Emergence of moiré magnetic chaos in twisted bilayer CrI3

Abstract: The study of magnetic chaos has traditionally focused on macroscopic variables under external driving. Here we demonstrate a new type of magnetic chaos, termed moiré magnetic chaos, associated with mesoscopic magnetic domain variables in twisted bilayer CrI3 without external driving. The domains are stabilized by a characteristic interlayer exchange frustration, which supplies the multiple dynamical degrees of freedom required for autonomous chaos. Through micromagnetic simulations, we show that relaxation toward moiré magnetic textures is extremely sensitive to minute local perturbations of the initial state, characterized by substantial finite-time Lyapunov exponents and a final-state sensitivity that persists over five decades of perturbation amplitude. Statistical analysis further reveals that the resulting domain configurations are stochastic and pairwise uncorrelated. Our results identify a form of microscopic, undriven chaos in twisted magnets that extends nonlinear magnetism beyond the conventional driven regime.

• The paper identifies transient, autonomous magnetic chaos arising during undriven relaxation, with a maximal Lyapunov exponent of 290.8 ± 0.1 ns⁻¹ and a Lyapunov time of about 3.4 ps.
• Moiré-frustrated interlayer exchange generates nearly degenerate metastable states, causing perturbations as small as 10⁻⁸ to produce approximately 50% domain decorrelation across the magnetic pattern.
• Across 1,000 simulations, domain states show 99.9% of maximum single-patch entropy and negligible pairwise correlations, supporting potential applications in hardware random-number generation and magnetic information technologies.

Overview and motivation

This paper reports a form of magnetic chaos that is autonomous, high-dimensional, and microscopic in origin. Prior studies of magnetic chaos have relied on macroscopic variables—uniform magnetization, vortex core position, skyrmion trajectory—driven by external forcing such as spin-transfer torque or time-dependent fields, because fewer than three dynamical degrees of freedom (DOF) cannot sustain autonomous chaos under the Poincaré–Bendixson theorem. The authors identify twisted bilayer CrI₃ as a system that circumvents this constraint intrinsically: the moiré-modulated interlayer exchange frustrates individual moments so that they act as independent DOFs, satisfying the dimensionality requirement without any drive. They term the resulting phenomenon moiré magnetic chaos.

The physical setting is a twisted van der Waals bilayer at twist angle $\theta = 1.61^\circ$, where the local stacking alternates between ferromagnetic (FM) and antiferromagnetic (AFM) interlayer coupling across the moiré superlattice. Each AFM patch admits two degenerate polarizations related by time reversal, giving $2^N$ nearly degenerate metastable configurations for $N$ patches. The model Hamiltonian combines intralayer FM exchange ($J_{\mathrm{intra}} = 2.12$ meV), single-ion anisotropy ($A = 0.087$ meV), a first-principles stacking-dependent interlayer exchange map spanning $-0.47$ to $+0.50$ meV, and dipole–dipole interactions. Dynamics are simulated with deterministic Landau–Lifshitz–Gilbert (LLG) equations using the mumax⁺ solver on a $588 \times 595$ grid covering $4 \times 7$ moiré cells.

High metastability from frustrated interlayer exchange

The energetic foundation for chaos is established by re-evaluating the total energy of relaxed configurations drawn from 1000 independent quenches. The entire metastable manifold spans only $E_{\max} - E_{\min} \approx 48$ meV—a relative spread $2^N$0, roughly $2^N$1 meV per patch—with an accurately Gaussian density of states ($2^N$2 meV), consistent with a central-limit-theorem sum of nearly independent per-patch contributions. Because flipping a single patch costs a fraction of a meV—far below the exchange and anisotropy scales stabilizing each domain—no configuration is energetically singled out, and infinitesimal changes to the initial condition can steer relaxation into any member of the manifold. This near-degeneracy is the direct mechanism linking frustration to extreme sensitivity: the authors note explicitly that quantitative outcomes such as the Lyapunov magnitude scale with the chosen parameters, while the statistical conclusions concern ensemble structure.

The relaxed textures consist of reversed domains nucleating at complementary AFM patches in the two layers, classified topologically into skyrmion-like domains ($2^N$3, about 7% of the population) and bubble-like domains ($2^N$4, about 93%). Which patches reverse differs run to run and is set entirely by the initial condition.

Damage spreading and structural chaos

A damage-spreading protocol quantifies sensitivity to initial conditions. A reference state is relaxed for 2 ns; perturbed trajectories tilt a single centre spin by angles $2^N$5 logarithmically spaced over five decades, $2^N$6 to $2^N$7. Even at $2^N$8—a tilt below $2^N$9 degrees—about half of the reference domains reverse their binary state, yielding a Hamming fraction $N$0 against $N$1 reference domains. This value coincides with the maximal-decorrelation limit for two statistically independent binary strings, and it remains pinned at $N$2 across the full five-decade sweep, in contrast to the $N$3 scaling of non-chaotic dynamics. Evaluated over all $N$4 patches, the perturbation-averaged Hamming fraction is $N$5, confirming the result is not an artifact of the reference subset.

The finite-amplitude Lyapunov exponent decreases monotonically from $N$6 at $N$7 to $N$8 at $N$9. The authors show this decrease follows quantitatively from saturation of $J_{\mathrm{intra}} = 2.12$0 at the geometric bound set by $J_{\mathrm{intra}} = 2.12$1: since complete decorrelation is reached regardless of amplitude, no perturbation is too small to be amplified to macroscopic scale within nanoseconds. A supplementary Benettin tangent-space calculation yields a rigorous maximal exponent of $J_{\mathrm{intra}} = 2.12$2 (Lyapunov time $J_{\mathrm{intra}} = 2.12$3 ps), roughly thirty times larger than the finite-amplitude estimate, which is therefore a saturation-limited lower bound. Importantly, the authors concede that because damped LLG dynamics relax to static states, the asymptotic exponent is non-positive: the phenomenon is transient chaos—final-state (basin) sensitivity generated by an exponential transient during texture formation—rather than a sustained chaotic attractor.

