Atomistic Spin-Lattice Dynamics (ASLD)

• Atomistic Spin-Lattice Dynamics (ASLD) is a framework that simultaneously evolves spin and lattice degrees of freedom to capture feedback between magnetic fluctuations and atomic displacements.
• It integrates atomistic spin dynamics with molecular or Langevin lattice dynamics using first-principles, phenomenological, or data-driven parameterizations to model coupling effects.
• ASLD enables detailed analysis of angular-momentum transfer, non-adiabatic phonon-magnon interactions, and temperature-dependent material properties in complex systems.

Atomistic Spin-Lattice Dynamics (ASLD) is an atomistic framework in which magnetic moments or spins and lattice degrees of freedom are evolved together in time, rather than treated as separate or adiabatically decoupled subsystems. In the literature, ASLD appears both as a microscopic electronically mediated theory—where spin-spin, lattice-lattice, and spin-lattice couplings are obtained after integrating out the electrons—and as a practical simulation methodology that combines atomistic spin dynamics with molecular dynamics or Langevin lattice dynamics using first-principles, phenomenological, or data-driven parameterizations (Fransson et al., 2015, Hellsvik et al., 2018). Its central purpose is to resolve feedback between magnetic fluctuations, phonons, strain, and defects, including regimes where non-adiabatic scattering, magneto-elastic renormalization, or angular-momentum exchange are essential (Perera et al., 2016, Stockem et al., 2018).

1. Conceptual foundations

A microscopic formulation of ASLD treats localized magnetization $\mathbf M_i$ and ionic displacement $\mathbf Q_i$ on the same footing. In the electronically mediated theory, the starting point is a local exchange coupling between electron spin and magnetic moment together with a local coupling between electronic charge and lattice displacement. After integrating out the electrons, one obtains an effective action containing three interaction classes: spin-spin, lattice-lattice, and spin-lattice couplings (Fransson et al., 2015). In discrete form, the resulting minimal bilinear Hamiltonian is

$$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$

This construction is notable because the spin-lattice term is not introduced as a post hoc correction. It is an electronically mediated susceptibility, alongside the more familiar interatomic force constants and spin-spin interactions, and it is explicitly constrained by time-reversal and inversion symmetry (Fransson et al., 2015).

A computationally efficient first-principles version of ASLD reformulates the same idea as a coupled classical dynamics problem. In that setting, atomistic spin dynamics provides the spin evolution, molecular dynamics provides the atomic motion, and a spin-lattice Hamiltonian couples the two through displacement-dependent magnetic interactions (Hellsvik et al., 2018). This formulation generalizes separate ASD and MD treatments into a single framework that can be parameterized from density functional theory rather than fitted phenomenologically (Hellsvik et al., 2018).

An important conceptual revision came from work showing that the conventional combined molecular and spin dynamics framework, with exchange interactions depending on atomic positions, still lacks a channel for transferring angular momentum between spins and lattice. The proposed remedy is to encode spin-orbit physics through local magnetic anisotropies induced by symmetry breaking from phonons or defects, thereby extending ASLD from an energy-coupled model to one that also supports angular-momentum exchange (Perera et al., 2016).

2. Hamiltonian structure and coupling mechanisms

A widely used decomposition writes the total Hamiltonian as

$$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$

where $\mathcal H_{\rm S}$ is the spin Hamiltonian, $\mathcal H_{\rm L}$ the lattice Hamiltonian, and $\mathcal H_{\rm LS}$ the spin-lattice coupling (Hellsvik et al., 2018). In first-principles ASLD for solids, the bilinear magnetic term is expanded in atomic displacements, producing a hierarchy of couplings. The leading term is the first-order spin-lattice or exchange-striction term,

$$\mathcal H_{\rm SSL} = -\frac{1}{2}\sum_{ijk} A_{ijk}\cdot \mathbf{u}_k\,(\mathbf{m}_i\cdot \mathbf{m}_j),$$

with $A_{ijk}$ obtained from derivatives of the exchange tensor with respect to displacement (Hellsvik et al., 2018). This makes explicit the reciprocal relation between magnetism and structure: atomic motion modulates magnetic exchange, and spin correlations generate forces on atoms.

