---
title: Umbrella-like Spin Structures in Magnetism
url: https://www.emergentmind.com/topics/umbrella-like-spin-structure
type: topic
---

# Umbrella-like Spin Structures in Magnetism

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 [1510.07969] [1701.06645] [2207.00017] [2304.10695] [1908.09725].

## 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 \(\mathbf{k}\), not tangentially around a contour [1510.07969] [1701.06645] [2207.00017] [1908.09725].

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
\[
\hat{\mathcal{H}}=
J\sum_{\langle i,j\rangle}\!\left(\hat{S}_i^x\hat{S}_j^x+\hat{S}_i^y\hat{S}_j^y\right)
+J_z\sum_{\langle i,j\rangle}\hat{S}_i^z\hat{S}_j^z
-H\sum_i \hat{S}_i^z
+J^\perp\!\sum_{\langle i,j\rangle_\perp}\!\left(\hat{S}_i^x\hat{S}_j^x+\hat{S}_i^y\hat{S}_j^y\right)
+J_z^\perp\sum_{\langle i,j\rangle_\perp}\hat{S}_i^z\hat{S}_j^z.
\]
The easy-plane regime is \(J>J_z\), or \(\Delta \equiv J_z/J < 1\). 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
\[
E_{\rm umb}(\theta)/N
=3S^2\left[-\frac{J}{2}\sin^2\theta+J_z\cos^2\theta\right]-HS\cos\theta,
\]
and minimization gives
\[
\cos\theta=\frac{H}{H_S^{\rm cl}},\qquad
H_S^{\rm cl}=6S\left(J_z+\frac{J}{2}\right).
\]
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 \(\pm \mathbf{Q}\). The effective condensate energy density is
\[
\frac{E_0}{M}=-\mu\left(|\psi_{\mathbf Q}|^2+|\psi_{-\mathbf Q}|^2\right)
+\frac{\Gamma_1}{2}\left(|\psi_{\mathbf Q}|^4+|\psi_{-\mathbf Q}|^4\right)
+\Gamma_2|\psi_{\mathbf Q}|^2|\psi_{-\mathbf Q}|^2,
\]
with \(\mu=H_S-H\). The phase criterion is explicit: \(\Gamma_1<\Gamma_2\) selects the umbrella state as a single-mode BEC, whereas \(\Gamma_1>\Gamma_2\) 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 \(J/J_z=1\) to \((J/J_z)_c=2.218\) for \(S=1/2\), \(1.554\) for \(S=1\), \(1.361\) for \(S=3/2\), \(1.269\) for \(S=2\), \(1.214\) for \(S=5/2\), \(1.178\) for \(S=3\), and \(1.152\) for \(S=7/2\). The monotonic approach to the classical boundary quantifies how increasing \(S\) reduces fluctuation corrections. In layered systems, antiferromagnetic interlayer coupling shrinks the coplanar pocket and can restore umbrella order; for \(S=1/2\) with \(J=J_z\), an interlayer XY coupling of order \(J^\perp/J \simeq 0.209\) is sufficient to favor the stacked umbrella over the stacked \(0\)-coplanar state. These results are directly relevant to Ba\(_3\)CoSb\(_2\)O\(_9\), 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 [1510.07969].

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

In \(R_3\)Fe\(_5\)O\(_{12}\) garnets with \(R=\) 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 \(Ia\bar{3}d\). Fe\(^{3+}\) ions occupy tetrahedral \(d\) and octahedral \(a\) sites and form a ferrimagnet through antiferromagnetic \(J_{ad}\), with net Fe moment along a cubic \([111]\) axis at low temperature. Rare-earth ions occupy dodecahedral \(c\)-sites with local \(D_2\) 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-\(d\) sublattice along \([111]\), yielding a non-collinear umbrella triad while the net magnetization remains along \([111]\) [2207.00017].

