Topological Ferroelectric Vortex States

• Topological ferroelectric vortex states are ordered textures where the order parameter winds around a core, producing quantized circulation and distinct winding numbers.
• They are characterized by diagnostic quantities such as winding numbers and toroidal moments, with variations across hexagonal manganites, perovskites, and other nanostructures.
• These states underpin innovative applications, enabling reconfigurable electronic channels and memory devices through controlled switching and dynamic manipulation.

Topological ferroelectric vortex states are ordered textures in which a ferroelectric, structural, or coupled structural–polar order parameter winds around a core, producing quantized circulation, flux closure, or vortex–antivortex networks. In two-dimensional cuts they appear as point defects, whereas in three dimensions they occur as lines or tubes. Their microscopic origin is not unique: in hexagonal manganites the relevant phase is the trimerization angle interlocked with out-of-plane polarization, while in perovskites and related nanosystems the polarization vector itself curls under electrostatic, elastic, gradient, and sometimes flexoelectric constraints. Across these settings, vortex states are diagnosed by winding number, toroidal moment, and characteristic real-space or reciprocal-space signatures, and they can organize into cloverleaf defects, vortex crystals, quasi-one-dimensional arrays, or moiré-registered lattices (Chae et al., 2013, Rijal et al., 2023, Sanchez-Santolino et al., 2023).

1. Order-parameter topology and diagnostic quantities

The minimal topological descriptor of a ferroelectric vortex is a winding number defined on a closed contour around a core. In hexagonal manganites, the phase variable is the trimerization angle $\theta$, and the winding number is

$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$

with $n=+1$ for a vortex and $n=-1$ for an antivortex. In polarization-curl systems, the corresponding quantity is usually written in terms of the in-plane polarization angle $\phi$,

$w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$

again giving $w=\pm 1$ for vortex and antivortex cores. This distinction is fundamental: the topological variable can be a structural phase, a polarization phase, or a coupled multicomponent field, depending on the material class (Chae et al., 2013, Sanchez-Santolino et al., 2023, Rijal et al., 2023).

A second widely used descriptor is the toroidal moment, which quantifies circulation rather than net polarization. In twisted freestanding BaTiO$_3$, the $z$-directed toroidal moment is defined as

$Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$

and is nonzero at vortex sites but zero at antivortex sites. In confined PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$0 geometries, a related quantity is

$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$1

which is finite for flux-closure states and vanishes for uniform monodomains. These observables are especially useful when the local polarization magnitude remains finite away from a narrow core and the texture is better regarded as a rotational dipole pattern than as a conventional domain arrangement (Sanchez-Santolino et al., 2023, Kondovych et al., 2021).

Ferroelectric vortices must also be distinguished from neighboring topological textures. Skyrmion-like or meron-like states can be characterized by

$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$2

with $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$3, but not every swirling polarization field is a skyrmion. In mechanically manipulated PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$4–SrTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$5 superlattices, the observed objects are vortices with $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$6 winding around cores rather than full skyrmions, because skyrmion-number quantization is not evidenced there. In ultra-thin Pb(Zr$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$7,Ti$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$8)O$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$9 films, meron crystals and bubble-like textures appear transiently during switching, but the equilibrium state studied is a vortex crystal of periodic vortex tubes (Chen et al., 2019, Rijal et al., 2023).

2. Canonical realization in hexagonal manganites

Hexagonal rare-earth manganites $n=+1$0-REMnO$n=+1$1 provide the canonical improper-ferroelectric vortex system. Their ferroelectricity is driven by a structural trimerization with amplitude $n=+1$2 and phase $n=+1$3, selecting three crystallographic antiphase variants $n=+1$4, $n=+1$5, and $n=+1$6, while the ferroelectric polarization points along the $n=+1$7 axis with $n=+1$8. The six resulting states $n=+1$9 realize an emergent $n=-1$0 symmetry, or equivalently a $n=-1$1 clock manifold. Because the ferroelectric walls are interlocked with structural antiphase boundaries, six interleaved domains meet at each core in a cloverleaf or kaleidoscopic junction, and circling the core advances the trimerization phase by $n=-1$2 (Chae et al., 2013).