The decorrelation is spatially structured rather than homogeneous. A fixed subset of 22 pinned domains remains locked to the reference across all perturbation amplitudes, 24 flip under every perturbation, and 43 switch stochastically. The authors label this partition structural chaos, organized by the moiré geometry itself, while noting that whether the pinned/active partition is intrinsic to the superlattice or specific to the chosen reference attractor requires multi-reference analysis.

Statistical characterization of the attractor ensemble

Ensemble statistics over 1000 independent quenches establish that the deterministic dynamics produce outputs statistically indistinguishable from fair coin flips. Each patch reverses with probability $J_{\mathrm{intra}} = 2.12$4, and the per-patch binary Shannon entropy reaches $J_{\mathrm{intra}} = 2.12$5 nats—99.9% of the maximum $J_{\mathrm{intra}} = 2.12$6. The pairwise Hamming-distance distribution matches $J_{\mathrm{intra}} = 2.12$7 with mean 0.500 and standard deviation 0.040, both matching ideal values to three decimals. Across all 12,246 patch pairs, Pearson correlation coefficients are centred at zero with width matching the finite-sample null $J_{\mathrm{intra}} = 2.12$8, the largest coefficient being only $J_{\mathrm{intra}} = 2.12$9, and no pair survives Bonferroni correction. A mutual-information analysis against an independent-Bernoulli permutation null corroborates these findings.

The authors are careful about scope: only pairwise independence is tested, higher-order dependence is not excluded, and because each patch's local state is ternary (reversed domain in top layer, bottom layer, or neither) while the recorded bit captures only top-layer occupancy, they refrain from asserting exhaustive access to all $A = 0.087$0 configurations. A null control strengthens the causal claim: shuffling the spatial structure of $A = 0.087$1 while preserving its value distribution eliminates domain formation entirely (zero reversed-area fraction in 100/100 runs versus a mean of 0.14 for the moiré map), demonstrating that the chaotic selection requires spatially coherent moiré frustration rather than generic coupling disorder.

Thermal robustness and field persistence

Two robustness tests address experimental relevance. Under stochastic LLG heating from $A = 0.087$2, a domain-order parameter $A = 0.087$3 (the spatial standard deviation of the time-averaged magnetization) stays pinned at its zero-temperature value $A = 0.087$4 below $A = 0.087$5 K, then declines steeply near 20–23 K and falls to $A = 0.087$6 by 29 K—well below the monolayer Curie temperature of 45 K. The melting scale is attributed, via a single-parameter mean-field estimate ($A = 0.087$7 K), to thermal disordering of intralayer ferromagnetic order rather than to moiré-specific physics; the authors flag that a controlled parameter sweep would be needed to establish this directly. Since standard cryogenic operating temperatures lie within the robust regime, the chaotically selected bit pattern constitutes a long-lived observable accessible to slow probes such as scanning quantum magnetometry. Thermal locking also cleanly separates roles: the deterministic flow generates randomness during relaxation, while noise merely preserves the result.

Under applied fields, quantified through a normalized pattern-survival correlation $A = 0.087$8, an out-of-plane field erodes the pattern only gradually ($A = 0.087$9 at 2 T), while an in-plane field drives a partial collapse near $-0.47$0 T—as the background cants into the plane—to a field-resilient plateau $-0.47$1 up to 2 T. The authors caution that $-0.47$2 is a spatial-correlation coefficient, not a per-patch survival count. Together, the two tests establish the selected pattern as robust to both perturbations relevant to readout.

Limitations and open questions

Several limitations are conceded explicitly. The chaos is transient rather than attractor-based, appropriate to relaxational dissipative dynamics but distinct from sustained chaos. The finite-amplitude Lyapunov exponent understates the true expansion rate by a factor of thirty, and the sub-$-0.47$3 perturbation points approach the single-precision floating-point noise floor. The statistical characterization records only binary top-layer occupancy of ternary local states, tests pairwise rather than mutual independence, and does not demonstrate access to the full $-0.47$4 manifold. Whether the pinned-domain partition is intrinsic to the superlattice or reference-specific, whether the melting-temperature attribution survives a systematic parameter sweep, and how the phenomenon extends to other twist angles and materials (e.g., twisted CrSBr) remain open questions.

Conclusion

This work identifies moiré magnetic chaos—an autonomous, high-dimensional sensitivity to initial conditions arising during undriven relaxation in twisted bilayer CrI₃—supported by three convergent lines of evidence: a maximal Lyapunov exponent of order $-0.47$5 with amplitude-independent decorrelation over five decades of perturbation; per-patch Shannon entropy at 99.9% of maximum across 1000 quenches; and pairwise correlations statistically indistinguishable from independence. The periodic moiré order is constitutive of the chaos rather than parasitic to it, making the phenomenon reproducible and diagnostic of twist engineering. The authors propose applications to hardware random-number generation, estimating $-0.47$6 Gbit s⁻¹ of uncorrelated bits per flake at quench cycles of order 10 ns before reset and readout limits, and note that the predicted domain patterns at smaller twist angles are directly resolvable by existing scanning quantum magnetometry.