Spiral Spin Liquid (SSL) Overview

• Spiral spin liquids are frustrated magnetic systems characterized by a continuous manifold of degenerate spin spiral states in reciprocal space, leading to distinctive diffuse scattering signatures.
• They are analyzed using classical Heisenberg models and Luttinger–Tisza formalism, revealing a subextensive degeneracy controlled by exchange ratios and lattice geometry.
• Experimental realizations on honeycomb, diamond, and triangular lattices showcase thermal, transport, and scattering signatures that illuminate defect dynamics and emergent gauge fields.

{"command":["bash","-lc","python - <<'PY'\nimport urllib.parse, urllib.request\nquery='all:\"spiral spin liquid\"'\nurl='http://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=5'\nprint(url)\nprint(urllib.request.urlopen(url,timeout=20).read().decode('utf-8')[:2000])\nPY"]} {"command":["bash","-lc","python - <<'PY'\nimport urllib.parse, urllib.request\nquery='all:\"spiral spin liquid\"'\nurl='https://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=10'\nprint(urllib.request.urlopen(url,timeout=20).read().decode('utf-8')[:5000])\nPY"],"timeout_ms":40000} {"command":["bash","-lc","python - <<'PY'\nimport urllib.parse, urllib.request\nquery='all:\"spiral spin liquid\"'\nurl='https://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=10'\ntry:\n data=urllib.request.urlopen(url,timeout=20).read().decode('utf-8')\n print(data[:5000])\nexcept Exception as e:\n print('ERR',repr(e))\nPY"],"timeout_ms":40000} {"command":["bash","-lc","python - <<'PY'\nprint('hi')\nPY"]} {"command":["bash","-lc","echo hello"]} Spiral spin liquid (SSL) denotes a correlated paramagnetic regime of a frustrated magnet in which the low-energy states are coplanar spin spirals whose propagation vectors do not collapse to isolated ordering wavevectors, but instead form a continuous contour in two dimensions or a continuous surface in three dimensions. The defining feature is therefore a subextensively degenerate manifold in reciprocal space: the system fluctuates cooperatively among many nearly or exactly degenerate spirals, producing diffuse scattering concentrated on a ring, line, or surface rather than conventional Bragg peaks. Across the recent literature, SSLs are treated as a distinctive form of classical spin liquid governed by exchange frustration, emergent momentum-space $U(1)$ structure, and unusual defect physics, with representative realizations now reported on honeycomb, diamond, triangular, stacked-triangular, and metallic platforms (Yao et al., 2020, Gao et al., 2021, Zhao et al., 26 Aug 2025).

1. Reciprocal-space definition and degeneracy structure

A standard starting point is the classical Heisenberg model with competing exchanges,

$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$

or straightforward lattice-specific extensions. In Luttinger–Tisza form, the Fourier-space interaction matrix acquires bands whose lower branch can be minimized along a continuous locus in momentum space rather than at isolated $\mathbf q$. Generic analyses on bipartite lattices identify the condition

$\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$

with $z$ the coordination number, as the threshold beyond which the minimum of the lower band expands into a spiral contour or spiral surface. On the honeycomb lattice this gives $1/6$; on the diamond lattice, $1/8$ (Yao et al., 2020).

This reciprocal-space degeneracy is subextensive rather than extensive. In the two-dimensional square-lattice SSL analyzed via a local-momentum formulation, the number of discrete Bragg-like states scales as $O(L)$ rather than $O(L^2)$, because the minimizing states lie on a one-dimensional ring of wavevectors. This distinction separates SSLs from classical spin liquids built from locally flippable real-space constraints, such as Coulomb-phase systems with extensive degeneracy (Yan et al., 2021).

A recurrent clarification in the literature is that SSL is not usually presented as a separate thermodynamic phase in the same sense as a symmetry-broken ordered state. Rather, it is a cooperative paramagnetic regime controlled by a continuous manifold of soft spiral modes. In three-dimensional materials, weak perturbations or fluctuation effects often select discrete ordering vectors at low temperature, leaving an approximate SSL in an intermediate window above the ordering transition. In strictly two-dimensional continuous-spin models, by contrast, the contour itself can govern the low-temperature regime without conventional long-range order (Yao et al., 2020, Iqbal et al., 2018).