A minimal Hamiltonian is written as \(\mathcal{H}=\mathcal{H}^0+\mathcal{H}^1\), with
\[
\mathcal{H}^{0}=- K_{e} \sum_{i} \left(S_{i}^{\alpha_{i}}\right)^{2}
+ J_{cd} \sum_{\langle i j\rangle} \mathbf{S}_{i}\cdot \mathbf{s}^{d}_{j}
+ J_{ad} \sum_{\langle j k\rangle} \mathbf{s}^{d}_{j}\cdot \mathbf{s}^{a}_{k},
\]
and extended terms
\[
\mathcal{H}^{1}=K_{h} \sum_{i} \left(S_{i}^{\beta_{i}}\right)^{2}
+ J_{dd} \sum_{\langle j j' \rangle} \mathbf{s}^{d}_{j}\cdot \mathbf{s}^{d}_{j'}
+ J_{aa} \sum_{\langle k k' \rangle} \mathbf{s}^{a}_{k}\cdot \mathbf{s}^{a}_{k'}.
\]
The microscopic origin of \(K_e\) is the rare-earth crystal-field Hamiltonian
\[
\mathcal{H}_{\mathrm{CF}}=\sum_{l,m} B_l^m O_l^m(\hat{\mathbf{J}}),
\]
with rank-2 Stevens terms dominating the uniaxial anisotropy. When the small Fe canting is neglected, the canting angle \(\theta\) of a rare-earth moment relative to \([111]\) is governed by
\[
E(\theta)=-K_e S^2 \cos^2(\alpha_o-\theta)-J_{\mathrm{eff}}\cos\theta,
\qquad
J_{\mathrm{eff}}\equiv |J_{cd}|\,S\,s_d,
\]
with \(\alpha_o=\arccos(1/\sqrt{3})\approx 54.7356^\circ\), and stationarity requires
\[
K_e S^2\sin\big(2(\alpha_o-\theta)\big)=J_{\mathrm{eff}}\sin\theta.
\]
This reproduces the correct limits: \(\theta\to\alpha_o\) as \(J_{cd}\to 0\), and \(\theta\to 0\) as \(J_{cd}\to\infty\). The adopted Fe–Fe exchange baseline is \(J_{ad}\simeq 6.8\ {\rm meV}\), \(J_{dd}\simeq 0.52\ {\rm meV}\), and \(J_{aa}\simeq 1.1\ {\rm meV}\), with a representative rare-earth crystal-field gap \(\Delta_{\rm CF}\approx 4.2\ {\rm meV}\).

A complementary planar “flatland umbrella” model reduces the ferrimagnet to one effective Fe block \(A\) and two symmetry-related rare-earth ribs \(B\) and \(C\). Its free-energy density is
\[
F(\phi)= -M_A^2 + 2 j M_A M_B \cos\phi - 2 d M_B^2(\cos\phi-\sin\phi)^2,
\]
and the equilibrium angle satisfies
\[
-j M_A(T)\sin\phi(T)+2 d M_B(T)\cos 2\phi(T)=0.
\]
The compensation condition is
\[
M_A(T_{\rm comp})=2M_B(T_{\rm comp})\cos\phi(T_{\rm comp}).
\]
For \(j=0.05\), \(d=0.025\), \(S_A=1\), and \(S_B=S_C=2\), the model gives \(T_{\rm comp}/T_N\approx 0.25\), with \(\phi(T)\) decreasing near compensation yet remaining finite up to \(T_N\) in mean field. Experimentally, the temperature dependence of the Tb canting angle in Tb\(_3\)Fe\(_5\)O\(_{12}\) is reproduced without a transition to a collinear arrangement at \(150\ {\rm K}\), and the Tb moment versus temperature is fit by a crystal-field Hamiltonian in an effective molecular field \(B_{111}\approx 13\ {\rm T}\). Finite low-temperature umbrella angles have likewise been reported in Dy\(_3\)Fe\(_5\)O\(_{12}\), Ho\(_3\)Fe\(_5\)O\(_{12}\), and Er\(_3\)Fe\(_5\)O\(_{12}\) [2304.10695].

