Spin-Wave Turbulence: Dynamics & Scaling

• Spin-wave turbulence is a state in magnetic systems and spinor condensates characterized by broadband, disordered spin fluctuations and scale-invariant power laws.
• Nonlinear spin interactions triggered by instabilities, such as parameter quenches or counterflow, drive cascades that deviate from classical Kolmogorov turbulence.
• Experimental diagnostics, including high-resolution imaging and spin-glass order parameters, reveal isotropic spin domains and distinct magnetic relaxation behaviors.

Spin-wave turbulence refers to nonequilibrium states in quantum or classical magnets and multicomponent superfluids where collective spin degrees of freedom exhibit broadband, disordered spatial fluctuations and cascade processes characterized by scale-invariant power laws distinct from hydrodynamic turbulence. The canonical platforms for spin-wave turbulence are spinor Bose–Einstein condensates (BECs), quantum magnets, and magnon condensates, where spin-wave modes interact nonlinearly and energy/“particle” fluxes range across an inertial spectrum. Spin-wave turbulence is distinguished by the specific dynamics of spin–spin interactions, conservation laws, and underlying symmetry (e.g., SU(2) or easy-plane), resulting in anomalous scaling exponents, glass-like behavior, and statistical isotropy in the spin sector (Jung et al., 2023, Fujimoto et al., 2016, Liu et al., 4 Dec 2025).

1. Governing Models and Instabilities

Spin-wave turbulence arises from the nonlinear dynamics of coupled spinor fields or spin chains as governed by multi-component Gross–Pitaevskii equations (BECs), the Landau–Lifshitz equation (ferromagnets), or lattice Heisenberg models. In spin-1 BECs, the mean-field Hamiltonian comprises density-density and spin-spin interactions,

$$H = H_0 + H_{\rm int} = \int d^d r\, \Psi^\dagger \left(-\frac{\hbar^2}{2m}\nabla^2 + q f_z^2 - \hbar\Omega f_x \right)\Psi + \int d^d r\, \left[\frac{g_n}{2} n^2 + \frac{g_s}{2} |\mathbf F|^2 \right]$$

where $\Psi=(\psi_{+1},\psi_0,\psi_{-1})^T$, $n=\Psi^\dagger\Psi$, $\mathbf{F} = \Psi^\dagger \mathbf{f} \Psi$ is the spin-density vector, and $q$ the quadratic Zeeman term. Ferromagnetic ($g_s<0$) and antiferromagnetic ($g_s>0$) cases yield distinct turbulence phenomenology. The central mechanism for generating turbulence is a dynamical instability of a highly symmetric or ordered initial state, triggered by counterflow, spin-helix preparation, quantum/thermal noise, or sudden parameter quenches (e.g., sign change of $q$ in spin-1 BECs) (Jung et al., 2023, Kang et al., 2017, Tsubota et al., 2013).

Linearization about a polarized (e.g., $m_F=0$) state yields the Bogoliubov spectrum for spin waves, e.g.: $$( \hbar \omega( k ) )^2 = [ \epsilon_k + q ] [ \epsilon_k + q + 2 g_s n_0 ]$$ where $\epsilon_k = \hbar^2 k^2 / 2 m$. Regions of imaginary $\omega(k)$ indicate momenta with exponential growth, seeding broadband fluctuations and cascading energy. Continuous spin driving (rf fields or dynamic noise) sustains a nonequilibrium steady spin-turbulent state by arresting semiclassical relaxation and supporting persistent mode-mixing (Jung et al., 2023, Hong et al., 2023).

2. Spectral Cascades and Scaling Laws

A primary feature of spin-wave turbulence is the presence of non-Kolmogorov cascades in the spin-dependent interaction energy spectrum: $$E_s(k) = \frac{g_s}{2} \sum_{k<|\mathbf{k}|<k+\Delta k} | \widetilde{\mathbf{F}}( \mathbf{k} ) |^2$$ where $| \widetilde{\mathbf F} ( \mathbf k ) |^2$ denotes the shell-averaged spin-density Fourier amplitude.

The fundamental spectral law observed in spinor BECs and ferromagnets is a $k^{-7/3}$ scaling of $E_s(k)$ in the inertial range: $E_s(k) \propto k^{-7/3}, \qquad k_b < k < k_s$ with $k_s=2\pi/\xi_s$ (spin healing length inverse), $k_b$ marking the low-$k$ boundary of the inertial subrange. This is theoretically derived via Kolmogorov-type dimensional analysis applied to the Landau–Lifshitz or spin-hydrodynamics equations, with the structure of the four-wave interaction vertex determining the exponent (Fujimoto et al., 2013, Fujimoto et al., 2016, Liu et al., 4 Dec 2025).

Additional regimes include a $k^{-1}$ law at lower $k$ in small-spin-magnitude turbulence (antiferromagnetic condensates or ferromagnets with $q>0$) (Fujimoto et al., 2013), and dual cascades:

• Direct energy cascade: $E_s(k) \propto k^{-7/3}$
• Inverse “wave-action” (particle number) cascade: $E_s(k) \propto k^{-5/3}$ The precise realization depends on conservation laws and system symmetry (e.g., XXZ versus SU(2)), with dual fluxes evident in easy-plane models (Fujimoto et al., 2016, Rodriguez-Nieva, 2020).