2. Microscopic constructions on honeycomb and related lattices

The honeycomb lattice has become a central SSL setting because the $J_1$–$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$0 model admits both exact and material-specific constructions. For antiferromagnetic couplings $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$1, the classical honeycomb model becomes frustrated at $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$2, and the minimizing wavevectors $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$3 form a one-dimensional contour given implicitly by

$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$4

with $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$5. As $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$6 increases, the contour evolves from a ring to a hexagon and then to pockets for $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$7 (Cônsoli et al., 2023).

A material-specific honeycomb realization is GdZnPO, described by the easy-plane Hamiltonian

$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$8

Fits yield $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$9, $\mathbf q$0, $\mathbf q$1, and $\mathbf q$2. The corresponding ground-state manifold is a contour around the honeycomb $\mathbf q$3 point, determined by

$\mathbf q$4

with $\mathbf q$5, well inside the classical SSL regime (Wan et al., 16 Mar 2025).

The same organizing principle extends beyond bipartite $\mathbf q$6–$\mathbf q$7 models. A line-graph construction maps honeycomb $\mathbf q$8–$\mathbf q$9 physics onto kagome $\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$0–$\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$1, and diamond $\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$2–$\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$3 onto pyrochlore $\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$4–$\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$5, broadening the candidate pool to non-bipartite lattices with approximate rather than exact degeneracy (Gao et al., 2022). On the triangular lattice, an ideal SSL occurs on the line $\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$6 of the $\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$7–$\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$8–$\left(\frac{J_2}{J_1}\right)_c=\frac{1}{2z},$9 model, where $z$0 has a continuous circle of minima. AgCrSe$z$1 was analyzed as a near-realization of this condition, with $z$2, sufficiently close that thermal fluctuations flatten the residual anisotropy and produce a broad SSL regime (Andriushin et al., 2024).

A distinct route is provided by stacking frustration. In an ABC-stacked multilayer triangular antiferromagnet with intralayer $z$3 and interlayer $z$4, the zero-temperature minima satisfy

$z$5

together with a phase condition fixing $z$6. The result is a one-dimensional manifold of spiral lines in three-dimensional reciprocal space, which again separates low-temperature spiral order from a high-temperature paramagnet through an intermediate SSL regime (Liu et al., 2022).

3. Collective fluctuations, momentum vortices, and emergent gauge structure

A central conceptual problem for SSLs is that moving along the degenerate spiral manifold generally requires a collective rearrangement of the entire spin configuration. The local description introduced for the square-lattice XY SSL defines a coarse-grained phase $z$7 and local momentum field

$z$8

In the absence of ordinary spin vortices, $z$9, yet the direction of $1/6$0 can still wind around defects. These objects are momentum vortices, with winding number

$1/6$1

where $1/6$2 (Yan et al., 2021).

The low-energy continuum theory of these modes admits a rank-2 $1/6$3 gauge interpretation. Writing the Hessian $1/6$4 and defining a symmetric tensor $1/6$5, the curl-free constraint becomes

$1/6$6

which is the Gauss law of a scalar-charged rank-2 gauge theory. In this formulation, momentum vortices are equivalent to quadrupoles of fractons; equivalently, in the elasticity dual, they are quadrupoles of disclinations (Yan et al., 2021).

A direct diagnostic of this gauge structure is the fourfold pinch-point singularity in a generalized correlator. For the square-lattice model, the tensor correlator obeys

$1/6$7

and Monte Carlo plus molecular-dynamics studies located the sharpest fourfold pinch points in the SSL regime itself. The same work embedded the SSL into a broader phase diagram containing a four-state Potts transition into a colinear double-striped phase, re-entrance into a nematic spiral, and a regime in which the transition into the SSL breaks no symmetry and exhibits nonstandard critical behavior. The rapid reduction of momentum-vortex density at the SSL boundary was argued to be consistent with a Kosterlitz–Thouless-like scenario for momentum vortices (Gonzalez et al., 2024).

Honeycomb SSL theory has developed a related topological vocabulary. In GdZnPO, classical Monte Carlo on six triangular sublattices $1/6$8 found in-plane spin vortices and antivortices bound to momentum vortices in the sublattice-resolved structure factor, as anticipated by fracton-gauge arguments. The paper attributes their low energetic cost to the softness of the spiral contour itself, which allows such defects to proliferate to very low temperature (Wan et al., 16 Mar 2025). This suggests that topological defects in SSLs are not secondary perturbations but part of the organizing low-energy structure.

