Valley-Locked Spin Skyrmions in Topological Media

• Valley-locked spin skyrmions are topological textures whose spin structure is constrained by valley degrees, manifesting as either fixed valley backgrounds or spin-valley entangled states.
• They appear across diverse platforms, including graphene quantum Hall ferromagnets, twisted bilayer graphene with hBN, and valley-Hall photonic/acoustic devices, each involving unique dynamical fields.
• Experimental approaches such as spin-resolved STM and directional edge-mode transport measurements help identify their distinct signatures and practical implications.

Valley-locked spin skyrmions are topological spin textures whose spin structure is constrained by a valley degree of freedom, but the phrase does not denote a single universal object. In electronic graphene systems, it can mean either a spin skyrmion propagating in a fixed valley-polarized Chern background or a genuinely spin-valley entangled $\mathrm{CP}^3$ texture; in valley photonic crystal waveguides and surface phononic crystals, it denotes spin-skyrmion textures carried by valley-polarized topological edge states, with skyrmion polarity or skyrmion number locked to the valley index (Bömerich et al., 2020, Lian et al., 2016, Lian et al., 2017, He et al., 4 May 2026, Liu et al., 22 Apr 2026). The concept is therefore best understood as a family resemblance across multicomponent topological media rather than a single microscopic mechanism.

1. Meanings of valley locking

The central distinction is whether valley is a frozen background label, a dynamical component of the order parameter, or a momentum-space index of a topological edge mode. In twisted bilayer graphene aligned with hBN at $\nu=3$, the low-energy theory excludes valley textures and retains only a magnetization field $\hat m(\mathbf r)\in S^2$; there, “valley locked” is an inference from the parent anomalous quantum Hall insulator being fully valley polarized (Bömerich et al., 2020). In graphene quantum Hall ferromagnets at $\nu=\pm1$ or $\nu=-1$, the order parameter is a $\mathrm{CP}^3$ spinor, so spin and valley can become locally non-separable; there, the closest realization of a valley-locked spin skyrmion is the spin-valley entangled or “entanglement” skyrmion (Lian et al., 2016, Lian et al., 2017). In photonic and acoustic valley-Hall platforms, valley locking is explicit: the valley index of the edge state determines skyrmion polarity and propagation direction (He et al., 4 May 2026, Liu et al., 22 Apr 2026).

Setting	Dynamical field	Meaning of “valley locked”
TBG/hBN AQH state	Spin-only $O(3)$ magnetization	Spin texture lives in a fixed valley-polarized Chern band
Graphene QH $\mathrm{CP}^3$ ferromagnet	Spin $\otimes$ valley field	Spin and valley are locally entangled within the skyrmion
Valley-Hall photonic or acoustic edge mode	Local optical or acoustic spin density	Valley index fixes skyrmion polarity/sign and transport direction

This tripartite classification also resolves a recurring ambiguity. “Valley locked” does not always mean that valley itself winds. In some systems the valley degree of freedom is explicitly frozen, while in others it is the texture-bearing field or the topological label of a propagating eigenmode.

2. Spin skyrmions in a fixed valley-polarized Chern background

In twisted bilayer graphene aligned with hBN, the relevant regime is the anomalous quantum Hall state at filling $\nu=3$, where the insulating parent state has both valley- and ferromagnetic order and is described as a fully spin- and valley-polarized Chern insulator (Bömerich et al., 2020). The crucial restriction is that the valley degree of freedom is treated as gapped: continuous valley rotation changes the sign of $\nu=3$0 and closes the charge gap, so the low-energy theory retains only magnetization. The effective order parameter is therefore a unit vector $\nu=3$1, not a coupled spin-valley field.