A more direct realization of the same idea appears in paramagnetic CrN, where the magnetic subsystem is represented by a classical Heisenberg model with distance-dependent exchange,

$$\mathcal{H} = -\sum_{i\neq j} J_{ij}(R_{ij})\, \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j,$$

and the dependence of $\mathbf Q_i$0 on the Cr–Cr distance $\mathbf Q_i$1 is the essential magneto-lattice coupling channel (Stockem et al., 2018). In that material, the exchange is short-ranged and can be captured by first- and second-neighbor Cr–Cr terms (Stockem et al., 2018).

Not all coupling channels are equivalent. A spin-orbit-inspired route introduces local anisotropy terms that emerge from symmetry breaking in the atomic environment. In the anisotropy-enhanced framework, the additional contribution is

$\mathbf Q_i$2

where $\mathbf Q_i$3 and $\mathbf Q_i$4 are determined by the instantaneous local atomic environment through a symmetry descriptor $\mathbf Q_i$5 (Perera et al., 2016). A related microscopic coupling used in coupled spin and lattice dynamics is a two-site pseudo-dipolar term motivated by spin-orbit interaction,

$\mathbf Q_i$6

chosen so that the Curie temperature is preserved under changes in coupling strength, net anisotropy is removed on a cubic lattice, and a spurious uniform translation is avoided when the magnetization is saturated (Strungaru et al., 2020).

3. Equations of motion and computational realization

The coupled dynamics are typically written as simultaneous equations for positions, velocities, and spins. In one representative formulation,

$\mathbf Q_i$7

with forces and effective fields obtained from derivatives of the total Hamiltonian with respect to positions and spins (Strungaru et al., 2020). Other implementations replace the purely precessional spin equation by stochastic Landau–Lifshitz–Gilbert dynamics and use Langevin terms for both lattice and spin subsystems (Nikolov et al., 2021).

A first-principles realization of fully coupled ASLD in paramagnetic CrN alternates atomistic spin dynamics with ab initio molecular dynamics. The workflow begins with pre-equilibration by adiabatic DLM-AIMD, then computes distance-dependent $\mathbf Q_i$8, generates an initial spin configuration by Monte Carlo, and iterates a coupled loop in which the current spins are used in a Born–Oppenheimer AIMD step for $\mathbf Q_i$9 fs, the new atomic positions update $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$0, and the spins are then propagated for $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$1 fs using ASD (Stockem et al., 2018). The force calculations use spin-polarized, noncollinear DFT in VASP with constrained spin directions through the Ma–Dudarev constrained-moment method with Lagrange multipliers (Stockem et al., 2018).

The numerical integrator depends on the coupling structure. A Suzuki–Trotter decomposition is used in self-consistent spin-lattice dynamics with pseudo-dipolar coupling, including a Cayley transform for norm-conserving spin updates (Strungaru et al., 2020). When nonlinear anisotropy terms are added, the conventional Suzuki–Trotter scheme becomes inapplicable in its standard form, and a hybrid integrator combining Suzuki–Trotter decomposition with the iterative method of Krech et al. is used; in that case the timestep is reduced from $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$2 fs to $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$3 fs to maintain microcanonical energy conservation (Perera et al., 2016). Data-driven large-scale ASLD for $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$4-Fe is implemented in LAMMPS using the SPIN package and a massively parallel symplectic ASLD algorithm, with the total potential energy surface decomposed into a collective atomic spin model and a SNAP-based machine-learning interatomic potential (Nikolov et al., 2021).

4. Non-adiabaticity, damping, and angular-momentum exchange

A recurrent simplification in early coupled spin-lattice models is the assumption that coordinate-dependent exchange alone is sufficient to mediate full relaxation. The literature summarized here shows that this is not generally the case. In the conventional MD-SD Hamiltonian, exchange depending on positions allows energy transfer between lattice and spins, but because of rotational symmetry it does not allow the lattice and spin subsystems to exchange angular momentum. A heat bath connected only to the lattice can therefore thermalize the atoms while leaving the spins trapped (Perera et al., 2016). The anisotropy-based extension remedies this by introducing symmetry-breaking torques that preserve energy, linear momentum, and total angular momentum while allowing angular momentum to move between spin and lattice sectors (Perera et al., 2016).