4. Experimental signatures and representative realizations

The most direct SSL signature is diffuse magnetic scattering concentrated on a continuous manifold in reciprocal space. In MnSc$1/6$9S$1/8$0, neutron diffuse scattering revealed the spiral surface directly on the diamond lattice, providing the canonical three-dimensional example (Gao et al., 2016). In FeCl$1/8$1, neutron scattering above $1/8$2 showed a continuous ring of diffuse intensity in the $1/8$3 plane, centered on $1/8$4, demonstrating an approximate momentum-space $1/8$5 symmetry in a van der Waals honeycomb magnet (Gao et al., 2021). In LiYbO$1/8$6, reconstructed single-crystal diffuse maps from powder data displayed continuous spiral contours on an elongated diamond lattice (Graham et al., 2023). In AgCrSe$1/8$7, combined wide-angle and small-angle neutron scattering showed a nearly isotropic ring at $1/8$8 K, with sixfold modulation developing only on cooling (Andriushin et al., 2024).

Thermodynamic and transport measurements have become increasingly important because they probe the mobility and topology of low-lying SSL excitations. In GdZnPO, the magnetic heat capacity below about $1/8$9 K follows

$O(L)$0

with $O(L)$1 and $O(L)$2 for $O(L)$3 T, while the magnetic thermal conductivity below 1 K is extracted from

$O(L)$4

with $O(L)$5 and $O(L)$6 at zero field in the highest-quality crystal. The interpretation advanced in the transport study is that $O(L)$7 arises from gapless modes propagating along the spiral ring, whereas $O(L)$8 reflects low-energy excitations normal to the contour (Wan et al., 16 Mar 2025, Zhao et al., 26 Aug 2025).

The same GdZnPO work reports a positive magnetic thermal Hall effect. Using the bosonic-band expression

$O(L)$9

the observed $O(L^2)$0 is interpreted in terms of Berry curvature in topological magnon bands; a two-flat-band fit yields a lower-band Chern number $O(L^2)$1. Proposed microscopic sources include weak Dzyaloshinskii–Moriya terms, quantum-fluctuation-induced ring lifting, and topological spin or momentum vortices threading fictitious flux (Zhao et al., 26 Aug 2025).

A complementary diagnostic is spontaneous spin noise. In Ca$O(L^2)$2Cr$O(L^2)$3O$O(L^2)$4, the magnetization-noise power spectral density obeys $O(L^2)$5 with $O(L^2)$6, while the variance $O(L^2)$7 and correlation function $O(L^2)$8 both cross over near $O(L^2)$9 mK. Monte Carlo of a two-dimensional SSL reproduced the measured frequency and temperature dependence of $J_1$0, $J_1$1, and $J_1$2, and the paper argued that this phenomenology is inconsistent with the then-competing quantum-spin-liquid interpretations (Takahashi et al., 2024).

Material	Lattice/model setting	Principal SSL signature
MnSc$J_1$3S$J_1$4	Diamond $J_1$5–$J_1$6	Spiral surface in neutron diffuse scattering (Gao et al., 2016)
FeCl$J_1$7	Honeycomb $J_1$8–$J_1$9	Continuous ring of scattering above $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$00 (Gao et al., 2021)
LiYbO$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$01	Elongated diamond $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$02–$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$03	Reconstructed diffuse spiral contours (Graham et al., 2023)
AgCrSe$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$04	Triangular $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$05–$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$06–$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$07	Isotropic ring at 40 K, sixfold modulation on cooling (Andriushin et al., 2024)
GdZnPO	Easy-plane honeycomb SSL candidate	$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$08, $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$09, positive $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$10 (Zhao et al., 26 Aug 2025)

5. Order-by-disorder, disorder response, and spin-glass formation

SSL manifolds are generically sensitive to weak perturbations that lift the degeneracy only partially. Thermal order-by-disorder selects particular wavevectors by comparing Gaussian fluctuation free energies around candidate spirals. On the honeycomb lattice this selection depends on $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$11: for $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$12, thermal fluctuations select $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$13, whereas for $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$14 they select $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$15. In three-dimensional MnSc$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$16S$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$17, sizeable $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$18 induces strong spiral selection near the ordering temperature, but the modulation around the spiral surface becomes much weaker on heating, yielding an approximate SSL over roughly $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$19 (Cônsoli et al., 2023, Iqbal et al., 2018).

