Polar Bi-Merons: Composite Topological Textures

• Polar bi-merons are composite structures characterized by paired meron-antimeron states with fractional topological charges in polar vector fields.
• They are observed in moiré ferroelectrics and magnetic systems, where differing in-plane textures and strain conditions yield distinct Bloch and Néel configurations.
• Their study bridges polar skyrmion and meron phenomena, offering insights into the interplay of elastic, electrostatic, and gradient energies.

Polar bi-merons are not a uniformly standardized term in current literature. Closely related objects are reported as a “network of polar merons and antimerons” and a “single meron-antimeron pair” in strained and twisted bilayers (Bennett et al., 2022), as elongated skyrmion stripes “or bimerons” and a “two-dimensional, tetratic lattice of merons” in ferroelectric superlattice membranes (Shao et al., 2021), and as magnetic or optical bimerons understood as a “pair of two merons” or a “second-order meron (also referred to as bimeron)” (Göbel et al., 2018); (Król et al., 2020). This suggests that the topic is best understood at the intersection of polar vector textures, half-skyrmion topology, and composite two-meron states.

1. Terminology and scope

The exact term “bi-meron” is not used by the authors of “Polar meron-antimeron networks in strained and twisted bilayers”; that paper explicitly discusses a “network of merons and antimerons (half-skyrmions and half-antiskyrmions)” and a “single meron-antimeron pair” in a confined moiré cell (Bennett et al., 2022). “Unusual topological polar texture in moiré ferroelectrics” likewise reports a “network of polar merons and antimerons” rather than an isolated discrete bi-meron (Li et al., 2024). “Imaging topological polar structures in marginally twisted 2D semiconductors” provides experimental proof for meron/antimeron structures in bilayer WSe$_2$, but states that it does not explicitly study “polar bi-merons” as a named object (Vu et al., 2024).

A more direct ferroelectric connection appears in “Emergent chirality in a polar meron to skyrmion phase transition,” where the main text emphasizes polar skyrmions and merons, while the supplementary phase-field discussion explicitly states that low-temperature elongated textures can be regarded as “elongated skyrmion stripes, or bimerons” (Shao et al., 2021). By contrast, “Merons and Meroniums in Spin-Orbit Coupled Bose Gases” is only partially relevant to the phrase because it treats a two-component pseudospin-$\tfrac12$ Bose gas, does not use “polar” in the spin-1 sense, and names its zero-charge composites “meroniums” rather than bimerons (Chen et al., 4 Sep 2025).

This distribution of terminology is itself significant. In the polar-ferroelectric literature, the dominant language is polar merons, antimerons, and meron-antimeron networks/pairs. In the magnetic and optical literatures, bimeron is a more explicit and stabilized object name.

2. Order parameters and topological descriptors

Across these literatures, the relevant field is a normalized vector order parameter: polarization $\mathbf{P}$ in moiré ferroelectrics, magnetization $\mathbf{m}$ or Néel order $\mathbf{L}$ in magnets, and the Stokes or pseudospin vector $\mathbf{S}$ in optics. In “Polar meron-antimeron networks in strained and twisted bilayers,” the winding number is written as

$$Q = \frac{1}{4\pi} \int \mathbf{P}\cdot \partial_{s_x} \mathbf{P}\times \partial_{s_y} \mathbf{P}\, d\mathbf{s},$$

with individual moiré polar domains converging to $Q=\pm \frac12$ (Bennett et al., 2022). In marginally twisted hBN, the reconstructed unit polarization field gives

$$N = \frac{1}{4\pi}\int \mathbf{P}\cdot(\partial_x \mathbf{P}\times \partial_y \mathbf{P})\,ds = \pm \frac12,$$

which is identified as the signature of merons and antimerons (Li et al., 2024). In bilayer WSe$_2$, the local winding density is written as

$\tfrac12$0

and integrating over AB and BA domains yields $\tfrac12$1 and $\tfrac12$2 (Vu et al., 2024).