## 4. Semi-compact merons and skyrmions as Bloch umbrellas

A localized umbrella-like spin texture can be realized analytically by coupling a scalar field \(\phi(r)\) to the magnetization through an angle \(\Theta(r)\). In cylindrical coordinates the magnetization is taken as
\[
\mathbf{M}(r)=\hat{\theta}\cos\Theta(r)+\hat{z}\sin\Theta(r),\qquad
\Theta(r)=\frac{\pi}{2}\phi(r)+\delta.
\]
Because \(\mathbf{M}\cdot\hat{r}=0\), this is a Bloch-type texture: the in-plane spin component is tangent to circles rather than radial. The scalar model is
\[
\mathcal{L}=\frac{1}{2}\dot{\phi}^{\,2}-\frac{1}{2}\nabla\phi\cdot\nabla\phi-U(r,\phi),\qquad
U(r,\phi)=\frac{1}{2r^2}P(\phi),
\]
with field equation
\[
r^2\phi''+r\phi'-\frac{1}{2}P'(\phi)=0,
\]
and energy density
\[
\varepsilon(r)=\frac{1}{2}(\phi')^2+\frac{1}{2r^2}P(\phi).
\]
The radial weight \(P(\phi)/r^2\) is the mechanism that drives semi-compactness: as \(r\to 0\) it penalizes departures from selected vacua and suppresses the core deformation without changing the large-\(r\) topological tail [1701.06645].

The topological charge is
\[
Q=\frac{1}{4\pi}\int d^2r\,\mathbf{m}\cdot(\partial_x\mathbf{m}\times\partial_y\mathbf{m}),
\]
which for the Bloch umbrella ansatz reduces to
\[
Q=\frac{1}{2}\big[\cos f(0)-\cos f(\infty)\big]
=\frac{1}{2}\sin\Theta(0)-\frac{1}{2}\sin\Theta(\infty).
\]
Three analytic cases are constructed. The non-topological compact object has \(Q=0\), \(\delta=\pi/2\), \(\phi(0)=\phi(\infty)=0\), and
\[
P(\phi)=n^2\phi^2(\phi^{-2/n}-1),\qquad n=3,5,7,\ldots
\]
with compact support \(e^{-\pi/2}\le r\le e^{\pi/2}\). Its state is linearly unstable because \(E_4<0\). The semi-compact umbrella-like meron has \(Q=1/2\), \(\delta=\pi/2\), \(\phi(0)=0\), \(\phi(\infty)=1\), and
\[
P(\phi)=\frac{1}{(1-s)^2}\phi^2\ln^2(\phi^2),\qquad s\in[0,1),
\]
with solution
\[
\phi_s(r)=\exp\!\big(-r^{-2/(1-s)}\big).
\]
Its energy is \(E_s=\pi/[2(1-s)]\), the mean matter radius is \(\bar r_s = 2^{(1-s)/2}\Gamma((3+s)/2)\), and \(\bar r_s\) decreases with \(s\), so increasing \(s\) sharpens the semi-compact core. The state is stable against radial fluctuations since \(E_4=3\pi/[16(1-s)]>0\). Geometrically, the spins open from \(+z\) at the center to a purely in-plane azimuthal swirl at infinity: “core up, canopy in-plane.”

The semi-compact umbrella-like skyrmion has \(Q=1\), \(\delta=0\), \(\phi(0)=+1\), \(\phi(\infty)=-1\), and
\[
P(\phi)=\frac{1}{(1-s)^2}(1+\phi)^2(1-\phi^n)^2,\qquad n=1,3,5,\ldots
\]
In the limit \(n\to\infty\),
\[
\phi_s(r)=
\begin{cases}
1,& 0<r\le 1,\\[2mm]
2r^{-1/(1-s)}-1,& r>1,
\end{cases}
\]
and the energy density becomes
\[
\varepsilon_s(r)=
\begin{cases}
0,& 0<r\le 1,\\[2mm]
\frac{4}{(1-s)^2}r^{-2(2-s)/(1-s)},& r>1.
\end{cases}
\]
The interior is exactly uniform for \(r\le 1\), the total energy is \(E_s=4\pi/(1-s)\), the mean matter radius is \(\bar r_s=2/(1+s)\), and linear stability follows from \(E_2=\pi/(1-s)>0\). This is the most literal localized umbrella in the set: the texture has a sharp uniform core with \(m_z=+1\), then traverses an annulus where the in-plane component is maximal, and finally approaches \(m_z=-1\) 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 \(P3_121\) and left-handed \(P3_221\) structures built from helical chains running along the crystallographic \(z\)-axis. Near the Brillouin-zone corners \(H\) and \(H'\), 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 \(\mathbf{k}\). In Te the relation is radial: \(\mathbf{s}(\mathbf{k})\) points along \(\pm\mathbf{k}\), producing a hedgehog-like or umbrella-like profile around the zone corner [1908.09725].

