Topological protection breakdown: a route to frustrated ferroelectricity

TL;DR

The paper introduces the Topological Breakdown Model (TBM) to explain complex, history-dependent mesoscale domain patterns in disordered ferroelectrics such as KTN. It couples two topological interaction scales on a ferroelectric Supercrystal (SC) tiling lattice and analyzes them with diffusion-based Laplacian methods and real-space RG, yielding a temperature–field phase diagram and percolation signatures. It identifies a metastable frustrated phase where vortex and ferroelectric domains coexist and demonstrates percolation transitions that align with experiments in KTN:Li, linking topological protection breakdown to frustration. The work provides a minimal, analytically tractable framework that highlights how flux-closure constraints and competing interactions drive non-ergodic behavior, with implications for designing noise-resistant memory and energy-storage devices in polar materials.

Abstract

Phases manifesting topological patterns in functional systems, like ferroelectric and ferromagnetic vortex superlattices, can manifest intricate and apparently ungovernable behavior, typical of frustrated non-ergodic 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 non-collinear 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 non-ergodic arena.

Figures (9)

• Figure 1: (a) Different types of defects: type i) and type ii), and a flux-closure cell. (b) SC-tiling lattice and sketch of the energetic constants. We highlight the anisotropy in the Dzyaloshinskii–Moriya interaction. The upper left insets show a microscopic vortex, and the lower inset is an experimental far-field imaging of a SC state (right, see below). (c) Specific heat ($C$), versus the temporal resolution parameter, $\tau$, of 2D SC lattices of different sides $L$ (see legend, $N=4L^2$). Insets show the two possible ultraviolet cut-offs, $\Lambda_1$ and $\Lambda_2$, the smallest scales showing translational invariance. The black dashed line highlights the plateau at $C=1$.
• Figure 2: Topological symmetry-breaking. (a) Temperature phase diagram of Eq.\ref{['Hamiltonian']} computed using direct simulations on a SC lattice with $N=1024$ nodes ($J=1$). Polarized states are computationally detected with the polarization order parameter, $P$, while vortex phases are detected through the vortex order parameter $v$. The metastable phase corresponds to the broad region of divergent polarization susceptibility $\chi_P=N\sigma^2_P$. (b)-(c) Polarization ($P$) and vortex ($v$) order parameters versus temperature for different lattice sizes (see legend, $N=4L^2$) for (b) $J/D=1.4$ and (c) $J/D=0.8$. Inset: Rescaled temporal variance $\chi_P=N\sigma^2_P$ versus temperature, $T$ for different system sizes. The metastable phase also presents characteristic temporal oscillations in the global system polarization (see Appendix \ref{['TempOsc']}). (d) Top: Far-field imaging of a SC structure at $T$= $T_C$-2K. Bottom: Computational 2D Fourier transform of an ordered vortex phase ($J/D=0.8,~T=0.4,~N=1024)$. (e) Temperature phase diagram of the model versus electric field $E$ for $J/D=0.8$ and $N=1024$ nodes. The breakup of the vortex phase translates into the emergence of a metastable phase where the ferroelectric and vortex domains naturally coexist. (f)-(g) Polarization ($P$) and vortex ($v$) order parameters as a function of the electric field for different lattice sizes (as in (b) and (c)) for (f) $T=0.4$ and (g) $T=0.7$. The peak in the system susceptibility is indicated here as a vertical dashed line. Note how the critical field at low temperatures splits into two different phase transitions when T increases, with naturally coexisting vortices and ferroelectric domains. All curves have been averaged over $10^2-10^3$ independent realizations.
• Figure 3: Direct imaging of SC ferroelectric clusters. Binarized crossed-polarizer transmission microscopy images for (a)$T_C$-4.5K and increasing electric field (0 kV/cm, 2.86 kV/cm, and 2.91 kV/cm, respectively, using $\theta=25$). Light transmission shows an abrupt transition at $E_C\approx$2.9 kV/cm. (b) Various conditions of temperature and bias electric field to enhance the critical point for the first two temperatures and the metastable region for $T_C$-2.5K (see images, with $\theta=37$, $37$, and $35$, respectively). (c-e) Light intensity versus applied electric field ($E$) for three different temperatures and different threshold values ($\theta$, see legend): (c)$T_C-4.5K$. An abrupt phase transition is present at a field $E_C\approx2.9$ kV/cm for a wide range of $\theta$ values. (d)$T_C-3.5K$. The abrupt transition changes its nature to a continuous one with $E_C\approx$ 2.75 kV/cm and (e)$T_C-2.5K$. The system exhibits two different phase transitions at different critical fields, $E_C\approx$2.5 kV/cm and $E_C\approx$ 2.86 kV/cm (see also Appendix \ref{['PerAnalysis']}).
• Figure 4: Percolation phase transition. (a-b)$P_\infty$ as a function of the applied field, $E$, for different system sizes (in the new square lattice of side $L$ resulting from the coarse-graining process, see legend) for: (a)$T=0.4$ Inset: $\chi$ versus $E$. Note the divergence at a specific point as the system size increases. (b)$T=0.75$. Inset: $\chi$ versus $E$. Note the divergence of the entire metastable region as the system size increases. (c) Configurations of simulated light transmission images. From up to down, images show a Vortex state ($T=0.4,~E=0.2$), the bifurcation line to the metastable phase ($T=0.5,~E=0.29$), and the metastable phase ($T=0.5,~E=0.3$). For the sake of comparison, the inset shows a zoomed region of experimental light transmission at $T_C$-2.5K and $E=2.65$ kV/cm.
• Figure 5: Slim loop hysteresis.$P_\infty$ as a function of the applied field, $E$, for different temperatures, $T=0.4$ (orange line) and $T=0.6$ (violet line) for a lattice with $L=64$ coarse-grained units. The wide hysteretic behavior at low temperatures, characteristic of the first-order phase transition, becomes a slim loop hysteresis cycle in the intermediate phase. The break of domain walls on pinned sites originates the observed jumps in the hysteretic cycle, that are shown in the zoomed insets. This gives rise to the well-known Barkhausen noise.