This six-state topology has both local and macroscopic forms. In type-I patterns, the full $n=-1$3 symmetry is preserved. In type-II patterns, typically near surfaces or under poling, the $n=-1$4 sector is macroscopically broken while $n=-1$5 remains, so one polarization orientation is preferred and the disfavored orientation collapses into narrow two-gon domains connecting vortex–antivortex pairs. Graph-theoretically, the type-I state is a 6-valent planar graph with even-gon faces that is $n=-1$6-colorable, whereas the type-II state becomes effectively 3-valent when the narrow two-gons are treated as edges. Depth profiling of annealed ErMnO$n=-1$7 showed type-II patterns at the original surfaces and type-I patterns after about $n=-1$8 of etching, establishing a depth-driven “topological condensation” and reverse “topological evaporation” (Chae et al., 2013).

The mechanistic origin of this evolution is not primarily vortex–antivortex interaction. Each interlocked wall behaves as a partial dislocation carrying a Burgers vector with a polarization component and a structural phase-shift component, such as $n=-1$9 for the $\phi$0 wall. Like-signed partial dislocations repel, opposite-signed pairs attract and can annihilate, and the interaction energy is approximately $\phi$1. The observed wall-angle statistics near cloverleaf cores show an average of about $\phi$2, a median about $\phi$3, and suppressed low-angle occurrences, consistent with short-range repulsion among adjacent walls. By contrast, vortex and antivortex core positions remain nearly fixed during condensation and evaporation, and generation or annihilation of new core pairs occurs at a rate of less than one pair per $\phi$4 in the reported observations (Chae et al., 2013).

The same family of materials also reveals critical and switching physics of unusual clarity. Near the ferroelectric critical point, the microscopic $\phi$5 anisotropy becomes dangerously irrelevant and the transition is governed by an emergent $\phi$6 symmetry in the 3D XY universality class. In that regime, the transition can be recast in a dual description as vortex-line proliferation and “Higgs condensation of disorder,” with the vortex disorder field coupled to an emergent $\phi$7 gauge field. Cooling-rate studies across the series YMnO$\phi$8, ErMnO$\phi$9, TmMnO$w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$0, and LuMnO$w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$1 yielded a defect-density scaling exponent of $w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$2, in excellent agreement with the Kibble–Zurek prediction $w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$3 for 3D XY criticality (Lin et al., 2015).

At the level of field-driven domain kinetics, in situ electron microscopy on ErMnO$w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$4 showed that switching proceeds by “topologically guided partner changing.” The vortex core remains immobile, while the six emanating walls reorganize into three closely spaced pairs. Neutral paired walls enclose domains about $w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$5 wide, whereas oppositely charged paired walls can stabilize domains only one unit cell wide, about $w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$6. Pair annihilation is prevented because the partial unit-cell-shift vectors carried by the interlocked walls sum to an incommensurate translation, so complete poling is topologically obstructed under ordinary biasing protocols (Han et al., 2013).

3. Perovskite, freestanding, and multiferroic realizations

Outside hexagonal manganites, ferroelectric vortices arise when homogeneous polarization is frustrated by electrostatic, elastic, and geometrical constraints. In perovskite superlattices this often produces flux-closure or vortex-tube states; in freestanding twisted membranes it yields moiré-registered vortex crystals; in nanocylinders it leads to geometry-selected vortex-core orientations; and in multiferroic superlattices it can produce ordered vortex phases with additional symmetry labels (Chen et al., 2019, Rijal et al., 2023, Sanchez-Santolino et al., 2023, Kondovych et al., 2021, Mei et al., 2018, Hong et al., 2017).

Platform	Stabilizing ingredients	Representative characteristics
$w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$7-REMnO$w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$8	Improper ferroelectricity from structural trimerization; interlocked structural and ferroelectric walls	Six-state $w = \frac{1}{2\pi}\oint \nabla \phi \cdot d\ell,$9 or $w=\pm 1$0 cloverleaf vortices
PbTiO$w=\pm 1$1-based superlattices and ultra-thin PZT	Depolarization, gradient and elastic energies, epitaxial boundary conditions	Ordered vortex lattices; period about $w=\pm 1$2 or $w=\pm 1$3
Twisted freestanding BaTiO$w=\pm 1$4	Open-circuit-like interfaces, moiré shear gradients, flexoelectric coupling	Two-dimensional alternating vortex–antivortex lattice with twist-set period
PbTiO$w=\pm 1$5 nanocylinders	Open-circuit confinement and geometry $w=\pm 1$6	$w=\pm 1$7-vortex or $w=\pm 1$8-vortex selected by geometry and temperature
TbScO$w=\pm 1$9/BiFeO$_3$0 and PTO/BFO/STO superlattices	Symmetry templating, nonlocal dipolar interactions, interfacial coupling	Ordered $_3$1–$_3$2 vortex phase or switchable spiral/semivortex textures

