Spiral Spin Liquid

• Spiral spin liquid is a frustrated magnetic state characterized by a continuous manifold of nearly degenerate spiral configurations in reciprocal space due to competing interactions.
• It exhibits unique thermodynamic and spectroscopic signatures, such as ring-shaped neutron scattering and anomalous specific heat from low-energy modes.
• Emergent momentum vortices and gauge structures in spiral spin liquids open pathways for exploring topological excitations and novel spin transport applications.

A spiral spin liquid is a class of frustrated magnetic state characterized by a continuous manifold of nearly degenerate spiral spin configurations in reciprocal space. In this regime, the magnetic system fails to select a unique ordering wave vector, instead displaying cooperative fluctuations across an extended ring or surface of propagation vectors. This leads to distinctive thermodynamic, spectroscopic, and dynamic properties, as well as providing a natural platform for exotic topological excitations and emergent gauge structures.

1. Definition and General Properties

The spiral spin liquid (SSL) is defined by a ground-state manifold consisting of a continuous set of spiral (helical or coplanar) spin configurations whose propagation (ordering) wave vectors $\mathbf{q}$ form a closed contour (“spiral contour”) or surface in reciprocal space. In the classical limit ($S\to\infty$), this degeneracy arises from competing interactions (such as ferromagnetic $J_1$ and antiferromagnetic $J_2$, $J_3$ couplings) in extended Heisenberg models on various lattices:

$$H = \sum_{\langle ij\rangle} J_1\, \mathbf{S}_i\cdot \mathbf{S}_j + \sum_{\langle\langle ij\rangle\rangle} J_2\, \mathbf{S}_i\cdot \mathbf{S}_j + \sum_{\langle\langle\langle ij\rangle\rangle\rangle} J_3\, \mathbf{S}_i\cdot \mathbf{S}_j + \ldots$$

Spiral spin liquids are “classical spin liquids” insofar as their extensive (or subextensive) degeneracy is not associated with fractionalization or topological order in the strict $\mathbb{Z}_2$/U(1) quantum sense, but with emergent cooperative paramagnetic behavior driven by frustration and entropic effects (Yao et al., 2020).

Key features include:

• Magnetic correlations exhibit sharp, ring- or surface-shaped maxima in the static spin structure factor $S(\mathbf{q})$;
• No unique ordering wave vector is selected in the absence of further perturbations (“order-by-disorder” or quenched disorder);
• Entropic and quantum fluctuations can partially or fully lift the degeneracy, selecting spiral orders or quantum paramagnetic states;
• The degenerate manifold can be classified by its dimension $d_s$ and codimension $d_c$ (spatial dimension minus $d_s$) (Yao et al., 2020);
• The ground-state entropy and finite-temperature specific heat acquire anomalous forms; e.g., $C_v = C_0 + C_1T$ for 1D manifolds in 2D systems, reflecting zero-energy “Goldstone” modes along the contour.

2. Microscopic Origins and Classification

The spiral spin liquid emerges typcially in lattices (square, honeycomb, triangular, diamond, etc.) with competing further-neighbor exchanges such that the spin-wave (Luttinger–Tisza) analysis yields a continuous set of minimizing $\mathbf{q}$. For a general bipartite lattice, the Luttinger–Tisza condition is:

$$f(\mathbf{q}) = p \left(\frac{J_1}{4J_2}\right)^2 - \frac{z}{p}$$

where $f(\mathbf{q})$ is determined by lattice geometry and $p,z$ are structure-dependent constants (Niggemann et al., 2019).

Spiral manifolds are classified by:

• $d_s$ (dimension of manifold): 1D contours in 2D (e.g., square/honeycomb $J_1$–$J_2$) or in 3D (e.g., ABC-stacked systems);
• $d_c = d - d_s$ (codimension): governs singularity of $S(\mathbf{q})$ and thermodynamic properties (Yao et al., 2020, Liu et al., 2022).

Variants include “codimension two” spiral spin liquids, where a 1D degenerate line is hosted within a 3D reciprocal space (e.g., Cs$_3$Fe$_2$Cl$_9$) (Gao et al., 29 May 2024).