The magnetic bimeron literature supplies the complementary integer-charge construction. “Magnetic bimerons as skyrmion analogues in in-plane magnets” defines the local topological charge density as

$\tfrac12$3

with a full bimeron carrying $\tfrac12$4 and the constituent meron and antimeron each carrying $\tfrac12$5 (Göbel et al., 2018). In synthetic antiferromagnets, the topology is written in terms of winding $\tfrac12$6, core polarity $\tfrac12$7, and helicity $\tfrac12$8, with

$\tfrac12$9

so that winding, polarity, and chirality are explicitly distinct descriptors (Bhukta et al., 2023). In optical microcavities, the Stokes-field charge is

$\mathbf{P}$0

and the second-order meron/antimeron carry $\mathbf{P}$1 (Król et al., 2020).

3. Polar merons and paired motifs in moiré ferroelectrics

The most direct polar setting is the moiré bilayer literature. In 3R-stacked bilayer hBN and similar inversion-symmetry-broken bilayers, the out-of-plane polarization $\mathbf{P}$2 and the in-plane component $\mathbf{P}$3 are both symmetry-allowed, with the important relation

$\mathbf{P}$4

The resulting real-space texture forms a network of merons and antimerons with winding numbers $\mathbf{P}$5; in twisted bilayers the merons are of Bloch type, whereas in strained bilayers they are of Néel type, and a confined moiré cell can host a single meron-antimeron pair (Bennett et al., 2022).

“Unusual topological polar texture in moiré ferroelectrics” provides direct experimental reconstruction in R-type marginally twisted hBN by vector PFM. The observed texture combines alternating out-of-plane polarizations at domain regions with in-plane vortex-like polarization patterns along domain walls, and the out-of-plane polarization reverses three times across a DW from AB to BA stackings. The authors attribute this unusual profile to the competition between moiré ferroelectricity and piezoelectricity, and they report similar polar textures in marginally twisted MoSe$\mathbf{P}$6 and WSe$\mathbf{P}$7 homobilayers (Li et al., 2024).

“Imaging topological polar structures in marginally twisted 2D semiconductors” extends this picture to bilayer WSe$\mathbf{P}$8 using angle-resolved high-resolution vector PFM. It resolves both Bloch-type and Néel-type merons and thereby differentiates moiré superlattices formed due to twist or heterogeneous strain (Vu et al., 2024). This suggests that the nearest bi-meron-like motif in these moiré ferroelectrics is a neighboring meron-antimeron pair embedded in a reconstructed domain-wall network rather than an isolated particle-like bimeron.

4. Ferroelectric superlattices: skyrmion–meron–bimeron continuity

A different polar route appears in freestanding ferroelectric superlattices. In $\mathbf{P}$9 lifted-off membranes, varying temperature and elastic boundary conditions drives a reversible transition from a skyrmion state with topological charge $\mathbf{m}$0 to a two-dimensional, tetratic lattice of merons with topological charge $\mathbf{m}$1. The same work shows that the transition is accompanied by a change in chirality, from zero-net chirality in the meronic phase to net-handedness in the skyrmionic phase, and the supplementary phase-field discussion states that at 223 K “elongated skyrmion stripes, or bimerons, are formed along X-axis.” The stabilization mechanism is not Dzyalozhinskii–Moriya interaction; it is the interplay of elastic, electrostatic and gradient energies, with strain acting as a crucial order parameter (Shao et al., 2021).

This ferroelectric result is important because it supplies an explicit bridge between polar skyrmions, polar merons, and bimeron-like elongated textures. A reasonable interpretation is that polar bi-merons appear here as elongated or paired half-skyrmion textures lying between circular skyrmions and ordered meron lattices.