The Einstein–de Haas effect provides a direct benchmark. In simulations of a freely rotating prolate spheroidal Fe nanocluster containing $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$5 Fe atoms, a modified version of SPILADY with the dynamic anisotropy correction of Perera et al. shows that the model conserves total angular momentum while spin and lattice angular momenta are not conserved separately. The rate of angular momentum transfer is proportional to the anisotropy strength $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$6, and the added anisotropy allows full spin-lattice relaxation on a timescale of approximately $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$7 ps (Dednam et al., 2022).

ASLD also supplies a microscopic route to effective damping without imposing a phenomenological Gilbert term on the spin subsystem. In the pseudo-dipolar coupling model, the effective damping parameter increases with both coupling strength and temperature; the reported dependence on coupling is approximately quadratic, and at low temperature damping values on the order of $$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$8 are obtained (Strungaru et al., 2020). On the spin side, a generalized atomistic equation with a damping tensor and a moment-of-inertia tensor,

$$\mathcal H_{\text{MQ}}= -\frac{1}{2}\sum_{ij}\Bigl(\mathbf Q_i\cdot[T^{cc}_{ij}\cdot \mathbf Q_j + T^{cs}_{ij}\cdot \mathbf M_j] + \mathbf M_i\cdot[T^{sc}_{ij}\cdot \mathbf Q_j + T^{ss}_{ij}\cdot \mathbf M_j]\Bigr).$$9

extends standard atomistic spin dynamics into the femtosecond regime and provides a first-principles foundation for damping and nutational effects, although it does not itself constitute a full spin-lattice model (Bhattacharjee et al., 2011).

5. Materials results and experimental observables

One of the clearest demonstrations of ASLD’s non-adiabatic content is paramagnetic CrN above the magnetic ordering temperature. In fully coupled ASD-AIMD, the dynamic coupling between spin fluctuations and lattice vibrations strongly broadens phonon spectra at $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$0 relative to the adiabatic DLM-AIMD control, whereas at $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$1 the difference is much smaller. The effect is strongest for acoustic phonons, and correlation analysis shows that antiferromagnetically aligned nearest-neighbor Cr pairs tend to have shorter distances while ferromagnetically aligned pairs tend to have longer distances; this correlation is strong at $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$2 and weaker at $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$3 (Stockem et al., 2018). The result is a microscopic explanation of the anomalous thermal conductivity of paramagnetic CrN: near the transition, dynamic magneto-lattice coupling shortens acoustic phonon lifetimes, while farther above the transition the coupling weakens and the extra spin-induced phonon scattering diminishes (Stockem et al., 2018).

In bcc Fe, first-principles ASLD based on a displacement expansion of the exchange tensor produces coupling-induced modifications of both magnon and phonon spectra, and the same framework exhibits dissipation-free coupled motion in small magnetic clusters such as dimers, trimers, and quadmers (Hellsvik et al., 2018). A data-driven extension for $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$4-iron couples a collective spin Hamiltonian with a SNAP machine-learning interatomic potential and yields quantitative predictions of bulk modulus, magnetization, and specific heat across the ferromagnetic-paramagnetic phase transition. Under fixed-volume conditions the Curie temperature is predicted near $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$5 K, while pressure-controlled and pressure- and magnetization-controlled protocols recover thermal softening and the specific-heat anomaly more faithfully (Nikolov et al., 2021).