Quantum spiral spin liquids—or spiral quantum spin liquids—emerge when quantum fluctuations melt long-range spiral order, favoring extended paramagnetic states that preserve the “spiral” character of correlations (Niggemann et al., 2019).

3. Experimental Realizations and Observational Signatures

Direct experimental signatures of spiral spin liquids include:

• Continuous rings/surfaces in diffuse magnetic neutron scattering: observed in MnSc$_2$S$_4$ (Gao et al., 2016, Iqbal et al., 2018), LiYbO$_2$ (Graham et al., 2023), FeCl$_3$ (Gao et al., 2021), GdZnPO (Wan et al., 16 Mar 2025, Zhao et al., 26 Aug 2025), AgCrSe$_2$ (Andriushin et al., 7 Oct 2024), CaMn$_2$P$_2$ (Islam et al., 3 Jan 2025), Cs$_3$Fe$_2$Cl$_9$ (Gao et al., 29 May 2024).
• Anomalous thermodynamic responses: large residual low-$T$ specific heat ($C_0$), unusual temperature dependencies ($C_v \sim C_0 + C_1T$), and giant low-temperature thermal conductivity (Zhao et al., 26 Aug 2025).
• Persistent dynamic spin noise with $1/f$-like power spectral density, observed via SQUID-based spin noise spectroscopy (Takahashi et al., 3 May 2024).
• Thermal Hall effect and evidence for topological magnon excitations (Zhao et al., 26 Aug 2025).
• Real-space vortex-like topological defects in the local momentum field, evidenced by Monte Carlo simulations and inferred from magnetization noise and theoretical models (Yan et al., 2021, Gonzalez et al., 29 Mar 2024, Takahashi et al., 3 May 2024).

Table 1 summarizes select material examples:

Material	Lattice/Class	Spiral Manifold	Signature Observed
MnSc$_2$S$_4$	Diamond (3D)	Spiral surface (2D)	Neutron ring (Gao et al., 2016)
FeCl$_3$, LiYbO$_2$, GdZnPO	Honeycomb (2D)	Spiral ring (1D)	Neutron ring (Gao et al., 2021, Graham et al., 2023, Wan et al., 16 Mar 2025)
AgCrSe$_2$	Triangular	Spiral ring (1D)	Diffraction/SANS (Andriushin et al., 7 Oct 2024)
CaMn$_2$P$_2$	Honeycomb	Spiral ring (1D)	Ring-like $S(q)$, 3 domains (Islam et al., 3 Jan 2025)
Cs$_3$Fe$_2$Cl$_9$	Honeycomb (3D)	Spiral line (1D, codim-2)	Phase-tuned ring (Gao et al., 29 May 2024)

4. Theoretical Excitations: Momentum Vortices and Emergent Gauge Structures

A distinctive theoretical insight is the role of momentum vortices—topological defects in the local spiral wave vector field $\mathbf{q}(\mathbf{r}) = \nabla\Phi(\mathbf{r})$ (Yan et al., 2021, Gonzalez et al., 29 Mar 2024). Unlike ordinary spin vortices, these defects only appear in the momentum field, respect the curl-free constraint except at vortex cores, and connect domains of constant spiral direction via straight-line domain walls:

• Such momentum vortices allow the system to fluctuate between degenerate spiral states via localized, rather than global, spin rearrangements.
• At intermediate temperatures, the proliferation of these defects leads to the “liquid” character of the SSL regime; at low $T$, a rigid network of domain walls and pinned vortices produces a “glassy” or kinetically arrested state (Yan et al., 2021).
• The low-energy effective field theory can be mapped onto a rank-2 U(1) gauge theory (with elasticity analogs), predicting four-fold pinch-point singularities in correlation functions (Yan et al., 2021, Gonzalez et al., 29 Mar 2024).

Characteristic formulas include:

$q(\mathbf{r}) = \nabla\Phi(\mathbf{r})$

$$H_{pot} = \int d^2r\, \left[\frac{1}{16}|q|^4 - 2\delta |q|^2 + \ldots \right]$$

$$\mathcal{C}_{EE}(\mathbf{q}) \propto \frac{q_x^2 q_y^2}{|\mathbf{q}|^4}$$

The mapping to fracton physics (fracton quadrupoles) links spiral spin liquids to the burgeoning field of constrained mobility and topological multipole conservation (Yan et al., 2021, Gonzalez et al., 29 Mar 2024).