5. Bimeron definitions in magnetic, antiferromagnetic, and photonic systems

The magnetic literature provides the most explicit bimeron taxonomy. “Magnetic bimerons as skyrmion analogues in in-plane magnets” defines a magnetic bimeron as a pair of two merons and treats it as the in-plane-magnetized version of a skyrmion; the full object carries $\mathbf{m}$2 while the constituent meron and antimeron each carry $\mathbf{m}$3 (Göbel et al., 2018). Near the Lifshitz point, “Bound states of skyrmions and merons near the Lifshitz point” shows that skyrmions and bi-merons are stable in a large part of the phase diagram, and that merons carrying fractional topological charge become deconfined as the stiffness $\mathbf{m}$4 tends to zero (Kharkov et al., 2017).

Easy-plane chiral-magnet analyses sharpen the internal structure. “Meron configurations in easy-plane chiral magnets” describes bimerons as vortex and antivortex of opposite polarities, each contributing one-half of the total topological charge, together giving $\mathbf{m}$5; stronger chirality induces different vortex and antivortex sizes and a detachment of merons (Bachmann et al., 2023). “Bubbling analysis of bimeron configurations” makes the same structure mathematically precise, describing a bound pair of merons with opposite in-plane winding and opposite polarity and identifying a Möbius-type core profile

$\mathbf{m}$6

in the bubbling limit (Bacho et al., 12 Dec 2025).

Synthetic antiferromagnets add a directly reconstructed polarity degree of freedom. In that setting, merons, antimerons, and bimerons are imaged through the Néel order parameter, with $\mathbf{m}$7, and the fully compensated synthetic antiferromagnets host homochiral Néel bimerons that are stable at room temperature (Bhukta et al., 2023). Related composite terminology also matters: a bimeronium is the in-plane analogue of a skyrmionium and exists as a combination of an inner bimeron with $\mathbf{m}$8 and an outer bimeron with $\mathbf{m}$9, giving total $\mathbf{L}$0 (Zhang et al., 2020). In optics, a second-order meron is “also referred to as bimeron” and carries $\mathbf{L}$1, while the second-order antimeron carries $\mathbf{L}$2 in the Stokes-pseudospin field of a liquid-crystal microcavity (Król et al., 2020).

For the topic of polar bi-merons, these magnetic and photonic works provide the most explicit definitions of two-meron composites, the clearest separation of winding, polarity, and helicity, and the strongest vocabulary for distinguishing bimeron from bimeronium.

6. Condensate analogues and present limits of the term

Spinor-condensate work is adjacent rather than direct. In the non-equilibrium condensation of spin-1 Bose gases with spin-orbit coupling, rapid quenches can generate a meron crystal / spin-vortex lattice in ferromagnetic $\mathbf{L}$3 and isolated inverted merons in spin-polarized antiferromagnetic $\mathbf{L}$4. The paper does discuss polar cores and antiferromagnetic/polar interactions, but it does not introduce the term “polar bi-meron”; the closest objects are the meron–antimeron paired building blocks of the $\mathbf{L}$5 spin-vortex lattice and the inverted merons with polar cores in $\mathbf{L}$6 (1111.07068).

The two-component Rashba-coupled Bose-gas literature is even more clearly terminological about its boundary. In that setting, the half vortex and spherical wave half vortex are merons with $\mathbf{L}$7, their time-reversed partners are antimerons with $\mathbf{L}$8, and the double-peak and spin-spiral phases are meron-antimeron superpositions with $\mathbf{L}$9 that the authors call meroniums. The paper explicitly states that it does not use the term “polar” in the sense of a spin-1 polar condensate, and it does not construct a direct same-sign $\mathbf{S}$0 bimeron (Chen et al., 4 Sep 2025).

Taken together, these boundaries clarify current usage. In the most literal sense, polar bi-merons are best reserved for topological textures in a polar vector field or in a polar-core spin texture that also possess a recognizable two-meron composite structure. The direct polar literature currently emphasizes merons, antimerons, networks, and meron-antimeron pairs, while the explicit bimeron language remains most developed in magnetic and optical pseudospin systems.