Surface-Confined Spiral State (SSS)

• SSS is a family of spiral configurations defined either by real-space surface localization or by confinement of spiral propagation vectors in reciprocal space.
• Methodologies include experimental techniques like REXS and neutron scattering alongside theoretical models such as Luttinger–Tisza analysis and micromagnetic simulations to characterize SSS behaviors.
• Implications span advanced nanomagnetism, spiral spin-liquid investigations, and symmetry-breaking in spherical-shell convection, highlighting challenges in understanding exchange interactions and boundary effects.

Surface-Confined Spiral State (SSS) denotes a family of spiral configurations whose defining confinement depends on context. In surface-supported and thin-film magnetism, it refers to a real-space spiral localized to a near-surface layer, an adatom chain, or an interface region; in frustrated magnetism, it refers to a manifold of degenerate spiral propagation vectors confined to a contour or surface in reciprocal space and realized thermally as a spiral spin-liquid; in spherical-shell convection, it maps to a single-arm spiral roll whose dominant morphology is defined on spherical surfaces as a rotating-wave solution (Tung et al., 2011, Andriushin et al., 2024, Baral et al., 13 Jul 2025, Ninomiya et al., 2014).

1. Terminological scope

In the literature considered here, “surface-confined” has two distinct technical meanings. One is spatial localization near a physical boundary: the spiral resides predominantly on a surface, in a thin near-surface layer, or on a deposited chain, while the substrate or bulk remains weakly involved. The other is confinement of allowed spiral wavevectors to a lower-dimensional manifold in reciprocal space: a ring in quasi-two-dimensional systems, a two-dimensional spiral surface in three-dimensional frustrated magnets, or line-confined manifolds in codimension-two analogs (Tung et al., 2011, Andriushin et al., 2024, Gao et al., 2024).

A third usage appears in non-rotating spherical-shell convection, where the single-arm spiral roll state is “surface-confined” because its planform is defined on spherical surfaces and the strongest thermal deviations are concentrated near the spherical boundaries, yielding an SSS-type morphology rather than a reciprocal-space degeneracy (Ninomiya et al., 2014).

Usage	Meaning of confinement	Representative systems
Surface-localized real-space spiral	Near-surface layer, interface, or adatom chain	V/Mn chains on Cu(001), Cu$_2$OSeO$_3$, conical-phase surface spirals
Reciprocal-space spiral manifold	Degenerate $\mathbf q$ confined to ring, surface, or lines	AgCrSe$_2$, MnSc$_2$S$_4$, FeCl$_3$, Cs$_3$Fe$_2$Cl$_9$
Curved-surface spiral roll	Spiral morphology defined on spherical surfaces	Double-spherical-shell Boussinesq convection

A common misconception is to treat these usages as interchangeable. They are not. In AgCrSe$_3$0, MnSc$_3$1S$_3$2, FeCl$_3$3, and Cs$_3$4Fe$_3$5Cl$_3$6, the “surface” is a degeneracy manifold in reciprocal space; in Cu$_3$7OSeO$_3$8, Fe/Cu(001), and supported transition-metal chains, the confinement is a real-space boundary effect (Andriushin et al., 2024, Gao et al., 2016, Gao et al., 2021, Baral et al., 13 Jul 2025).

2. Reciprocal-space SSS and spiral spin-liquids

In frustrated Heisenberg systems, the reciprocal-space form of SSS is defined by a continuous set of degenerate propagation vectors. The general criterion is that the exchange energy has minima on a manifold

$_3$9

or, equivalently in single-$\mathbf q$0 classical descriptions, on a contour or surface of minima of $\mathbf q$1. A coplanar spiral on that manifold has the standard form

$\mathbf q$2

and finite temperature populates the manifold according to

$\mathbf q$3

The spiral spin-liquid is therefore the finite-temperature state that samples the SSS manifold without selecting a single long-range-ordered $\mathbf q$4 (Andriushin et al., 2024).