3. Statistical Isotropy, Entanglement, and Spin-Glass Behavior

Statistical and dynamical analysis reveals that fully developed spin-wave turbulence supports isotropic spatial and spin composition, as demonstrated by:

• Magnetization correlation functions $G_\alpha(r)$ ($\alpha=x,y,z$) collapsing for all directions,
• Entanglement entropy between spin and spatial degrees of freedom reaching its maximum ($S\to\ln 3$ for spin-1) (Jung et al., 2023).

Temporal behavior is characterized by local freezing (spin-glass analogy) in the ferromagnetic case: the unit spin vector $\hat{\mathbf{s}}(\mathbf r,t)$ displays spatial randomness and temporal persistence (disorder but absence of time evolution) after the turbulent regime is established, quantified via the overlap order parameter,

$$q(t) = \bigl[ \langle \hat{\mathbf{s}}(\mathbf r,t) \rangle_T^2 \bigr]$$

which saturates near unity in the spin-glass–like phase (Tsubota et al., 2013).

Antiferromagnetic turbulent regimes lack such freezing, instead exhibiting persistent dynamic fluctuations and suppression of $q(t)$ (Tsubota et al., 2013).

4. Universality, Cascades Outside BECs, and Spectral Nonlocality

Weak wave turbulence theory predicts universal scaling for both direct and inverse cascades. In the Heisenberg ferromagnet and magnon gases, the KZ spectrum for wave action is $n_k \sim k^{-d-2/3}$. However, one-loop corrections and vertex renormalization in spin-wave models yield pronounced spectral nonlocality: the effective nonlinearity at low $k$ receives UV-enhanced corrections, and universality can break down far from the forcing scale. This leads to a transition from weak- to strong-turbulence regimes and, in the strong (critical-balance) regime, an amplitude $n_k\propto k^{2-d}/(\lambda^2 Q)$ that decreases as the pump amplitude $Q$ increases (Liu et al., 4 Dec 2025). Such behavior is in sharp contrast to local models (NSE) where universality persists and the spectrum remains independent of the pump.

Spin-wave turbulence and cascades extend beyond BECs to:

• Ultrathin ferromagnetic films and magnon condensates (e.g., $n_k\propto k^{-2}$ cascade in 2D magnon gas (Safonov et al., 3 Aug 2025)),
• Two-dimensional Heisenberg models after quantum quenches (Rodriguez-Nieva, 2020),
• Spin Seebeck systems and caloritronic devices, where wave turbulence underlies the delocalization of magnons and enables macroscopic spin transport (Wang et al., 2018).

5. Experimental Realizations and Diagnostics

Recent experiments and high-resolution simulations have confirmed key aspects of spin-wave turbulence:

• Stationary turbulence in antiferromagnetic spin-1 BECs generated by continuous rf driving and field fluctuations, with measured spin spectrum $E_s(k)\propto k^{-7/3}$, isotropic spin domains of size $d_\alpha \approx \xi_s$, and steady-state spin–energy fraction $E_s/E_{s0}\sim0.42$ (Jung et al., 2023, Hong et al., 2023);
• Universal scaling and time–space collapse of correlation functions in quenched AFM BECs, with underlying critical-point scaling $\ell_q \sim |q|^{-1/2}$ in spin-domain formation (Kang et al., 2017);
• Imaging of spin domain structures and spectral analysis of spin density by phase-contrast or Stern–Gerlach–separated absorption, bin-averaged over angular shells, and entanglement entropy estimation via Schmidt decomposition of the condensate wavefunction.

Recent methodologies involve construction of a projected local spin ensemble at each point, comparing the distribution with the Haar-random ensemble to verify the degree of local randomness and high-order quantum design (Jung et al., 2023). In ferromagnetic ST, the spin-glass order parameter is used as a diagnostic for temporal freezing, distinguishing it from ordinary disordered paramagnets (Tsubota et al., 2013).

6. Open Problems, Interdisciplinary Extensions, and Outlook

Unresolved theoretical questions include the precise analytical foundation for the $-7/3$ law in systems with strong nonlinearity, the effect of higher spin and anisotropy, and the crossover regime between weak and strong turbulence—especially in the presence of strong spectral nonlocality and vertex renormalization (Liu et al., 4 Dec 2025). The range of universal scaling is sensitive to the nature of the pumping (rf drive or quench) and dissipative mechanisms, as well as boundaries and finite-size effects.

Experimental extensions to condensed matter systems—magnonic crystals, ultrathin ferromagnetic films, and models of wave transport (spin Seebeck)—highlight the relevance of spin-wave turbulence to nonequilibrium magnonics, transport, and wave localization phenomena (Safonov et al., 3 Aug 2025, Wang et al., 2018). The interplay between wave localization, turbulence-induced delocalization, and topological defect formation (e.g., half-quantum vortices in quenched AFM BECs) reflects a broader set of universal mechanisms.

Spin-wave turbulence functions as a paradigmatic case of multicomponent wave turbulence, combining elements of nonequilibrium statistical mechanics, hydrodynamics, nonlinear optics (wave turbulence theory), and magnetic systems. Its unique scaling laws, spectral structure, and isotropy diagnostics position it as a benchmark for studies of quantum turbulence beyond vortex dynamics, and a platform for testing nonlocality- and critical-balance–dominated strong turbulence regimes (Jung et al., 2023, Liu et al., 4 Dec 2025).