In PbTiO$_3$3–SrTiO$_3$4 superlattices, the polarization rotates continuously inside each PbTiO$_3$5 layer, forming arrays of vortex–antivortex pairs with an in-plane period of about $_3$6, comparable to the superlattice periodicity of about $_3$7. Unit-cell-resolved imaging and phase-field simulations identify lattice–charge interactions, epitaxial constraint, and dielectric coupling as the relevant stabilizing ingredients, and the reported vortex-core size is typically $_3$8–$_3$9 (Chen et al., 2019). In ultra-thin Pb(Zr$z$0,Ti$z$1)O$z$2 films of thickness about $z$3 unit cells, the vortex crystal is instead described as a finite-$z$4 instability: a soft optical phonon at $z$5 with $z$6 condenses at $z$7, and the ordering can be formulated as an emergent SU(2) symmetry-breaking transition between two degenerate modulation directions (Rijal et al., 2023).

Twisted freestanding BaTiO$z$8 introduces a different mechanism. In $z$9-thick membranes stacked into twisted bilayers, the dielectric interface enforces open-circuit-like electrostatics favoring in-plane polarization, while the moiré interface generates a periodic modulation of symmetric shear strain $Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$0 and rotational strain $Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$1. STEM-derived strain maps show shear strain gradients up to $Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$2, and the flexoelectric relations

$Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$3

with $Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$4 describe the observed polarization curls. Vortex cores coincide with AA and AB coincidence regions of minimal shear and maximal lattice rotation, antivortex cores with S-sites of maximal shear, and the lattice is topologically neutral because vortices and antivortices alternate (Sanchez-Santolino et al., 2023).

Confinement alone can select vortex morphology. In PbTiO$Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$5 nanocylinders under open-circuit conditions, the system avoids depolarization charge by forming divergence-free flux-closure states, and the orientation of the vortex core is controlled by cylinder geometry and temperature. The phase-field and analytical treatment distinguishes an $Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$6-vortex favored in elongated cylinders from a $Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$7-vortex favored in disc-like cylinders, with the boundary determined by the competition of elastic, electrostatic, and gradient energies (Kondovych et al., 2021).

Multiferroic BiFeO$Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$8-based heterostructures add further symmetry structure. Coherent TbScO$Q_z = \frac{1}{2N}\sum (r_i \times P_i)_z,$9/BiFeO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$00 superlattices realize an ordered $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$01–$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$02 vortex phase with positive topological charge and chiral staggering, in which approximately $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$03-wide cores form a quasi-one-dimensional lattice. Its stabilization is attributed to symmetry templating by the orthorhombic substrate and a field model balancing local stiffness against nonlocal dipole–dipole interactions (Mei et al., 2018). In tricolor PTO/BFO/STO superlattices, phase-field simulations predict a polar spiral built from alternating semivortices whose cores undulate around the BFO layers; the spiral pitch is about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$04, the Curie temperature is about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$05, and the spiral-to-in-plane transition occurs near $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$06 (Hong et al., 2017).

4. Switching, nonequilibrium dynamics, and topological transitions

The dynamical response of ferroelectric vortices is strongly stimulus-dependent. In hexagonal manganites, electric fields chiefly rearrange wall connectivity while leaving topological cores pinned; in superlattices, mechanical loading can annihilate vortex order reversibly; in ultra-thin films, resonant driving can rotate the entire vortex-crystal orientation; and in model 2D films or atomically thin membranes, localized or nonuniform fields and strains can create or destroy specific vortex–antivortex motifs (Han et al., 2013, Chae et al., 2013, Chen et al., 2019, Rijal et al., 2023, Roy et al., 2011, Xu et al., 2022).

In ErMnO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$07, in situ biasing showed that the vortex core acts as a topological anchor. Increasing the field along the $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$08 axis expands domains aligned with the field and shrinks the antiparallel ones, but the core remains immobile and the six walls simply repartition into new bound pairs. Abrupt reconfigurations were reported at field steps $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$09, $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$10, and $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$11, and a built-in internal field near the surface was inferred from back-switching and loop shifts (Han et al., 2013). A related but more macroscopic control mode is topological condensation in $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$12-REMnO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$13: self-poling near surfaces, attributed to effective electric fields from surface chemistry and defect states acquired during post-growth annealing, biases one polarization orientation and stabilizes the type-II state; external poling of YMnO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$14 with Ag electrodes at $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$15 reproduces the same topology change (Chae et al., 2013).