The topology is that of an $\nu=3$2 nonlinear sigma model. The skyrmion number is

$\nu=3$3

with local topological density

$\nu=3$4

Because the texture lives in a Chern band, topological density binds electric charge according to

$\nu=3$5

This is the precise sense in which the skyrmion is valley locked in this problem: the dynamical field is spin-only, but the charge-topology relation is inherited from the selected valley-Chern sector (Bömerich et al., 2020).

The lowest-energy texture at $\nu=3$6 and small doping is not a conventional skyrmion lattice but a double-tetarton lattice. Its building block covers one quarter of the unit sphere twice, and unlike skyrmion lattices its net magnetization vanishes. At larger magnetic fields the system crosses over to more conventional skyrmion lattices, yet even there the cost of adding charge is strongly reduced by smooth spin textures; for estimated parameters $\nu=3$7 meV, $\nu=3$8 meV, and $\nu=3$9 T, adding one charge flips about $\hat m(\mathbf r)\in S^2$0 spins (Bömerich et al., 2020).

This restricted construction has an important interpretive consequence. The theory supports spin skyrmions and double-tetarton relatives in a valley-polarized anomalous Hall background, but it does not formulate an explicit spin-valley skyrmion theory with fluctuating valley order. A plausible implication is that the expression “valley-locked spin skyrmion” is accurate here only if “locked” means “embedded in a fixed valley-polarized Chern band,” not “co-winding in spin and valley.”

3. Spin-valley entangled skyrmions in graphene quantum Hall ferromagnets

In monolayer graphene at filling factor $\hat m(\mathbf r)\in S^2$1 or $\hat m(\mathbf r)\in S^2$2, the low-energy Hilbert space is four-dimensional,

$\hat m(\mathbf r)\in S^2$3

and the approximate internal symmetry is $\hat m(\mathbf r)\in S^2$4 (Lian et al., 2016, Lian et al., 2017). The ferromagnetic order parameter is therefore a normalized four-component spinor modulo phase, i.e. a point in $\hat m(\mathbf r)\in S^2$5. The $\hat m(\mathbf r)\in S^2$6 Landau level introduces a graphene-specific identification: valley and sublattice coincide, so $\hat m(\mathbf r)\in S^2$7 and $\hat m(\mathbf r)\in S^2$8 have support on opposite sublattices. As a result, a valley texture is simultaneously a sublattice-resolved density texture.

The continuum theory is a $\hat m(\mathbf r)\in S^2$9 nonlinear sigma model. Its $\nu=\pm1$0-symmetric sector contains the covariant gradient

$\nu=\pm1$1

and the topological density

$\nu=\pm1$2

which at $\nu=\pm1$3 is identified with the excess electric charge density (Lian et al., 2016). Symmetry breaking enters through Zeeman coupling and valley anisotropies $\nu=\pm1$4 and $\nu=\pm1$5, so the stabilization of different skyrmion families is not a consequence of bare $\nu=\pm1$6 symmetry alone.

A general local state can be parameterized as

$\nu=\pm1$7

For $\nu=\pm1$8 or $\nu=\pm1$9, the state factorizes into spin and valley sectors. For intermediate $\nu=-1$0, it is genuinely entangled, and $\nu=-1$1 define an “entanglement Bloch sphere” (Lian et al., 2016). This structure yields a taxonomy of skyrmions: pure spin skyrmions, pure valley pseudospin skyrmions, spin-valley entangled skyrmions, and deflated pseudospin skyrmions (Lian et al., 2016, Lian et al., 2017).

The strongest realization of a valley-locked spin skyrmion in this literature is the entanglement skyrmion. It is not merely a spin skyrmion in one fixed valley, nor a valley skyrmion with homogeneous spin. Rather, the local state cannot be specified as an independent spinor in spin space and an independent spinor in valley space. This means that opposite spin orientations can be tied to opposite valley, hence opposite sublattice, components. In easy-axis backgrounds the predicted lattice-scale signature is a sublattice-antiferromagnetic spin pattern: one sublattice carries spin up and the other spin down in the maximally entangled region (Lian et al., 2016). In the terminology of the $\nu=-1$2 analysis, entanglement skyrmions are the most faithful carriers of “valley-locked spin” because their texture lies in the space of spin-pseudospin entanglement rather than in spin or valley alone (Lian et al., 2017).

