Time-Dependent Spin-Wave Theory

• Time-dependent spin-wave theory is a family of dynamical methods that generalize static magnon models by incorporating rotating frames, magnetic textures, and non-equilibrium behaviors.
• It employs advanced techniques such as 1/S expansions, fluctuation Hamiltonians, and time-dependent density functional theories to capture collective excitations and soft modes.
• The theory bridges equilibrium and dynamic regimes, offering insights into dissipative, hydrodynamic, and many-body interactions in both localized and itinerant magnetic systems.

Time-dependent spin-wave theory denotes a family of dynamical extensions of spin-wave methods for quantum and classical magnets in which the relevant object is no longer only a static magnon band, but a time-resolved fluctuation problem around a polarized state, a rotating frame, a magnetic texture, or a self-consistent itinerant-electron response. Across the literature, it includes $1/S$ expansions in rotating coordinates, quadratic fluctuation Hamiltonians around domain walls, linear-response TD-SDFT and TDDFPT formulations in which magnons appear as poles of the transverse spin susceptibility, open-system master equations for magnon relaxation, hydrodynamic spin-polarization waves, and explicitly time-dependent many-body correlation methods beyond linear spin-wave theory (Rückriegel et al., 2011, Gorni et al., 2018, Gorni et al., 2022, Roscilde et al., 2023).

1. Conceptual scope

In equilibrium linear spin-wave theory, one first selects a static classical reference configuration, then bosonizes transverse fluctuations, and finally diagonalizes a quadratic bosonic Hamiltonian. The time-dependent generalization changes the first step. For explicitly driven magnets or rotating magnets, the relevant local quantization axes become time dependent; for itinerant systems, the central object becomes the dynamical spin susceptibility rather than a static quadratic boson Hamiltonian; for finite systems with continuous symmetry, the zero mode cannot be treated as a harmonic boson and must instead be identified with a rotor degree of freedom. This suggests that “time-dependent spin-wave theory” is best understood as a unifying label for methods that retain the spin-wave idea of small collective fluctuations while reformulating the dynamics in a basis adapted to non-equilibrium evolution or dynamical response (Rückriegel et al., 2011, Karimi et al., 2016, Roscilde et al., 2023).

A recurrent theme is the status of soft modes. In finite-size $U(1)$-symmetric systems, the $\mathbf q=0$ Goldstone mode produces divergences in ordinary linearization and reconstructs the Anderson tower of states as a quantum rotor. In texture problems, the wall-localized mode must be included alongside propagating magnons. In itinerant magnets, the long-wavelength transverse collective mode is constrained by Larmor’s theorem in spin-rotation-invariant settings, but this constraint is broken by spin-orbit coupling. In hydrodynamic formulations, only transverse spin-polarization components propagate. The common structure is therefore not a single universal Hamiltonian, but a hierarchy of dynamical reductions dictated by symmetry, collective coordinates, and the relevant response channel (Borys et al., 2015, Karimi et al., 2016, Ambrus et al., 2022, Roscilde et al., 2023).

2. Rotating frames, magnetic textures, and localized-spin formulations

A foundational formulation was given by the paper "Time-dependent spin-wave theory" (Rückriegel et al., 2011), which generalizes the $1/S$ expansion to explicitly time-dependent quantum spin Hamiltonians. Its central statement is that spin operators should be projected onto properly defined rotating reference frames before their components are bosonized using the Holstein–Primakoff transformation. If the exact evolution operator is factorized as ${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$, the effective Hamiltonian in the rotating frame is

$$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$

with

$$\tilde{\cal H}_A(t)= {\cal U}_0^\dagger(t)\,{\cal H}(t)\,{\cal U}_0(t), \qquad \tilde{\cal H}_B(t) = -i\,{\cal U}_0^\dagger(t)\,\partial_t {\cal U}_0(t).$$

The second term is the geometric or inertial contribution generated by the time-dependent basis. In the rotating-field example, this construction avoids the singular behavior that appears if one bosonizes directly in the laboratory frame. The same paper applies the method to a YIG-motivated model and predicts a characteristic dip in the magnetization (Rückriegel et al., 2011).

For textured magnets, the time-dependent theory is often formulated as a fluctuation expansion around a nonuniform static profile. In "Spin Wave Eigenmodes of Dzyaloshinskii Domain Walls" (Borys et al., 2015), the background is a Néel-type Dzyaloshinskii domain wall stabilized by interfacial DMI in an ultrathin perpendicularly magnetized ferromagnet. The fluctuation Hamiltonian is

$$\delta H=\frac{K_\perp}{\kappa}\int dx\; \delta\theta\, V_P(x)\,\delta\theta +\delta\phi\left[V_P(x)-\Delta(x)-\kappa\right]\delta\phi,$$

with

$V_P(x)=\left[-\lambda^2\partial_x^2+1-2\sech^2(x/\lambda)\right], \qquad \Delta(x)=\frac{D\,\kappa}{\lambda K_\perp}\sech(x/\lambda).$

The key result is that the DMI term destroys the reflectionless character of the Bloch-wall Pöschl–Teller problem, so magnons are partially reflected, the spectrum acquires a DMI-induced splitting, and periodic wall arrays develop band gaps (Borys et al., 2015).

