Layer-Pseudospin Skyrmion Textures

• Layer-pseudospin skyrmion textures are topological configurations where a discrete layer pseudospin winds across multilayer systems, establishing robust magnetic structures.
• They occur in diverse platforms—from interfacial-DMI multilayers to moiré graphene—where tuning interactions like DMI and dipolar coupling drives transitions between distinct magnetic states.
• These textures generate emergent gauge fields and Chern bands, offering insights into collective excitations, transport phenomena, and potential applications in advanced device technologies.

Layer-pseudospin skyrmion textures are topological textures in which the discrete layer degree of freedom of a multilayer, bilayer, or effectively layered system is treated as an internal pseudospin and develops a nontrivial spatial or momentum-space winding. In the simplest bilayer case, the pseudospin distinguishes top and bottom layers; in thicker stacks it is naturally generalized to a multicomponent order parameter in $\mathbb{C}P^{N-1}$ or to a discrete layer index coupled by dipolar, exchange, or moiré-mediated interactions. Across current literature, the term encompasses several closely related structures: vertically coherent skyrmion tubes and domain tubes in interfacial-DMI multilayers, skyrmion and bimeron strings in weakly coupled layered frustrated magnets, multilayer skyrmion crystals with layer-dependent Dzyaloshinskii–Moriya interaction, and real-space layer-skyrmion lattices in moiré graphene systems whose Berry phase generates emergent Chern bands (Pulecio et al., 2016, Zhang et al., 2021, Hayami, 2022, Guerci et al., 2024, Tan et al., 10 Nov 2025).

1. Conceptual and topological framework

The basic language of layer pseudospin follows the multicomponent Skyrmion formalism developed for quantum Hall ferromagnets with internal spin, valley, or layer indices. In that setting, the local many-body state is represented by a normalized $d$-component spinor

$$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$

and the low-energy theory is a $CP^{d-1}$ nonlinear sigma model with Coulomb interactions (Kovrizhin et al., 2012). For a pure bilayer, $d=2$, the two spinor components can be identified with top and bottom layers, and the layer pseudospin vector is

$$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$

so that a layer skyrmion is a configuration in which $\mathbf{P}(\mathbf{r})$ wraps the Bloch sphere.

The corresponding real-space topological charge is the standard skyrmion number,

$$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$

or, in multicomponent form, the $CP^{N-1}$ invariant written in terms of the projector $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$ (Tan et al., 10 Nov 2025). The same topological logic also admits a momentum-space version. For a two-level internal operator, including a layer pseudospin, one may define a Brillouin-zone texture $d$0 and its skyrmion number

$d$1

which underlies chiral and helical topological skyrmion phases of matter (Cook, 2019).

These constructions are realized in several distinct physical settings.

Platform	Layer degree of freedom	Characteristic texture
$d$2 multilayers	Discrete Co layers	Skyrmion tubes and coherent domain tubes
Centrosymmetric trilayer triangular magnet	Layers A/B/C with layer-dependent DM	Twisted surface SkX, anti-SkX, high-$d$3 SkX
Weakly coupled frustrated monolayers	Stack index of pancake skyrmions	3D skyrmion string and bimeron string
Ideal moiré Chern bands	Layer or color spinor	Real-space layer Skyrme texture with $d$4
Rhombohedral graphene multilayers	$d$5-component layer pseudospinor	Interaction-induced layer-skyrmion lattice

A recurrent distinction is between the “all-layers-aligned” limit, in which interlayer coupling locks nearly identical textures across depth, and genuinely nontrivial layer-space textures, in which helicity, topological charge, or even the presence of a skyrmion varies with layer index. This suggests a useful organizing principle: layer-pseudospin skyrmion textures interpolate between ordinary skyrmion tubes and full multicomponent topological textures in a synthetic layer dimension.

2. Multilayer micromagnetics and vertically coherent layer textures

A concrete micromagnetic realization appears in interfacial-DMI multilayers of

$d$6

where the Co layers are discrete, the coupling between them is purely magnetostatic, and increasing $d$7 increases the total dipolar energy (Pulecio et al., 2016). The effective energy density is of standard ultrathin-film form,

$d$8

with interfacial DMI

$d$9

Experiment and simulation identify single-domain states, isolated hedgehog skyrmion bubbles, transitional deformed bubbles, and cycloidal labyrinth domains. The key observation is that the line separating bubble-like textures from cycloidal textures shifts to lower $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$0 as $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$1 increases, because interlayer dipolar coupling promotes coherent domain formation. Matching the observed $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$2-driven transition yields $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$3.