The distinction between exact and induced locking is essential. The locking is not an exact microscopic constraint imposed by $\nu=-1$3; it is symmetry-enabled and anisotropy-induced. Near $\nu=-1$4, valley pseudospin textures soften, favoring pseudospin skyrmions; for positive $\nu=-1$5 and $\nu=-1$6, especially when both exceed $\nu=-1$7, partially entangled ferromagnetic backgrounds and entangled skyrmions become favorable (Lian et al., 2016, Lian et al., 2017).

4. Valley-locked spin skyrmions as topological edge-mode eigenstates

In wave-based valley-Hall systems, the phrase is used literally. The skyrmion texture is carried by a topological valley edge state, and the valley index determines the skyrmion polarity or skyrmion number. In a valley photonic crystal waveguide, the relevant spin field is the normalized optical spin angular momentum density,

$\nu=-1$8

and the skyrmion number is

$\nu=-1$9

The all-dielectric platform consists of two inversion-related valley photonic crystals with opposite valley Chern numbers, $\mathrm{CP}^3$0 and $\mathrm{CP}^3$1, forming a zigzag interface that supports a topological edge state below the light cone (He et al., 4 May 2026). Because the edge mode is evanescent, spin-orbit coupling in the near field generates nonzero in-plane spin components, while valley-dependent phase vortices determine the sign reversal of $\mathrm{CP}^3$2. The resulting texture is a Néel-type optical spin skyrmion with nearly quantized $\mathrm{CP}^3$3 or $\mathrm{CP}^3$4, and time-reversal-related valleys $\mathrm{CP}^3$5 and $\mathrm{CP}^3$6 carry opposite polarity and opposite skyrmion number (He et al., 4 May 2026).

The same logical structure appears in the acoustic quantum skyrmion-valley Hall effect. There the spin field is the acoustic spin angular momentum density

$\mathrm{CP}^3$7

with skyrmion number

$\mathrm{CP}^3$8

In a surface phononic crystal with a honeycomb lattice of resonant cavities, inversion breaking opens a valley Hall gap and produces orbital-valley locking. The reported correspondence is explicit: $\mathrm{CP}^3$9 At the domain wall, these valley-polarized edge states propagate directionally and carry acoustic Néel-type spin skyrmions with measured or estimated $O(3)$0 and $O(3)$1 near $O(3)$2 (Liu et al., 22 Apr 2026).

These two wave platforms differ from the electronic graphene cases in one decisive respect. Valley locking here is not inferred from a frozen background or from internal spin-valley entanglement; it is encoded directly by topological edge-state structure. The valley label determines transport direction, local orbital angular momentum, and the sign of the real-space skyrmion texture.

5. Multicomponent theory, adjacent valley-spin phenomena, and scope

The broadest formal precursor is the $O(3)$3 or $O(3)$4 theory of multicomponent quantum Hall skyrmions, where the internal components may represent spin, valley, layer, or combinations thereof (Kovrizhin et al., 2012). In that framework, the local state is a normalized $O(3)$5-component complex spinor, the ground state in the isotropic limit is a hexagonal skyrmion lattice, and the collective spectrum contains $O(3)$6 gapless acoustic magnetic modes plus a magnetophonon. For the $O(3)$7 AlAs example, the detailed anisotropy analysis addresses valley skyrmions with spin effectively quenched, so the work is foundational for multicomponent and valley skyrmion lattices but does not itself demonstrate spin-valley locking (Kovrizhin et al., 2012).

