Umbrella-like Spin Structures in Magnetism

• Umbrella-like spin structures are non-collinear arrangements where spins cant away from a common axis, forming conical or radial profiles in various magnetic systems.
• They include distinct variants such as the cone state in triangular-lattice XXZ antiferromagnets, Bloch-type merons/skyrmions, ferrimagnetic garnet textures, and momentum-space hedgehog textures in chiral Te.
• Analytical and numerical models reveal that stabilization mechanisms—ranging from exchange frustration and crystal-field anisotropy to spin–orbit coupling—govern field-dependent phase transitions and topological characteristics.

Umbrella-like spin structure denotes a non-collinear spin arrangement in which local moments tilt away from a common axis or center and form a conical, rib-like, or radial profile. The term is used across several technically distinct settings. In real space it describes the three-sublattice cone state of the triangular-lattice XXZ antiferromagnet, Bloch-type meron and skyrmion textures whose spins point out of plane at the core and cant toward the plane or the opposite background, and ferrimagnetic rare-earth iron garnets in which rare-earth moments tilt toward the iron molecular field along $[111]$. In momentum space it is also used for hedgehog-like spin textures with $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$ around Brillouin-zone corners in chiral tellurium (Marmorini et al., 2015, Bazeia et al., 2017, Tomasello et al., 2022, Mori et al., 2023, Sakano et al., 2019).

1. Geometric definition and principal variants

The defining geometric feature is canting: spins are neither strictly collinear nor confined to a single plane, but instead interpolate between a longitudinal component and a transverse component in a way that visually resembles the ribs of an opened umbrella. In the triangular XXZ model, the umbrella state is a three-sublattice texture with a uniform canting angle $\theta$ toward a longitudinal field and azimuthal phases $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$,

$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$

This is the standard cone-state realization of umbrella order. In Bloch-type skyrmion language, the corresponding texture is written as

$$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$

and for $n=1$ and $\gamma=\pi/2$ one obtains $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$, so the in-plane component is tangential rather than radial. In rare-earth iron garnets, the umbrella is a triad of rare-earth moments tilting coherently toward the ferrimagnetic $[111]$ direction while remaining tied to three mutually orthogonal crystal-field easy axes. In chiral Te, by contrast, the umbrella-like descriptor refers to a momentum-space hedgehog in which spin points predominantly parallel or antiparallel to $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$0, not tangentially around a contour (Marmorini et al., 2015, Bazeia et al., 2017, Tomasello et al., 2022, Sakano et al., 2019).

These uses establish several distinct variants. A cone umbrella is field-canted and periodic in real space. A Bloch umbrella is localized and topological or meron-like. A garnet umbrella is ferrimagnetic and crystal-field driven. A hedgehog umbrella is a reciprocal-space SOC texture. This suggests that “umbrella-like” labels a geometry of canting and radial or conical opening rather than a unique microscopic interaction.

2. Umbrella order in the triangular-lattice XXZ antiferromagnet

For the triangular-lattice XXZ antiferromagnet in a longitudinal field, the minimal Hamiltonian is

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$1

The easy-plane regime is $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$2, or $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$3. Classically, the umbrella state is the ground state throughout the field range in this regime. Its energy per site in the purely intralayer problem is

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$4

and minimization gives

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$5

Accordingly, the longitudinal magnetization grows linearly with field up to saturation.

Near saturation, the quantum problem is formulated as a dilute Bose gas with minima at $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$6. The effective condensate energy density is

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$7

with $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$8. The phase criterion is explicit: $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$9 selects the umbrella state as a single-mode BEC, whereas $\theta$0 selects a coplanar state with equal condensation at both valleys. In two dimensions, quantum fluctuations strongly suppress the umbrella near saturation. The critical anisotropy shifts from the classical Heisenberg value $\theta$1 to $\theta$2 for $\theta$3, $\theta$4 for $\theta$5, $\theta$6 for $\theta$7, $\theta$8 for $\theta$9, $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$0 for $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$1, $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$2 for $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$3, and $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$4 for $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$5. The monotonic approach to the classical boundary quantifies how increasing $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$6 reduces fluctuation corrections. In layered systems, antiferromagnetic interlayer coupling shrinks the coplanar pocket and can restore umbrella order; for $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$7 with $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$8, an interlayer XY coupling of order $\phi_\alpha \in \{0,2\pi/3,4\pi/3\}$9 is sufficient to favor the stacked umbrella over the stacked $$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$0-coplanar state. These results are directly relevant to Ba$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$1CoSb$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$2O$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$3, where the high-field phase near saturation is expected to be coplanar rather than umbrella for the experimentally inferred easy-plane anisotropy and small interlayer exchange (Marmorini et al., 2015).