Although that work does not explicitly introduce a pseudospin operator for the layer index, it contains the essential ingredients of a layer-pseudospin description. Each Co layer hosts a chiral texture $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$4, and the interlayer dipolar coupling acts as a nonlocal interaction between these layer-resolved textures. In the skyrmion-bubble regime the stable objects are effectively skyrmion tubes, while in the cycloidal regime they are vertically coherent domain tubes. The lowest-energy configurations therefore correspond to an “all layers in phase” state in layer space, with misalignment penalized by stray-field energy. In the paper’s own formulation, “Fully 3D micromagnetic models were employed… which we have found can lead to different ground states than those implemented with effective medium generalizations since coupling between discrete magnetic layers is treated more rigorously” (Pulecio et al., 2016).

A related but distinct multilayer mechanism arises in magnets with local inversion asymmetry, modeled as a stack of ferromagnetic layers with alternating DMI sign,

$$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$5

and interlayer exchange $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$6 (Walsem et al., 2020). In the strong-coupling limit the stack behaves as a single effective layer with

$$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$7

so that $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$8 for odd $$w(\mathbf{r})=\bigl(w_1(\mathbf{r}),\ldots,w_d(\mathbf{r})\bigr), \qquad \sum_n |w_n(\mathbf{r})|^2=1,$$9 and $CP^{d-1}$0 for even $CP^{d-1}$1. This produces a sharp even–odd effect: odd-layer stacks retain a reduced net chirality, whereas even-layer stacks suppress spirals in the strong-coupling regime. In layer-pseudospin language, the DMI-dominant regime is staggered in layer space, while the interlayer-dominant odd-$CP^{d-1}$2 regime behaves as a ferromagnetically aligned pseudospin state with reduced magnitude.

More generally, the transformation from stripy states to skyrmion crystals need not be a topological phase transition. In chiral films with interfacial DMI, stripy states and SkXs are described as skyrmion condensates that are topologically equivalent, and continuous changes in stripe width relative to skyrmion spacing interpolate smoothly between helical states, mazes, and crystals while conserving the total skyrmion number $CP^{d-1}$3 (Wang et al., 2022). This suggests that in multilayer layer-pseudospin systems, stripe-like layer textures and skyrmion-lattice textures can also belong to a common topological sector when the layer-resolved $CP^{d-1}$4 is preserved.

3. Layer-resolved skyrmion, anti-skyrmion, and meron phases

The most explicit realization of nontrivial textures in layer space is the centrosymmetric trilayer triangular-lattice magnet with layer-dependent polar Dzyaloshinskii–Moriya interaction (Hayami, 2022). Layers A and C have opposite DM vectors,

$CP^{d-1}$5

while an interlayer exchange $CP^{d-1}$6 couples adjacent layers. The natural object at each in-plane position is the three-component state

$CP^{d-1}$7

and the low-temperature phase diagram contains thirteen phases, nine of which have quantized layer-resolved skyrmion numbers.

The phase taxonomy is naturally expressed through the tuple

$CP^{d-1}$8

Representative examples include $CP^{d-1}$9, where only the outer layers host Néel skyrmion crystals; $d=2$0, a three-layer SkX; $d=2$1, in which the middle layer hosts an anti-SkX; $d=2$2, a high-topological-number SkX on the middle layer; and $d=2$3, a central-layer-only anti-SkX. One of the most characteristic states is the “twisted surface SkX,” in which all three layers have $d=2$4 but the helicity twists from Néel-like to Bloch-like across the stack. Here the nontriviality lies not in differing skyrmion numbers but in the layer dependence of the internal skyrmion helicity. In pseudospin terms, this is a texture in layer space even though the real-space topological charge is the same in every layer.

A closely related bilayer-like setting is the frustrated honeycomb-lattice antiferromagnet with weak next-nearest-neighbor DM interaction (Mohylna et al., 13 Feb 2025). Because the honeycomb lattice is bipartite, its two triangular sublattices A and B act as pseudolayers. At lower frustration and in a narrow field window, the model stabilizes antiferromagnetic meron–antimeron pair crystal and gas phases. The fundamental unit is a meron–antimeron pair residing on different sublattices, so the texture is intrinsically layer-resolved. At larger frustration, the system supports a “two-layer-like three-sublattice antiferromagnetic skyrmion crystal phase,” in which each triangular sublattice carries its own skyrmion crystal and the two are coupled antiferromagnetically. The meron and skyrmion phases connect at higher temperature to spiral spin liquid phases, so the layer-pseudospin topological textures emerge from multi-$d=2$5 ordering selected out of frustrated spiral manifolds.

Layer-resolved internal structure also appears in weakly coupled frustrated monolayers that form 3D skyrmion strings (Zhang et al., 2021). A stack of 11 frustrated ferromagnetic monolayers with weak ferromagnetic interlayer exchange supports straight skyrmion strings built from “pancake skyrmions” in each layer. The core positions are nearly aligned, but the helicity $d=2$6 is slightly layer dependent for Bloch-type strings because dipolar interaction distorts the in-plane spins near the surfaces. The same system supports a current-driven transformation from a skyrmion string to a bimeron string, and the layer-resolved dynamics remain nearly synchronous across the stack. This is not yet a strongly nonuniform layer-pseudospin texture, but it is already a layer-resolved internal mode of a topological string.