Several recent developments are closely related but are not skyrmion theories. In 2D altermagnets, helical and chiral edge channels can be spin-valley-momentum locked and characterized by a composite spin-valley Chern number $O(3)$8, but the relevant objects are momentum-space Berry-curvature textures and edge states, not real-space skyrmions (Chen et al., 6 Mar 2026). In monolayer WSe$O(3)$9, the valley Hall effect generates a pure valley current which, because of robust spin-valley locked valence states, acts as an effectively pure spin current with out-of-plane polarization and produces a damping-like torque $\mathrm{CP}^3$0 on an adjacent ferromagnet; this is an enabling mechanism for future skyrmionics rather than a skyrmion result (Sousa et al., 2022). In BaMnBi$\mathrm{CP}^3$1, stacked quantum Hall effect and nonlinear Hall effect provide transport evidence for a spin-valley locked Dirac state with spin-valley degeneracy $\mathrm{CP}^3$2 and valley-contrasted Berry curvature, again without a skyrmion texture being established (Mali et al., 30 Dec 2025).

A common misconception follows from this overlap of language. Not every valley-locked spin phenomenon is a valley-locked spin skyrmion. The latter requires an actual topological spin texture—either in a real-space magnetization or in a local optical or acoustic spin field—whereas spin-valley-momentum-locked transport, valley Hall torques, and Berry-curvature dipoles are adjacent but distinct phenomena (Chen et al., 6 Mar 2026, Sousa et al., 2022, Mali et al., 30 Dec 2025).

6. Experimental signatures and conceptual distinctions

The observable content of valley-locked spin skyrmions depends strongly on the platform. In the TBG/hBN anomalous Hall system, the proposed discriminator is magnetization as a function of doping and field: at $\mathrm{CP}^3$3, the undoped parent is fully polarized, while infinitesimal doping yields a double-tetarton lattice with zero net magnetization; spin-polarized STM is proposed as the most direct probe (Bömerich et al., 2020). In graphene quantum Hall ferromagnets, the decisive observables are lattice-resolved spin densities. Because valley maps onto sublattice in the $\mathrm{CP}^3$4 Landau level, a genuine entanglement skyrmion is expected to display a sublattice-antiferromagnetic spin pattern, making spin-resolved STM or STS an appropriate probe (Lian et al., 2017).

In the wave analogues, the skyrmion itself is part of the propagating eigenmode. In the photonic system, the preserved quantity is the local optical SAM texture and its skyrmion number under directional transport through a sharp Z-shaped bend and two defect classes defined by either reduced side length or a $\mathrm{CP}^3$5 twist of triangular rods (He et al., 4 May 2026). In the acoustic system, directional transmission is selected by orbital angular momentum $\mathrm{CP}^3$6, and the skyrmion sign follows the valley label through the chain

$\mathrm{CP}^3$7

with spin-texture-selective excitation demonstrated numerically through the overlap rule

$\mathrm{CP}^3$8

(Liu et al., 22 Apr 2026).

Three conceptual distinctions are therefore necessary. First, a valley-locked spin skyrmion need not involve a dynamical valley texture; the TBG/hBN problem is explicitly spin-only at low energy (Bömerich et al., 2020). Second, when valley is dynamical, the most characteristic objects are not pure spin skyrmions in a fixed valley but entangled $\mathrm{CP}^3$9 textures in which spin and valley cannot be separated (Lian et al., 2016). Third, in valley-Hall photonic and acoustic platforms, the “spin” is optical or acoustic spin angular momentum rather than microscopic electron spin, yet the skyrmion number is defined by the same Pontryagin-type real-space texture integral and is topologically locked to the valley index of the edge mode (He et al., 4 May 2026, Liu et al., 22 Apr 2026).

Under this broader but technically precise usage, valley-locked spin skyrmions designate a class of topological textures whose internal spin structure is inseparable from valley selection, valley polarization, or valley-Hall propagation. The exact mechanism—frozen valley background, $\otimes$0 entanglement, or valley-protected edge-state transport—depends on the microscopic platform.