3. Crystal-field-driven umbrella states in rare-earth iron garnets

In $$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$4Fe$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$5O$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$6 garnets with $$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$7 Tb, Dy, Ho, or Er, umbrella-like order originates from competition between strong rare-earth crystal-field anisotropy and exchange coupling to the iron ferrimagnetic sublattice. The crystal structure is $$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$8. Fe$$\vec{S}_\alpha=S\left[\sin\theta\cos\phi_\alpha,\;\sin\theta\sin\phi_\alpha,\;\cos\theta\right].$$9 ions occupy tetrahedral $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$0 and octahedral $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$1 sites and form a ferrimagnet through antiferromagnetic $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$2, with net Fe moment along a cubic $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$3 axis at low temperature. Rare-earth ions occupy dodecahedral $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$4-sites with local $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$5 symmetry, which produces three mutually orthogonal crystal-field easy axes in the minimal unit. The rare-earth moments therefore tilt away from their local easy axes toward the molecular field generated by the Fe-$$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$6 sublattice along $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$7, yielding a non-collinear umbrella triad while the net magnetization remains along $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$8 (Tomasello et al., 2022).

A minimal Hamiltonian is written as $$\mathbf{m}(r,\theta)=\big(\sin f(r)\cos(n\theta+\gamma),\;\sin f(r)\sin(n\theta+\gamma),\;\cos f(r)\big),$$9, with

$n=1$0

and extended terms

$n=1$1

The microscopic origin of $n=1$2 is the rare-earth crystal-field Hamiltonian

$n=1$3

with rank-2 Stevens terms dominating the uniaxial anisotropy. When the small Fe canting is neglected, the canting angle $n=1$4 of a rare-earth moment relative to $n=1$5 is governed by

$n=1$6

with $n=1$7, and stationarity requires

$n=1$8

This reproduces the correct limits: $n=1$9 as $\gamma=\pi/2$0, and $\gamma=\pi/2$1 as $\gamma=\pi/2$2. The adopted Fe–Fe exchange baseline is $\gamma=\pi/2$3, $\gamma=\pi/2$4, and $\gamma=\pi/2$5, with a representative rare-earth crystal-field gap $\gamma=\pi/2$6.

A complementary planar “flatland umbrella” model reduces the ferrimagnet to one effective Fe block $\gamma=\pi/2$7 and two symmetry-related rare-earth ribs $\gamma=\pi/2$8 and $\gamma=\pi/2$9. Its free-energy density is

$$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$0

and the equilibrium angle satisfies

$$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$1

The compensation condition is

$$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$2

For $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$3, $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$4, $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$5, and $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$6, the model gives $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$7, with $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$8 decreasing near compensation yet remaining finite up to $$\mathbf{m}(r,\theta)=\hat{\theta}\sin f(r)+\hat{z}\cos f(r)$$9 in mean field. Experimentally, the temperature dependence of the Tb canting angle in Tb$[111]$0Fe$[111]$1O$[111]$2 is reproduced without a transition to a collinear arrangement at $[111]$3, and the Tb moment versus temperature is fit by a crystal-field Hamiltonian in an effective molecular field $[111]$4. Finite low-temperature umbrella angles have likewise been reported in Dy$[111]$5Fe$[111]$6O$[111]$7, Ho$[111]$8Fe$[111]$9O$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$00, and Er$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$01Fe$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$02O$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$03 (Mori et al., 2023).

4. Semi-compact merons and skyrmions as Bloch umbrellas

A localized umbrella-like spin texture can be realized analytically by coupling a scalar field $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$04 to the magnetization through an angle $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$05. In cylindrical coordinates the magnetization is taken as

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$06

Because $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$07, this is a Bloch-type texture: the in-plane spin component is tangent to circles rather than radial. The scalar model is

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$08

with field equation

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$09

and energy density

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$10

The radial weight $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$11 is the mechanism that drives semi-compactness: as $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$12 it penalizes departures from selected vacua and suppresses the core deformation without changing the large-$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$13 topological tail (Bazeia et al., 2017).

The topological charge is

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$14

which for the Bloch umbrella ansatz reduces to

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$15

Three analytic cases are constructed. The non-topological compact object has $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$16, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$17, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$18, and

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$19

with compact support $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$20. Its state is linearly unstable because $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$21. The semi-compact umbrella-like meron has $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$22, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$23, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$24, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$25, and

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$26

with solution

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$27

Its energy is $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$28, the mean matter radius is $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$29, and $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$30 decreases with $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$31, so increasing $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$32 sharpens the semi-compact core. The state is stable against radial fluctuations since $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$33. Geometrically, the spins open from $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$34 at the center to a purely in-plane azimuthal swirl at infinity: “core up, canopy in-plane.”

The semi-compact umbrella-like skyrmion has $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$35, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$36, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$37, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$38, and

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$39

In the limit $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$40,

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$41

and the energy density becomes

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$42

The interior is exactly uniform for $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$43, the total energy is $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$44, the mean matter radius is $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$45, and linear stability follows from $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$46. This is the most literal localized umbrella in the set: the texture has a sharp uniform core with $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$47, then traverses an annulus where the in-plane component is maximal, and finally approaches $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$48 in the background.