4. Moiré layer skyrmions, emergent gauge fields, and Chern bands

In moiré quantum materials, the layer index becomes an explicit two-component pseudospin field whose smooth real-space texture can generate topology. A central result for ideal $d=2$7 Chern bands is the factorization

$d=2$8

where $d=2$9 is a generalized lowest-Landau-level wavefunction and $$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$0 is a normalized layer spinor (Guerci et al., 2024). Bloch periodicity requires the spinor to obey a screening boundary condition,

$$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$1

and its real-space Berry connection

$$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$2

carries a real-space Chern number

$$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$3

For a two-component layer spinor, the Bloch vector

$$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$4

therefore forms a real-space skyrmion with Pontryagin index $$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$5 per unit cell. For $$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$6, the ideal band decomposes into $$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$7 color Landau levels carried by $$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$8 layer spinors, and the real-space connection becomes non-Abelian in color space. Even then, the total real-space Chern number remains $$\mathbf{P}(\mathbf{r}) = w^\dagger(\mathbf{r})\,\boldsymbol{\sigma}\,w(\mathbf{r}),$$9, so the layer or color texture screens the fictitious magnetic phase irrespective of $\mathbf{P}(\mathbf{r})$0 and of the number of layers.

This real-space picture persists in realistic moiré graphene. The topologically robust Skyrme texture remains intact in twisted bilayer graphene “even far from the chiral limit, and for realistic values of corrugation,” and analogous structures are verified at the first magic angle of twisted bilayer, trilayer, and monolayer-bilayer graphene (Guerci et al., 2024). A central design rule is $\mathbf{P}(\mathbf{r})$1: in single-Dirac models, ideal bands require the number of layers to exceed the Chern number.

The same mechanism becomes interaction driven in the ideal limit of rhombohedral graphene multilayers (Tan et al., 10 Nov 2025). There the parent conduction-band layer spinor is

$\mathbf{P}(\mathbf{r})$2

and short-range repulsion $\mathbf{P}(\mathbf{r})$3 enforces purely chiral two-body zeros. For a Slater determinant this forces a factorization

$\mathbf{P}(\mathbf{r})$4

so that the many-body state separates into a lowest-Landau-level Slater determinant and a common layer-pseudospin texture. The latter generates an emergent magnetic field

$\mathbf{P}(\mathbf{r})$5

and when translation symmetry is reduced to a lattice with one flux quantum per unit cell, the normalized projector $\mathbf{P}(\mathbf{r})$6 carries a $\mathbf{P}(\mathbf{r})$7 skyrmion number equal to the flux. At filling one electron per skyrmion unit cell, the interaction-induced skyrmion lattice produces a $\mathbf{P}(\mathbf{r})$8 Chern band.

A complementary controlled continuum theory for moiré pseudospin textures starts from

$\mathbf{P}(\mathbf{r})$9

where $$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$0 is the local layer pseudospin texture (Du, 16 Feb 2026). A local SU(2) rotation yields a non-Abelian gauge field $$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$1, whose diagonal part is an emergent U(1) Berry connection $$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$2. Its flux is exactly the skyrmion density,

$$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$3

so that

$$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$4

For large branch splitting $$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$5, a Schrieffer–Wolff expansion produces a controlled single-branch Hamiltonian beyond strict adiabaticity, and the leading non-adiabatic corrections are fixed by the real-space quantum geometric tensor

$$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$6

In that formulation, the layer-skyrmion texture is simultaneously a source of emergent flux, a Chern-band generator, and a real-space quantum-geometric background.

5. Dynamics, strings, and collective excitations

Layer-pseudospin skyrmion textures support both rigid-body and internal collective modes. In layered frustrated magnets, a Bloch-type skyrmion string can be driven by a dampinglike spin–orbit torque and undergo a current-induced transformation into a dynamically stable bimeron string (Zhang et al., 2021). The dynamics follow an LLG equation with dampinglike torque,

$$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$7

and for currents $$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$8 the skyrmion string translates with a damping-dependent skyrmion Hall effect, whereas for $$Q = \frac{1}{4\pi}\int d^2r\; \mathbf{n}(\mathbf{r})\cdot \left(\partial_x \mathbf{n}(\mathbf{r}) \times \partial_y \mathbf{n}(\mathbf{r})\right),$$9 it transforms into a bimeron string that rotates stably around its center. When the current is switched off, the bimeron string relaxes back to a Bloch-type skyrmion string. The transformation is essentially synchronous across layers, which indicates a collective mode in layer space rather than sequential nucleation.

