---
title: 'Polar Bi-Merons: Composite Topological Textures'
url: https://www.emergentmind.com/topics/polar-bi-merons
type: topic
---

# Polar Bi-Merons: Composite Topological Textures

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 [2210.10786], as elongated skyrmion stripes “or bimerons” and a “two-dimensional, tetratic lattice of merons” in ferroelectric superlattice membranes [2101.04545], and as magnetic or optical bimerons understood as a “pair of two merons” or a “second-order meron (also referred to as bimeron)” [1811.07068]; [2010.01691]. 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 [2210.10786]. “Unusual topological polar texture in moiré ferroelectrics” likewise reports a **“network of polar merons and antimerons”** rather than an isolated discrete bi-meron [2412.18763]. “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** [2405.15126].

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”** [2101.04545]. 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 [2509.03849].

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 \(Q=\pm \frac12\) [2210.10786]. 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 [2412.18763]. In bilayer WSe\(_2\), the local winding density is written as
\[
q(\mathbf{x}) = \mathbf{P}(\mathbf{x}) \cdot \left(\partial_x \mathbf{P}(\mathbf{x}) \times \partial_y \mathbf{P}(\mathbf{x})\right),
\]
and integrating over AB and BA domains yields \(Q_{\mathrm{AB}}=+\tfrac12\) and \(Q_{\mathrm{BA}}=-\tfrac12\) [2405.15126].

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
\[
n_\mathrm{Sk}(\mathbf{r})=\frac{1}{4\pi}\,\mathbf{m}(\mathbf{r})\cdot\left[\frac{\partial \mathbf{m}(\mathbf{r})}{\partial x}\times \frac{\partial \mathbf{m}(\mathbf{r})}{\partial y}\right],
\]
with a full bimeron carrying \(N_\mathrm{Sk}=\pm1\) and the constituent meron and antimeron each carrying \(\pm \tfrac12\) [1811.07068]. In synthetic antiferromagnets, the topology is written in terms of winding \(w\), core polarity \(L_z|_{\textrm{core}}\), and helicity \(\gamma\), with
\[
Q = w \cdot L_z|_{\textrm{core}},
\]
so that winding, polarity, and chirality are explicitly distinct descriptors [2303.14853]. In optical microcavities, the Stokes-field charge is
\[
Q = \frac{1}{4\pi} \int \mathbf{S} \cdot \left( \partial_x \mathbf{S} \times \partial_y \mathbf{S} \right) dxdy,
\]
and the second-order meron/antimeron carry \(Q=\pm 1\) [2010.01691].

## 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 \(P_{\perp}\) and the in-plane component \(\mathbf{P}_{\parallel}\) are both symmetry-allowed, with the important relation
\[
\mathbf{P}_{\parallel}(\mathbf{s}) \propto \nabla_{\mathbf{s}} P_{\perp}(\mathbf{s}).
\]
The resulting real-space texture forms a **network of merons and antimerons** with winding numbers \(\pm \frac12\); 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** [2210.10786].

“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\(_2\) and WSe\(_2\) homobilayers [2412.18763].

“Imaging topological polar structures in marginally twisted 2D semiconductors” extends this picture to bilayer WSe\(_2\) 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 [2405.15126]. 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 \([(\mathrm{PbTiO}_3)_{16}/(\mathrm{SrTiO}_3)_{16}]_8\) lifted-off membranes, varying temperature and elastic boundary conditions drives a reversible transition from a **skyrmion state** with topological charge \(-1\) to a **two-dimensional, tetratic lattice of merons** with topological charge \(-\tfrac12\). 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 [2101.04545].

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 \(N_\mathrm{Sk}=\pm1\) while the constituent meron and antimeron each carry \(\pm \tfrac12\) [1811.07068]. 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 \(\rho\) tends to zero [1703.09173].

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 \(Q=\pm1\); stronger chirality induces different vortex and antivortex sizes and a detachment of merons [2304.14314]. “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
\[
f(z)=c\,\frac{z-a}{z+a}
\]
in the bubbling limit [2512.11400].

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 \(Q=w\cdot L_z|_{\textrm{core}}\), and the fully compensated synthetic antiferromagnets host **homochiral Néel bimerons that are stable at room temperature** [2303.14853]. 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 \(Q=-1\) and an outer bimeron with \(Q=+1\), giving total \(Q=0\) [2010.10822]. In optics, a **second-order meron** is “also referred to as bimeron” and carries \(Q=+1\), while the second-order antimeron carries \(Q=-1\) in the Stokes-pseudospin field of a liquid-crystal microcavity [2010.01691].

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 \(^{87}\mathrm{Rb}\) and **isolated inverted merons** in spin-polarized antiferromagnetic \(^{23}\mathrm{Na}\). 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 \(^{87}\mathrm{Rb}\) spin-vortex lattice and the inverted merons with polar cores in \(^{23}\mathrm{Na}\) [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 \(Q=\tfrac12\), their time-reversed partners are antimerons with \(Q=-\tfrac12\), and the double-peak and spin-spiral phases are meron-antimeron superpositions with \(Q=0\) 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 \(Q=\pm1\) bimeron [2509.03849].

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.

Source: https://www.emergentmind.com/topics/polar-bi-merons