AgCrSe$\mathbf q$5 realizes this physics on a quasi-two-dimensional triangular lattice. For the $\mathbf q$6 model with ferromagnetic $\mathbf q$7 and antiferromagnetic $\mathbf q$8, the line $\mathbf q$9 yields degeneracy of the incommensurate Phase II and Phase III states for all in-plane orientations of $_2$0 at fixed $_2$1, so the minima of $_2$2 form a continuous ring. Experimentally, at $_2$3 K the magnetic intensity forms a continuous, radially sharp ring in the $_2$4 plane; on cooling from $_2$5 K to $_2$6 K, the ring develops six broad maxima along $_2$7 but no resolution-limited magnetic Bragg peaks appear (Andriushin et al., 2024).

MnSc$_2$8S$_2$9 provides the three-dimensional spiral-surface realization on the diamond lattice. In the frustrated $_2$0 model, the spiral spin-liquid emerges for classical spins when $_2$1, and MnSc$_2$2S$_2$3 realizes the strongly frustrated case $_2$4. Neutron diffuse scattering directly images the spiral surface as a “squared-ring” of intensity in the $_2$5 plane, and residual diffuse weight persists even below the ordering transition, evidencing surviving spiral-surface fluctuations inside the ordered regime (Gao et al., 2016).

FeCl$_2$6 realizes the honeycomb-lattice analog, where the reciprocal-space manifold is a ring rather than a surface. For a bipartite honeycomb lattice, the spiral spin-liquid condition is $_2$7; experimentally, diffuse neutron scattering above $_2$8 shows a continuous ring of intensity centered at $_2$9, with SCGA fits giving $_4$0 meV, $_4$1 meV, $_4$2 meV, and $_4$3. The ring therefore reflects an approximate $_4$4 symmetry in momentum space rather than a physical surface localization (Gao et al., 2021).

Cs$_4$5Fe$_4$6Cl$_4$7 sharpens the distinction by exhibiting a codimension-two analog rather than a canonical SSS. Its AB-stacked triangular bilayers generate an effective honeycomb model with $_4$8 and $_4$9, placing the system at $_3$0 inside the spiral regime. The experimentally realized manifold consists of one-dimensional spiral lines confined to integer-$_3$1 planes, with complementary visibility at $_3$2 and $_3$3 due to sublattice interference,

$_3$4

This suggests that “surface-confined” in the reciprocal-space literature can itself admit higher-codimension generalizations (Gao et al., 2024).

3. Surface-localized spiral states in supported and ultrathin magnets

In surface-supported magnets, SSS denotes a real-space noncollinear texture localized near a boundary or on an adatom structure. For V, Cr, and Mn atomic chains on Cu(001), the spiral is a noncollinear transverse spin spiral whose magnetization resides predominantly on the deposited one-dimensional transition-metal chain and shows negligible propagation into the metallic substrate. Operationally, the induced Cu magnetization is negligible enough that only the chain atoms are retained when extracting exchange parameters from frozen-magnon dispersions (Tung et al., 2011).

The magnetic energetics of these chains are captured by

$_3$5

with $_3$6 Å. The substrate tunes both confinement and instability. V on atop sites exhibits $_3$7, corresponding to $_3$8 Å $_3$9 lattice spacings; Mn on hollow sites has a shallow minimum near $_3$0; Mn on atop sites has a more pronounced spiral at $_3$1, giving $_3$2 Å $_3$3 spacings; Cr remains antiferromagnetic with $_3$4. Because SOC is weak in these $_3$5 chains, the spin moments remain almost unchanged on including SOC, orbital moments and MAEs are small, and the spirals are driven primarily by frustrated exchange rather than Dzyaloshinskii–Moriya interaction (Tung et al., 2011).

Ultrathin Fe/Cu(001) realizes a different surface-confined spiral mechanism. In the 5–11 monolayer window, the film stabilizes a spin spiral while a ferromagnetic top bilayer coexists on the surface. Soft-x-ray momentum-resolved photoemission shows that the instability originates in nested regions confined to out-of-plane Fermi surfaces, whereas the in-plane Fermi surfaces display no discernible nesting. Autocorrelation analysis yields an ordering vector

$_3$6

corresponding to a spiral period $_3$7, and reciprocal-space mapping gives a compressive $_3$8-axis strain of $_3$9. Here the SSS is not frustration-driven but nesting-driven and strain-selected (0911.2300).