ASLD has also been propagated into electron microscopy observables. In body-centered cubic iron at $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$6 K, an extension of TACAW to ASLD produces momentum-resolved EELS from unified finite-temperature trajectories containing both atomic displacements and spin configurations. The resulting “Full” spectrum is not the sum of separate “Phonons (ASLD)” and “Magnons (ASLD)” signals; instead it shows interference and energy redistribution in $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$7-space. With an annular dark-field detector of inner and outer collection angles $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$8 and $$\mathcal H_{\rm SLD}=\mathcal H_{\rm S}+\mathcal H_{\rm L}+\mathcal H_{\rm LS},$$9 mrad, phonons dominate below $\mathcal H_{\rm S}$0 meV, magnetic intensity becomes visible above approximately $\mathcal H_{\rm S}$1 meV, and energy-resolved EELS improves the signal-to-noise ratio by roughly three orders of magnitude relative to a no-energy-resolution configuration (Castellanos-Reyes et al., 9 Aug 2025). This establishes that phonon and magnon excitations in the coupled system are not generally additive observables.

6. Software, scaling, and methodological boundaries

ASLD practice spans full spin-lattice codes, spin-only frameworks that supply the magnetic subsystem, and accelerator-oriented infrastructure. UppASD is used as the ASLD engine in coupled phonon-magnon EELS simulations, where it supplies unified spin-lattice trajectories through coupled Langevin equations (Castellanos-Reyes et al., 9 Aug 2025). SPILADY has been modified to include dynamic anisotropy for Einstein–de Haas simulations in Fe nanoclusters (Dednam et al., 2022). LAMMPS, through the SPIN package and a massively parallel symplectic ASLD algorithm, supports data-driven magneto-elastic simulations with machine-learning interatomic potentials (Nikolov et al., 2021). Spirit provides the atomistic spin side of the ecosystem, with an extended Heisenberg-type spin-lattice Hamiltonian, Monte Carlo, LLG solvers, geodesic nudged elastic band, minimum mode following, harmonic transition-state theory, and CPU/GPU parallelization, but it does not propagate atomic coordinates in the cited work (Müller et al., 2019). SpinX supplies a JAX-based atomistic spin framework with dense, sparse, FFT-based, pair-list, and reciprocal-space dipolar backends, plus deterministic and stochastic LLG, Monte Carlo, spectroscopy, and string/GNEB calculations; performance benchmarks report peak throughput exceeding $\mathcal H_{\rm S}$2 billion spin-site operations per second on a single accelerator and aggregate single-node workloads of over $\mathcal H_{\rm S}$3 billion atomic spins (Xu et al., 25 May 2026).

Post-processing and spectroscopy are themselves major computational bottlenecks. For real-space dynamical spin-spin correlations, GPU kernel fusion reformulates the pair-correlation construction as GEMM, yielding up to $\mathcal H_{\rm S}$4 speedup over a baseline GPU code and up to $\mathcal H_{\rm S}$5 over a $\mathcal H_{\rm S}$6-core CPU GEMM implementation. Fusing inner-product work into the GEMM kernel with CUTLASS improves performance by $\mathcal H_{\rm S}$7 over a cuBLAS-plus-Thrust implementation, while on-the-fly epilogue computation of the phase matrix enables larger systems; on four A100 GPUs, reported speedups range from $\mathcal H_{\rm S}$8 to $\mathcal H_{\rm S}$9 over one GPU (Chen et al., 2023).

Several methodological boundaries recur across the literature. Most implementations remain semiclassical, with classical spins and nuclei and electrons integrated out (Hellsvik et al., 2018). Phenomenological ingredients persist where first-principles extraction is difficult, particularly for local anisotropy parameters and some spin-orbit-mediated couplings (Perera et al., 2016). Fixed-magnitude spin approximations can limit accuracy at strong spin disorder or elevated pressure (Nikolov et al., 2021). Explicit quantum spin effects are commonly neglected, although path-integral approaches have been proposed to encode quantum corrections as effective anisotropy-like fields within ASD-style sampling (Nussle et al., 2023). Three simplifications are shown to fail in specific contexts: isotropic exchange alone does not enable spin-lattice angular-momentum transfer (Perera et al., 2016), adiabatic magnetic disorder misses non-adiabatic phonon lifetime broadening in CrN (Stockem et al., 2018), and separate phonon and magnon spectra need not add to the full signal in EELS (Castellanos-Reyes et al., 9 Aug 2025). Together, these results define ASLD less as a single algorithm than as a class of coupled atomistic theories and simulation strategies designed to resolve the mutual dynamics of magnetic and structural degrees of freedom.