Mechanical control is particularly explicit in PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$16–SrTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$17 superlattices. A tungsten nanoindenter inside the TEM applied forces up to about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$18, corresponding via the Hertz contact model to a peak pressure of about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$19 for a contact radius of about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$20. Under this load, the in-plane vortex reflections in SAED disappear, the out-of-plane $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$21 parameter decreases from about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$22 to about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$23, the $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$24 ratio drops from about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$25 to about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$26, and the system transforms into an $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$27-domain state with uniform in-plane polarization. Upon unloading, the vortex reflections and original structure recover. Time-resolved imaging further showed that individual vortex cores do not translate before collapse, and once a distorted core is broken the entire vortex annihilates within about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$28 (Chen et al., 2019).

In ultra-thin PZT, the primary dynamical variable is the orientation of the vortex tubes. Effective-Hamiltonian molecular dynamics predicts that a homogeneous in-plane ac field resonant with a low-frequency $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$29 polar mode can rotate the vortex tubes by $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$30. At $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$31, a drive $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$32 with $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$33 and threshold $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$34 converts the state from $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$35-oriented to $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$36-oriented vortex tubes between about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$37 and $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$38. During this process, transient meron crystals, disclinations, and bubble-like geometries appear (Rijal et al., 2023).

Model systems make the role of spatially structured forcing especially transparent. In a 2D LGD treatment of ultrathin ferroelectric films, boundary conditions imposing zero net polarization at the edges stabilize a single vortex or antivortex, while two localized in-plane fields applied at diagonally separated points generate a vortex–antivortex–vortex triplet that relaxes back toward a single vortex when the fields are removed (Roy et al., 2011). In monolayer PbX ($n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$39), a multiscale workflow combining Berry-phase DFT, DeePMD molecular dynamics, and finite-element modeling shows that a spherical indenter of radius about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$40 and depth about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$41 can nucleate a vortex-like polar texture in PbTe, and that membrane geometry and loading select antivortex or flux-closure states at pressures between $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$42 and $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$43 (Xu et al., 2022).

5. Electronic consequences and functional uses

Topological ferroelectric vortices are not only structural textures; in some platforms they also define electronically distinct channels, collective excitations, or controllable state variables. The most direct evidence comes from BiFeO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$44 nanoislands, where topological cores support quasi-one-dimensional metallic conduction, but related functionality is also anticipated in domain-wall networks, toroidal-moment lattices, and switchable chiral textures (Yang et al., 2020, Chae et al., 2013, Sanchez-Santolino et al., 2023, Mei et al., 2018, Hong et al., 2017).

In patterned rhombohedral BiFeO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$45 nanoislands, vector PFM reconstructs two electrically writable topological states: a quadrant vortex state with four neutral $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$46 walls meeting at a core, and a quadrant center-convergent state with four head-to-head charged walls converging on a monopole-like core. Conductive AFM at $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$47 reveals bright nanoscale core channels with currents of about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$48 for vortex cores and about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$49 for center cores, whereas charged domain walls carry about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$50–$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$51 and neutral walls only a few pA. Simulations and profile analysis give channel full width at half maximum below about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$52–$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$53, and the current density is of order $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$54. Both core currents decrease with temperature from $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$55 to $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$56, consistent with metallic conduction, and the electrically written high-conduction states exhibit an on/off resistance ratio higher than $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$57, stability for about $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$58 minutes, and retention over $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$59 tested cycles without apparent fatigue (Yang et al., 2020).

The microscopic origin of this conduction depends on core type. Center cores are intrinsically conductive because the head-to-head polarization converges screening electrons and charged defects, locally lowering the conduction band below the Fermi level and confining carriers into a quasi-1D metallic channel. Vortex cores are different: at zero bias they are not equivalently charged, but local tip bias twists the core into a center-like convergent configuration while preserving the surrounding flux closure, thereby making the core metallic during readout. This distinction is important because it shows that topological equivalence in winding number does not imply identical electronic structure (Yang et al., 2020).