Spin- and angle-resolved photoemission establishes this structure directly. Along a \(k\)-space trajectory nearly parallel to \(k_z\), the highest valence band shows strong \(S_z\) polarization that reverses sign on crossing \(H\), while \(S_x\) and \(S_y\) are nearly zero. Quantitatively, \(|P_z|\approx 55\%\) near the valence-band maximum at \(|k-H|\approx 0.02\ {\rm \AA}^{-1}\), and \(|P_z|\approx 90\%\) at the Fermi level, approximately \(20\ {\rm meV}\) 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, \(H\) and \(H'\) carry the same radial sense, consistent with time-reversal symmetry and the \(D_3\) constraints at the zone corner.

The symmetry-based SOC description is
\[
H_{\rm SOC}=\lambda\,\mathbf{s}\cdot\mathbf{g}(\mathbf{k}),
\]
with
\[
\mathbf{g}_\chi(\mathbf{k})=\chi\,\mathbf{k}
\]
near \(H\) or \(H'\), where \(\chi=\pm 1\) indexes crystal chirality. The resulting spin expectation value is radial,
\[
\mathbf{s}(\mathbf{k})=s_0\hat{\mathbf{k}},
\]
with inward or outward sense set by chirality and band index. Because \(\mathbf{s}(\mathbf{k})\parallel\mathbf{k}\), 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 \(H/H'\) 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 \(f_k=(b_k+c_k)/\sqrt{2}\) and an odd mode \(g_k=(b_k-c_k)/\sqrt{2}\). Only the even mode hybridizes with the dispersive Fe-like mode, while the odd mode remains a non-bonding flat crystal-field branch. The \(k=0\) hybridization gap is approximated by
\[
\Delta_0(T)\simeq 2\sqrt{2}\,j\cos^2\!\left[\frac{\phi(T)}{2}\right]\sqrt{M_A(T)M_B(T)},
\]
and the model finds a finite low-temperature gap, collapse of the gap at \(T_{\rm comp}\), and weak reopening as \(T\to T_N\). The dynamical chirality of a mode can be expressed through the spin-flip susceptibilities,
\[
\mathcal{X}(\mathbf{q},\omega)=
\frac{\chi^{+-}(\mathbf{q},\omega)-\chi^{-+}(\mathbf{q},\omega)}
{\chi^{+-}(\mathbf{q},\omega)+\chi^{-+}(\mathbf{q},\omega)},
\]
or, in the planar umbrella treatment, through
\[
\chi_k(\omega)=\frac{1}{2}\sum_l
\langle S_{l,k}^+(\omega)S_{l,-k}^-(-\omega)-S_{l,-k}^-(-\omega)S_{l,k}^+(\omega)\rangle.
\]
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
\[
j_s \propto - \sum_n \int d^3k\,(2\pi)^{-3}\,s_n(k)\,v_n(k)\,\frac{\partial f}{\partial \omega}\cdot\nabla T.
\]
These predictions imply gapped or depolarized acoustic modes, hybridization-controlled chirality reversal, and compensation-point anomalies in spin-current generation [2207.00017] [2304.10695].

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 \(U\propto P(\phi)/r^2\) 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 [1510.07969] [1701.06645] [1908.09725].

Source: https://www.emergentmind.com/topics/umbrella-like-spin-structure