The construction is intentionally phenomenological. The model is scale-invariant and lacks Dzyaloshinskii–Moriya interaction, so the semi-compact profiles are not derived from a micromagnetic Hamiltonian with exchange, anisotropy, and DMI. Instead, they demonstrate analytically how the inner core of a Bloch-type umbrella can be tailored without altering the far-field meron or skyrmion character.

5. Momentum-space umbrella textures in chiral tellurium

In elemental tellurium, the umbrella-like descriptor refers to a reciprocal-space hedgehog or radial spin texture rather than to a real-space magnetic texture. Trigonal Te is chiral and noncentrosymmetric, with right-handed $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$49 and left-handed $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$50 structures built from helical chains running along the crystallographic $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$51-axis. Near the Brillouin-zone corners $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$52 and $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$53, the highest valence band forms a small dumbbell-shaped Fermi pocket and carries spin polarization parallel to momentum. This differs qualitatively from Rashba and Dresselhaus systems, where spin is tangential to constant-energy contours and typically orthogonal to $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$54. In Te the relation is radial: $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$55 points along $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$56, producing a hedgehog-like or umbrella-like profile around the zone corner (Sakano et al., 2019).

Spin- and angle-resolved photoemission establishes this structure directly. Along a $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$57-space trajectory nearly parallel to $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$58, the highest valence band shows strong $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$59 polarization that reverses sign on crossing $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$60, while $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$61 and $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$62 are nearly zero. Quantitatively, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$63 near the valence-band maximum at $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$64, and $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$65 at the Fermi level, approximately $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$66 below the valence-band maximum in the highest-doped sample. The second-highest valence band has the opposite radial orientation. The chirality dependence is equally direct: the left-handed crystal exhibits an outward radial spin texture, whereas the right-handed crystal exhibits the inverted inward texture under identical momentum-space conditions. Within a fixed chirality, $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$67 and $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$68 carry the same radial sense, consistent with time-reversal symmetry and the $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$69 constraints at the zone corner.

The symmetry-based SOC description is

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$70

with

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$71

near $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$72 or $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$73, where $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$74 indexes crystal chirality. The resulting spin expectation value is radial,

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$75

with inward or outward sense set by chirality and band index. Because $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$76, a charge current shifts the Fermi surface and generates a net spin and magnetization along the current direction, providing a microscopic basis for current-induced spin polarization, current-induced bulk magnetization, and nonreciprocal transport in chiral media. Under hydrostatic pressure, first-principles calculations predict band inversion near $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$77 and a Weyl semimetal phase, so the hedgehog texture also connects naturally to strong Berry-curvature responses in the tuned system.

6. Excitations, chirality, transport, and conceptual scope

Umbrella states are not defined solely by their static spin arrangement; their dynamical spectra carry equally distinctive signatures. In rare-earth iron garnets, linear spin-wave theory is performed after local rotations that align the quantization axes with the non-collinear sublattice moments. In the planar model this yields a quadratic bosonic Hamiltonian in which the symmetry-related rare-earth modes combine into an even mode $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$78 and an odd mode $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$79. Only the even mode hybridizes with the dispersive Fe-like mode, while the odd mode remains a non-bonding flat crystal-field branch. The $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$80 hybridization gap is approximated by

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$81

and the model finds a finite low-temperature gap, collapse of the gap at $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$82, and weak reopening as $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$83. The dynamical chirality of a mode can be expressed through the spin-flip susceptibilities,

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$84

or, in the planar umbrella treatment, through

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$85

Because different sublattices are quantized along different local axes, a single normal mode can carry mixed chirality. The lowest hybridized band changes chirality sign near the avoided crossing, the non-bonding odd rare-earth branch retains negative chirality, and the resulting mode polarization directly enters the spin Seebeck response through

$\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$86

These predictions imply gapped or depolarized acoustic modes, hybridization-controlled chirality reversal, and compensation-point anomalies in spin-current generation (Tomasello et al., 2022, Mori et al., 2023).

The broader comparative picture is that umbrella-like structure does not imply a single stabilization mechanism or even a single physical space. In triangular XXZ magnets, umbrella order is selected or suppressed by the balance between easy-plane anisotropy, magnetic field, and quantum fluctuations. In rare-earth garnets, Dzyaloshinskii–Moriya interactions are not required; non-collinearity is driven by crystal-field anisotropy competing with rare-earth–iron exchange. In the semi-compact Bloch skyrmion models, the inner canopy is engineered by a radial potential $\mathbf{s}(\mathbf{k}) \parallel \mathbf{k}$87 rather than by microscopic DMI, and the results are therefore qualitative with respect to materials length scales. In chiral Te, the umbrella-like object is not a magnetic ground state in real space at all, but a momentum-space SOC texture. A common misconception is therefore to treat “umbrella” as synonymous with a fixed cone state or with DMI-stabilized skyrmions. The comparative evidence indicates instead that the term identifies a family of canting geometries whose microscopic origin may be exchange frustration, crystal-field anisotropy, tailored scalar potentials, or chiral spin–orbit coupling (Marmorini et al., 2015, Bazeia et al., 2017, Sakano et al., 2019).