In $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$60-REMnO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$61, the emphasis is less on core conduction than on network morphology. Because type-I vortex–antivortex networks and type-II condensed stripe-like states differ in wall connectivity, narrow two-gon prevalence, and macroscopic polarization bias, boundary conditions, strain, and poling provide handles for engineering domain-wall networks. The underlying study explicitly notes that domain walls in hexagonal manganites can exhibit distinct electronic or conductive behavior, suggesting that topology control may be used to define reconfigurable conduction pathways or nonvolatile states tied to wall geometry (Chae et al., 2013).

Other platforms contribute complementary functional variables. Twisted freestanding BaTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$62 offers a high-density two-dimensional vortex crystal whose period is set by twist angle and whose local chirality and toroidal moment alternate with moiré registry; the reported work mentions prospects for nonvolatile memory, reconfigurable metasurfaces, neuromorphic elements, and electromechanical transducers, although it does not provide bit-density, switching-energy, or speed estimates (Sanchez-Santolino et al., 2023). In coherent TbScO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$63/BiFeO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$64 superlattices, the ordered $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$65–$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$66 phase combines multiferroicity with a chiral vortex lattice, while in PTO/BFO/STO tricolor superlattices the semivortex spiral carries a net in-plane polarization switchable by an experimentally feasible irrotational in-plane field of $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$67–$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$68 through a reversible spiral $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$69 vortex-like $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$70 spiral pathway (Mei et al., 2018, Hong et al., 2017).

6. Theoretical synthesis, misconceptions, and open problems

Despite their material diversity, ferroelectric vortex states are consistently described as compromises between local condensation of order and nonlocal penalties associated with depolarization, strain, and spatial gradients. Recent GLD-based “soft-domain” theory for strained PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$71 films makes this explicit by enforcing the near-divergence-free condition $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$72 and reducing the texture problem to a small set of variational amplitudes. In that framework, compressive epitaxial strain stabilizes a rotated vortex phase for thicknesses $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$73, while helix, wave, and mixed phases become nearly degenerate near the phase boundaries, providing a compact theoretical explanation for experimentally observed labyrinthine mixtures in PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$74-based systems (Boron et al., 8 Sep 2025).

Several terminological confusions recur in the literature. First, “ferroelectric vortex” does not denote a single microscopic object. In hexagonal manganites the topological variable is the trimerization phase interlocked with $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$75, not simply the local polarization angle (Chae et al., 2013). In most perovskite vortex lattices, by contrast, the relevant winding is directly that of the polarization field (Rijal et al., 2023). Second, not every rotational texture is a skyrmion: the mechanically switched PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$76–SrTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$77 structures are vortices rather than full skyrmions because skyrmion-number quantization is not demonstrated there (Chen et al., 2019). Third, the phrase can become misleading in adjacent fields. In ferroelectric superconductors such as polar SrTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$78, the ferroelectric order fixes the Rashba axis, but the “vortices” discussed in the topological-superconductivity literature are superconducting vortices with Majorana states, not polarization vortices (Yerzhakov et al., 2022).

The space of candidate materials is also still expanding. A Landau theory for the iron-based metal-organic framework (DMA)Fe$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$79(COOH)$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$80 identifies a primary two-component nonpolar order parameter $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$81, a secondary polarization $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$82, and a trilinear coupling $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$83, predicting $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$84 vortex topology and unusually wide, sometimes nested, domain walls. This extends the improper-ferroelectric vortex paradigm beyond inorganic oxides and suggests that molecular ordering can play the same symmetry-breaking role that structural trimerization plays in hexagonal manganites (Foggetti et al., 2022).

Open problems remain system-specific and substantial. In $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$85-REMnO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$86, a microscopic quantitative theory of partial-dislocation interactions is still needed, particularly because the observed $n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$87 phenomenology and the inferred short-range repulsion must be reconciled with first-principles suggestions of weak wall–wall interactions, and the origin and tunability of self-poling fields remain unresolved (Chae et al., 2013). In PbTiO$n = \frac{1}{2\pi}\oint \nabla \theta \cdot d\ell,$88-based superlattices, chirality control, fatigue under repeated mechanical cycling, and layer-resolved manipulation beyond proof-of-principle nanoindentation are open engineering challenges (Chen et al., 2019). In tricolor spiral systems and 2D PbX membranes, experimental verification of the predicted textures, switching pathways, and temperature windows is still required (Hong et al., 2017, Xu et al., 2022). More broadly, the field continues to move from identifying vortex topology toward understanding how topology, kinetics, and electronic functionality can be co-designed in the same material platform.