Quenched disorder produces a different mechanism, order-by-quenched-disorder (ObQD), and can drive the system into qualitatively new states. In the honeycomb $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$20–$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$21 model, a weak defect generates long-ranged textures with Friedel-like oscillations,

$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$22

whose period directly encodes the geometry of the spiral contour. In two dimensions with a one-dimensional contour, the asymptotic decay is $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$23, much slower than the $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$24 decay of ordinary electronic Friedel oscillations. At finite defect density, incompatible local selections of $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$25 destroy long-range spiral order and produce a spiral spin glass with diffuse scattering still concentrated on the original contour (Cônsoli et al., 2023).

The same work argues that, in two dimensions, random bonds act as random fields in a low-temperature Potts-like description of the selected $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$26 axes, so that any defect density $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$27 destroys discrete long-range order by the Imry–Ma mechanism. Vacancies and randomly oriented bond defects therefore extend the SSL-like fluctuation regime to lower temperature while simultaneously promoting glassiness (Cônsoli et al., 2023).

Finite spin quantization can also weakly lift the manifold. In GdZnPO, $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$28P NMR down to $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$29 mK found three regimes below 1 K: a weak stripe-ordered phase for $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$30 T and $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$31 K; an SSL regime for $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$32 T $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$33 T with spatially uniform magnetization and persistent low-energy dynamics; and a polarized ferromagnet above $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$34 T. The spin-lattice relaxation rate follows $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$35 with $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$36 in the liquid regime, while the ordered moment in the stripe phase remains far below the ideal stripe value. The interpretation is that $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$37 quantization produces a weak order-by-quantum-disorder correction without destroying the underlying SSL phenomenology (Chen et al., 22 Jun 2026).

6. Broader significance and current directions

One strand of current work treats classical SSLs as precursors of more strongly quantum nonmagnetic states. PFFRG studies on square and hcp models deliberately constructed to host classical spiral manifolds found that substantial parts of the classical SSL regime become paramagnetic for $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$38, with the momentum-resolved susceptibility retaining clear remnants of the classical spiral contours or surfaces. The authors therefore argued that classical SSLs can provide a route toward quantum spin-liquid behavior, although the exact nature of the quantum phases remains model dependent (Niggemann et al., 2019).

Another strand concerns electronic mediation of SSL physics. In EuAg$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$39Sb$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$40, diffuse magnetic neutron scattering above $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$41 K revealed a ring of fluctuating spin modulations at $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$42, matching $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$43 of a quasi-two-dimensional hole pocket. The resulting picture is a Fermi-surface-driven approximate SSL whose near-$$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$44-symmetric manifold emerges from conduction-electron-mediated interactions rather than purely local superexchange. This extends SSL design beyond insulating frustrated magnets to metallic systems with intertwined magnetic and electronic structure (Neves et al., 3 Mar 2026).

The experimental frontier is correspondingly broad. For GdZnPO, proposed next steps include inelastic neutron or resonant X-ray scattering to image the ring-shaped low-energy minima, field- and angle-dependent thermal Hall and spin Seebeck measurements to probe chiral magnon transport, and chemical or strain tuning of $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$45 or $$H = J_1 \sum_{\langle i,j\rangle}\mathbf S_i\!\cdot\!\mathbf S_j + J_2 \sum_{\langle\!\langle i,j\rangle\!\rangle}\mathbf S_i\!\cdot\!\mathbf S_j,$$46 to deform the contour and explore adjacent ordered states (Wan et al., 16 Mar 2025, Zhao et al., 26 Aug 2025). For triangular and stacked systems, the observed sensitivity to thermal flattening of small anisotropies suggests that controllable tuning across the border between exact, approximate, and selected manifolds will remain central (Andriushin et al., 2024, Liu et al., 2022).

Several misconceptions are explicitly corrected by the recent literature. SSL behavior does not require geometric frustration in the narrow sense; it can arise on bipartite lattices from competing exchanges and can even be engineered through line-graph mappings or stacking frustration (Yao et al., 2020, Gao et al., 2022). Nor is an SSL defined solely by diffuse elastic scattering: transport, noise, NMR, and Hall measurements now probe whether the manifold supports mobile, topological, or long-lived low-energy excitations (Takahashi et al., 2024, Zhao et al., 26 Aug 2025, Chen et al., 22 Jun 2026). A plausible implication is that the field is shifting from identification of reciprocal-space manifolds to a more dynamical program centered on itinerancy, topology, and defect-mediated collective behavior.