5. Stability Against Fluctuations, Disorder, and Emergent Ordered Phases

Stability and transitions out of the SSL regime are sensitive to both thermal/quantum fluctuations and to perturbations such as higher-order neighbor exchanges, Dzyaloshinskii–Moriya interaction, and disorder:

• Thermal fluctuations lift degeneracies by the “order-by-disorder” mechanism, selecting states with special propagation directions, usually on points or segments of the spiral manifold (e.g., selection of spiral vectors by entropy maximization in ABC-stacked systems) (Liu et al., 2022).
• Quantum fluctuations can melt spiral order, converting the SSL into a quantum spin liquid, as demonstrated by pseudofermion functional RG studies (Niggemann et al., 2019, Iqbal et al., 2018).
• Quenched disorder acts via order-by-quenched-disorder (ObQD), favoring different spiral states depending on the impurity type/orientation, yielding Friedel-like oscillatory textures and, at finite defect concentration, spiral spin glass states (Cônsoli et al., 2023).
• Proximity to commensurate or collinear phases can result in multicritical points (e.g., Lifshitz transition) and the formation of gapped spin-liquid windows similar to the Haldane phase (Kharkov et al., 2018).
• In many systems, the SSL only exists within an extended but finite temperature window above the ordered ground state and below a trivial paramagnetic regime (Iqbal et al., 2018, Islam et al., 3 Jan 2025).

6. Topological Excitations, Spin Transport, and Thermodynamics

The low-energy landscape of SSLs is governed by mobile and topological excitations:

• Excitations along the spiral contour cost zero or near-zero energy in the classical limit; excitations off-contour cost finite energy, giving rise to scaling laws in the specific heat ($C_m \sim C_0 + C_1T$) and magnetic thermal conductivity ($\kappa_{xx}^m \sim \kappa_0 + \kappa_1T$) (Zhao et al., 26 Aug 2025).
• Thermal Hall measurements in SSL candidates such as GdZnPO demonstrate positive $\kappa_{xy}/T$, indicating topological magnon-like transport and possible Chern-number-carrying triplet excitations (Zhao et al., 26 Aug 2025).
• The residual entropy, giant low-$T$ specific heat, and broad distribution of relaxation times (logarithmic decay of magnetization autocorrelation) observed in Ca$_{10}$Cr$_7$O$_{28}$ (Takahashi et al., 3 May 2024) and GdZnPO (Wan et al., 16 Mar 2025, Zhao et al., 26 Aug 2025) are direct signatures of the subextensive degeneracy and dynamic nature of the SSL regime even in high-quality insulators.

7. Engineering and Applications

The spectrum of spiral spin liquids can be engineered through a variety of “knobs”:

• Lattice stacking geometry (ABC or AB stacking) to control frustration and manifold codimension (Liu et al., 2022, Gao et al., 29 May 2024);
• Spin anisotropy (Ising, XY, Heisenberg) and field orientation to tune between reciprocal “kagomé” structures and spiral contours (Huang et al., 2021);
• Control of further-neighbor exchange by chemical substitution, pressure, or stacking control, mirroring magic-angle strategies in 2D materials (Liu et al., 2022);
• Doping and disorder as probes of glassy SSL behavior and order-by-quenched-disorder selection (Cônsoli et al., 2023).

Potential applications include antiferromagnetic spintronics (where topologically protected spin/thermal currents and vortex excitations could encode information), unconventional quantum refrigeration (magnetocaloric effects near the SSL regime), and platforms for exploring fracton and tensor gauge theories in a condensed matter context.

---

In summary, spiral spin liquids constitute an intricate class of paramagnetic correlated states stabilized by frustration, characterized by continuous spiral manifolds in reciprocal space, cooperative fluctuations, unconventional excitations, and a rich phenomenology governed by the interplay of entropy, topology, and external control parameters. The recent convergence of high-resolution experimental probes and advanced theoretical techniques has led to the direct identification and functional exploration of these phases in honeycomb, triangular, diamond, and other lattice materials, making the SSL a central paradigm at the intersection of frustrated magnetism, topological matter, and emergent gauge structures.