In interaction-induced moiré layer-skyrmion lattices, time-dependent Hartree–Fock reveals a hierarchy of collective excitations that are naturally interpreted as skyrmion-lattice dynamics (Tan et al., 10 Nov 2025). The spectrum contains two gapless acoustic phonon branches associated with lattice translations and a ladder of weakly dispersing gapped modes. A variational $CP^{N-1}$0-TDVP formulation identifies these gapped branches as chiral shape modes of the layer-skyrmion cores. The $CP^{N-1}$1 branch is a quadrupolar deformation rotating in time and is described as analogous to the “chiral graviton” mode of fractional quantum Hall systems, while higher $CP^{N-1}$2 yield higher-multipole shape oscillations. A breathing mode with $CP^{N-1}$3 modulates the skyrmion radius and layer polarization.

At the level of a gapped Hall phase built from a skyrmion Chern band, the collective dynamics admit a universal magneto-elastic effective theory (Du, 16 Feb 2026). If the texture $CP^{N-1}$4 forms a skyrmion lattice and $CP^{N-1}$5 denotes the displacement field, the Berry-phase term is

$CP^{N-1}$6

and implies

$CP^{N-1}$7

The resulting magnetophonon is therefore noncommutative and type-B. For short-range elasticity its dispersion is

$CP^{N-1}$8

while unscreened Coulomb interactions produce a $CP^{N-1}$9 crossover. Weak moiré pinning generates a pinned resonance at $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$0.

These results place layer-pseudospin textures within the same dynamical universality class as magnetic skyrmion crystals, but with electric, optical, and moiré control replacing or supplementing magnetic field control.

6. Experimental relevance, observables, and open directions

The experimental manifestations of layer-pseudospin skyrmion textures are platform dependent. In interfacial-DMI multilayers, Lorentz TEM and room-temperature remanent imaging resolve large domains, isolated hedgehog skyrmion bubbles, and dense cycloidal labyrinths as $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$1 is varied in $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$2 (Pulecio et al., 2016). In magnets with alternating DMI sign across layers, the spiral pitch depends strongly on layer number and interlayer coupling, including an even–odd suppression of spirals for even $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$3 (Walsem et al., 2020). These results make the layer dependence itself a metrological handle on DMI.

In moiré graphene, the most direct real-space signature is layer-resolved density modulation. For ideal and realistic layer-skyrmion lattices, the top-layer density has strong minima at skyrmion cores whereas the total density is much smoother, so top-layer-sensitive probes such as STM are natural candidates (Tan et al., 10 Nov 2025). In ideal Chern-band constructions, the layer spinor’s real-space Chern number $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$4 and the associated non-Abelian color connection suggest directly measurable real-space Berry-phase structure in layer-resolved wavefunctions (Guerci et al., 2024).

Optical response provides a second diagnostic. In the controlled skyrmion-Chern-band theory, the long-wavelength structure factor defines a quantum-weight tensor $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$5 obeying

$$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$6

with excess optical weight above the topological lower bound generated by flux inhomogeneity and finite-$$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$7 non-adiabatic corrections (Du, 16 Feb 2026). The same theory predicts Umklapp-folded collective modes in optical channels and a low-frequency pinned magnetophonon resonance. In rhombohedral graphene, the gapped shape modes and the breathing mode supply additional Raman and THz targets (Tan et al., 10 Nov 2025).

Several open directions are explicit in the cited literature. One is true layer-pseudospin engineering by making $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$8, $$P(\mathbf{r})=\hat{\chi}(\mathbf{r})\hat{\chi}^\dagger(\mathbf{r})$$9, $d$00, or $d$01 layer dependent, or by adding interlayer exchange $d$02, so that skyrmions appear only in selected layers, shift laterally with depth, or vary in chirality across the stack (Pulecio et al., 2016). Another is the weak-locking regime of layered frustrated strings, where relative helicity shifts or core displacements between layers become appreciable (Zhang et al., 2021). A third is the extension from trilayers to thicker stacks and superlattices, where one could realize “domain wall” skyrmion structures in the synthetic layer dimension (Hayami, 2022). In moiré settings, the leading unresolved issues are disorder, finite temperature, non-Abelian textures with multiple valleys or layers, and quantitative extraction of elastic moduli and $d$03 corrections from microscopic continuum models (Du, 16 Feb 2026).

Taken together, current work supports a precise usage of the term. Layer-pseudospin skyrmion textures are topological textures of a layer-resolved internal order parameter, realized either as real-space skyrmions and skyrmion lattices distributed across physical layers or as multicomponent layer-spinor textures that generate emergent gauge fields and Chern bands. Their defining features are the coupling of real-space topology to the layer index, the possibility of coherent or staggered stacking across depth, and the existence of collective modes and transport responses controlled by interlayer coupling, chirality, and quantum geometry.