A third real-space mechanism appears in chiral magnets in the conical phase. Micromagnetic simulations show that surface spirals emerge adjacent to ferromagnetic–conical interfaces and at facets of skyrmion clusters, with penetration depth of order $_2$0. Their morphology is controlled by the conical phase factor $_2$1 through

$_2$2

For a one-dimensional interface at $_2$3, no surface spiral forms at $_2$4; a single positively charged spiral nucleates at $_2$5; multi-spiral states appear near $_2$6 and $_2$7 with periodicity $_2$8; and a discontinuity near $_2$9 flips the sign of the topological charge. This establishes $_9$0 as a control parameter for SSS polymorphism (Zhao et al., 6 Dec 2025).

4. Doubled-period SSS in Cu$_9$1OSeO$_9$2

Cu$_9$3OSeO$_9$4 hosts a recently identified SSS that is both surface-localized and wavevector-distinct from all previously known incommensurate phases of the material. The state is a metastable, chiral Bloch-type modulation confined to a thin near-surface layer, with real-space pitch $_9$5 nm, exactly twice the $_9$6 nm period of the conventional bulk helix, conical phase, and the fundamental of the field-induced chiral soliton lattice distortions. In reciprocal space, it appears at $_9$7, with a pronounced third harmonic at $_9$8 and a suppressed second harmonic (Baral et al., 13 Jul 2025).

The state was identified by resonant elastic x-ray scattering at the Cu $_9$9 edge. In reflection geometry near the anomalous $_3$00 reflection, the x-ray penetration depth is only $_3$01 nm, and half-order satellites appear only when the magnetic field is decreased just below the first-order conical-to-helical transition and only for field directions near $_3$02. At $_3$03 K, the SSS occurs in the narrow interval $_3$04 mT, disappears both at higher fields and lower fields, and is not seen for $_3$05 K. Its angular window is similarly narrow: the field must be tilted by $_3$06 away from a high-symmetry $_3$07 direction in the sample plane (Baral et al., 13 Jul 2025).

Surface localization is established by probe contrast. Reflection REXS shows the half-order satellites and the third harmonic, whereas SANS shows no corresponding satellites in the bulk and transmission REXS on a lamella also does not reveal the SSS. The same sign and magnitude of circular dichroism for CSL and SSS satellites show that the SSS retains Bloch-type chirality. The Q-space peaks are broader and weaker than CSL/helical satellites, indicating shorter coherence lengths, and scanning REXS further shows heterogeneous domain formation, preferentially near sample edges with typical lateral sizes of $_3$08 (Baral et al., 13 Jul 2025).

The stabilization mechanism differs from both pure DMI selection and demagnetization-only stacked spirals. The relevant free-energy form includes exchange, DMI, anisotropic exchange, cubic anisotropy, Zeeman energy, and an explicit surface term,

$_3$09

with the interpretation that surface axial anisotropy and demagnetization fields renormalize the near-surface balance so that $_3$10. The strong third harmonic and suppressed second harmonic are then consistent with a square-wave-like stripe modulation rather than a fan texture of alternating wall chirality (Baral et al., 13 Jul 2025).

5. Surface-confined spiral roll in spherical-shell convection

In thermal convection between non-rotating concentric spherical boundaries, the SSS concept maps to a hydrodynamic single-arm spiral roll state. The system is a Boussinesq fluid in a shell bounded by $_3$11 and $_3$12, subject to an adverse radial temperature gradient with hot inner boundary and cold outer boundary. In the Chandrasekhar framework used here, gravity scales linearly with radius and the conductive temperature profile is quadratic in $_3$13, so the nondimensional equations are

$_3$14

$_3$15

The observed spiral roll is non-axisymmetric, has $_3$16 symmetry under $_3$17, exhibits chirality, and is dominated by poloidal circulation (Ninomiya et al., 2014).

Its traveling-wave character is explicit:

$_3$18

with constant angular velocity $_3$19. The state therefore rotates even though the spherical boundaries are non-rotating. For $_3$20, $_3$21 is $_3$22; at $_3$23, two chiral solutions coexist with $_3$24 and $_3$25; and at Zhang et al.’s narrow-gap parameters $_3$26, $_3$27, explaining why earlier direct integrations described the state as steady (Ninomiya et al., 2014).

