathematical Society, Berkeley, CA, 2003), Vol. 346.
  62. J. C. Turner, P. Van De Griend, and C. Warner, History and Science of Knots (World Scientific, Singapore, 1996), Vol. 11.
  63. L. Faddeev and A. J. Niemi, Stable knot-like structures in classical field theory, Nature (London) 387, 58 (1997).
  64. U. Tkalec, M. Ravnik, S. Čopar, S. Žumer, and I. Muševič, Reconfigurable knots and links in chiral nematic colloids, Science 333, 62 (2011).
  65. D. Kleckner and W. T. Irvine, Creation and dynamics of knotted vortices, Nat. Phys. 9, 253 (2013).
  66. P. Sutcliffe, Skyrmion knots in frustrated magnets, Phys. Rev. Lett. 118, 247203 (2017).
  67. F. Zheng, N. S. Kiselev, F. N. Rybakov, L. Yang, W. Shi, S. Blügel, and R. E. Dunin-Borkowski, Hopfion rings in a cubic chiral magnet, Nature (London) 623, 718 (2023).
  68. K.-E. Kim, S. Jeong, K. Chu, J. H. Lee, G.-Y. Kim, F. Xue, T. Y. Koo, L.-Q. Chen, S.-Y. Choi, R. Ramesh et al., Configurable topological textures in strain graded ferroelectric nanoplates, Nat. Commun. 9, 403 (2018).
  69. Y. Tang, Y. Zhu, X. Ma, A. Y. Borisevich, A. N. Morozovska, E. A. Eliseev, W. Wang, Y. Wang, Y. Xu, Z. Zhang et al., Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films, Science 348, 547 (2015).
  70. X. Li, C. Tan, C. Liu, P. Gao, Y. Sun, P. Chen, M. Li, L. Liao, R. Zhu, J. Wang et al., Atomic-scale observations of electrical and mechanical manipulation of topological polar flux closure, Proc. Natl. Acad. Sci. USA 117, 18954 (2020).
  71. J. Seidel, R. K. Vasudevan, and N. Valanoor, Topological structures in multiferroics—domain walls, skyrmions and vortices, Adv. Electron. Mater. 2, 1500292 (2016).
  72. I. I. Naumov, L. Bellaiche, and H. Fu, Unusual phase transitions in ferroelectric nanodisks and nanorods, Nature (London) 432, 737 (2004).
  73. A. Fert, V. Cros, and J. Sampaio, Skyrmions on the track, Nat. Nanotechnol. 8, 152 (2013).
  74. J. Scott, Multiferroic memories, Nat. Mater. 6, 256 (2007).
  75. H. Du, X. Zhao, F. N. Rybakov, A. B. Borisov, S. Wang, J. Tang, C. Jin, C. Wang, W. Wei, N. S. Kiselev, Y. Zhang, R. Che, S. Blügel, and M. Tian, Interaction of individual skyrmions in a nanostructured cubic chiral magnet, Phys. Rev. Lett. 120, 197203 (2018).
  76. J. Wild, T. N. Meier, S. Pöllath, M. Kronseder, A. Bauer, A. Chacon, M. Halder, M. Schowalter, A. Rosenauer, J. Zweck et al., Entropy-limited topological protection of skyrmions, Sci. Adv. 3, e1701704 (2017).
  77. Y. Nahas, S. Prokhorenko, I. Kornev, and L. Bellaiche, Topological point defects in relaxor ferroelectrics, Phys. Rev. Lett. 116, 127601 (2016).
  78. D. J. Amit, H. Gutfreund, and H. Sompolinsky, Spin-glass models of neural networks, Phys. Rev. A 32, 1007 (1985).
  79. R. A. Goldstein, Z. A. Luthey-Schulten, and P. G. Wolynes, Optimal protein-folding codes from spin-glass theory, Proc. Natl. Acad. Sci. USA 89, 4918 (1992).
  80. P. Villegas, P. Moretti, and M. A. Muñoz, Frustrated hierarchical synchronization and emergent complexity in the human connectome network, Sci. Rep. 4, 5990 (2014).
  81. T. Itou, E. Watanabe, S. Maegawa, A. Tajima, N. Tajima, K. Kubo, R. Kato, and K. Kanoda, Slow dynamics of electrons at a metal-Mott insulator boundary in an organic system with disorder, Sci. Adv. 3, e1601594 (2017).
  82. L. E. Cross, Relaxorferroelectrics: An overview, Ferroelectrics 151, 305 (1994).
  83. G. Burns and F. Dacol, Crystalline ferroelectrics with glassy polarization behavior, Phys. Rev. B 28, 2527 (1983).
  84. P. Timonin, Griffiths’ phase in dilute ferroelectrics, Ferroelectrics 199, 69 (1997).
  85. D. Sherrington, Relaxors, spin, stoner and cluster glasses, Phase Transitions 88, 202 (2015).
  86. H. Zheng, T. Zhou, D. Sheyfer, J. Kim, J. Kim, T. D. Frazer, Z. Cai, M. V. Holt, Z. Zhang, J. Mitchell et al., Heterogeneous field response of hierarchical polar laminates in relaxor ferroelectrics, Science 384, 1447 (2024).
  87. S. Prosandeev, D. Wang, A. R. Akbarzadeh, B. Dkhil, and L. Bellaiche, Field-induced percolation of polar nanoregions in relaxor ferroelectrics, Phys. Rev. Lett. 110, 207601 (2013).
  88. A. L. Goodwin, Opportunities and challenges in understanding complex functional materials, Nat. Commun. 10, 4461 (2019).
  89. G. Nataf, M. Guennou, J. Gregg, D. Meier, J. Hlinka, E. Salje, and J. Kreisel, Domain-wall engineering and topological defects in ferroelectric and ferroelastic materials, Nat. Rev. Phys. 2, 634 (2020).
  90. R. De Vogli, J. Montomoli, G. Abu-Sittah, and I. Pappé, Break the selective silence on the genocide in Gaza, The Lancet 406, 688 (2025).
  91. L. Lo Presti, J. Parravicini, R. Soave, G. Parravicini, M. Mauri, L. Loconte, F. Di Mei, L. Falsi, L. Tartara, S. Binetti, A. J. Agranat, and E. DelRe, Observation of an exotic lattice structure in the transparent KTa1−xNbxO3 perovskite supercrystal, Phys. Rev. B 102, 214110 (2020).

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Phys. Rev. Research 7, 043038– Published 10 October, 2025
  • Received 6 February 2025
  • Accepted 1 August 2025
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