The onset mechanism is also distinctive. Neutral curves $_3$28 are labeled by spherical harmonic degree $_3$29, with $_3$30. At $_3$31, the paper gives $_3$32 and $_3$33. Spiral branches can bifurcate directly from the static conductive state at intersections of adjacent neutral curves, $_3$34, and the bifurcating state is a superposition of adjacent spherical harmonics stabilized by nonlinear interaction. The weakly nonlinear balance implies $_3$35 at birth, so the drift is finite already at onset (Ninomiya et al., 2014).

This hydrodynamic use of SSS is therefore surface-confined in morphology rather than in exchange-wavevector space. A plausible implication is that the article’s broader significance lies in symmetry-breaking without imposed global rotation: curvature and spherical isotropy alone permit an autonomous rotating-wave spiral pattern (Ninomiya et al., 2014).

6. Experimental diagnostics, computational frameworks, and open questions

Across these literatures, SSS is identified by a small set of high-specificity diagnostics. In reciprocal-space SSS/SSL systems, the central observable is the static structure factor

$_3$36

which appears as a ring in AgCrSe$_3$37, a squared-ring in MnSc$_3$38S$_3$39, a $_3$40-centered ring in FeCl$_3$41, and line-confined contours with $_3$42-dependent visibility in Cs$_3$43Fe$_3$44Cl$_3$45. The corresponding theoretical tools are exchange Fourier transforms, Luttinger–Tisza analysis, SCGA, and classical Landau–Lifshitz spin dynamics on large lattices (Andriushin et al., 2024, Gao et al., 2016, Gao et al., 2021, Gao et al., 2024).

For surface-localized real-space spirals, the key diagnostics are probe selectivity and mode decomposition. In supported atomic chains, generalized-Bloch spin-spiral DFT and exchange extraction from $_3$46 resolve frustration-driven minima while negligible induced Cu moments operationally establish confinement to the chain; in ultrathin Fe/Cu(001), energy-dependent SX-ARPES and autocorrelation analysis isolate the out-of-plane nesting vector; in Cu$_3$47OSeO$_3$48, the combination of reflection REXS, transmission REXS, and SANS separates near-surface order from bulk order; and in conical-phase chiral magnets, micromagnetic minimization quantifies surface-spiral size, morphology, and topological charge as functions of $_3$49, thickness, and interface geometry (Tung et al., 2011, 0911.2300, Baral et al., 13 Jul 2025, Zhao et al., 6 Dec 2025).

The hydrodynamic SSS is resolved by pseudo-spectral continuation rather than scattering. Spherical harmonics, modified Chebyshev polynomials, and Newton–Raphson solution of the steady rotating-wave equation in a rotating frame make it possible to solve simultaneously for the fields and the angular velocity $_3$50, thereby distinguishing an extremely slow traveling wave from a genuinely steady state (Ninomiya et al., 2014).

Several open problems recur. In reciprocal-space SSS systems, weak perturbations, order-by-disorder, and field response determine whether a degenerate manifold remains diffuse or selects isolated $_3$51-vectors; Cs$_3$52Fe$_3$53Cl$_3$54 further raises the problem of line-to-surface manifold transitions under altered bilayer locking (Gao et al., 2024). In surface-localized magnets, the exact microscopic route from boundary conditions to spiral pitch remains incomplete in Cu$_3$55OSeO$_3$56, and realistic low-field dipolar and anisotropy effects remain to be incorporated for phase-factor-controlled conical surface spirals (Baral et al., 13 Jul 2025, Zhao et al., 6 Dec 2025). In convection, formal center-manifold or normal-form treatments of the spiral-roll bifurcation remain undeveloped, and the neglected toroidal component away from onset remains an open issue (Ninomiya et al., 2014).

Taken together, these works show that SSS is best understood not as a single phase but as a structural principle: spiral order constrained by geometry, boundary conditions, or degeneracy manifolds. Depending on the system, the confinement may be to a physical surface, a near-surface layer, an adatom chain, a spherical boundary, a reciprocal-space ring, a spiral surface, or even codimension-two lines. That breadth is precisely what makes the term scientifically useful and technically nontrivial.