A related but distinct texture problem is magnon-driven domain-wall motion. In "Equations of Motion and Frequency Dependence of Magnon-Induced Domain Wall Motion" (Risinggård et al., 2017), the wall coordinates $X(t)$ and $U(1)$0 are treated as slow collective variables, while spin waves are first-order fluctuations on the wall background. The wall response enters only at second order in spin-wave amplitude, leading to explicit equations for $U(1)$1 and $U(1)$2. In the dissipationless limit, the theory gives $U(1)$3, which the authors use to argue that magnonic spin-transfer torque does not cause rotation-induced Walker breakdown. The same paper also shows that much of the puzzling frequency dependence reported in earlier simulations can be reproduced once the frequency dependence of the emitted spin-wave amplitude from a localized microwave source is included (Risinggård et al., 2017).

Not every localized-spin time-dependent theory is a linear magnon theory in the usual sense. "Spin waves in ferromagnetic thin films" (Sun et al., 2019) studies a nonlinear shape-preserving traveling texture of the Landau–Lifshitz–Gilbert equation rather than a full $U(1)$4 spectrum. Under the ansatz

$U(1)$5

the PDE reduces to two ODEs for $U(1)$6 and $U(1)$7. The resulting theory describes field-driven phase drift and damping-induced relaxation of a coherent nonlinear wave, but the paper explicitly distinguishes this from a general linear magnon dispersion theory (Sun et al., 2019).

3. Itinerant-electron and susceptibility-based formulations

In itinerant systems, time-dependent spin-wave theory is naturally formulated in terms of a dynamical transverse spin response. "Spin precession and spin waves in a chiral electron gas: beyond Larmor's theorem" (Karimi et al., 2016) treats a quasi-two-dimensional partially spin-polarized electron gas with Rashba and Dresselhaus spin-orbit coupling by linear-response TDDFT. Without spin-orbit coupling, the uniform transverse mode satisfies

$U(1)$8

which is Larmor’s theorem. With spin-orbit coupling, the small-$U(1)$9 dispersion becomes

$\mathbf q=0$0

where $\mathbf q=0$1 is the chiral momentum shift and $\mathbf q=0$2 acquires a second-order SOC correction, thereby quantifying the breakdown of Larmor’s theorem. The paper’s central message is that the collective mode frequency is determined by SOC acting inside a Coulomb-renormalized, time-dependent many-body response, not by SOC alone (Karimi et al., 2016).

A first-principles implementation of this general perspective is given in "Ab initio calculation of spin fluctuation spectra using time dependent density functional perturbation theory, planewaves, and pseudopotentials" (Cao et al., 2017). There the central observable is the generalized susceptibility tensor $\mathbf q=0$3, and the transverse channel

$\mathbf q=0$4

determines the neutron-scattering intensity through $\mathbf q=0$5. The method solves the time-dependent Sternheimer equation rather than summing over empty states, and it captures collective magnons, Stoner excitations, Landau damping, and soft paramagnon instabilities within one framework. The calculations for Fe, Ni, and Cr demonstrate, respectively, well-defined magnons, magnons strongly affected by the underlying exchange splitting, and a soft-paramagnon instability associated with the incommensurate spin-density-wave tendency in Cr (Cao et al., 2017).

The Liouville-Lanczos version of TDDFPT goes one step further by replacing frequency-by-frequency linear solves with a Krylov-space evaluation of the Liouvillian resolvent. In "Spin dynamics from time-dependent density functional perturbation theory" (Gorni et al., 2018), the response density matrix obeys

$\mathbf q=0$6

and the spin susceptibility is written as a matrix element of $\mathbf q=0$7. This yields an ab initio time-dependent spin-wave theory in which magnons and Stoner excitations appear on the same footing, without explicit empty-state sums and without solving separate response equations for every frequency (Gorni et al., 2018).

The same architecture is systematized in the code "turboMagnon" (Gorni et al., 2022). In that work, the dynamical spin susceptibility matrix is computed in a noncollinear spin-polarized framework with self-consistent SOC, and the pseudo-Hermitian Lanczos algorithm is reported to require only one Liouvillian application per iteration, giving an approximate factor-of-two speedup. The CrI$\mathbf q=0$8 monolayer example shows transverse magnon peaks in $\mathbf q=0$9 and $1/S$0, a SOC-induced magnon gap of about $1/S$1 meV, and the nontrivial appearance of mixed longitudinal-transverse susceptibility components once SOC is included (Gorni et al., 2022).

A related TD-SDFT construction for a Dirac system appears in "Spin waves in doped graphene: a time-dependent spin-density-functional approach to collective excitations in paramagnetic two-dimensional Dirac fermion gases" (Anderson et al., 2021). There the collective transverse mode is determined by

$1/S$2

and the spin wave is shown to be an xc effect absent in RPA. Using Slater and modified STLS kernels, the low-$1/S$3 dispersion takes the form

$1/S$4

with a pseudo-Larmor frequency $1/S$5 and a density- and polarization-dependent stiffness (Anderson et al., 2021).

4. Open-system, transport, and hydrodynamic generalizations

One major extension of time-dependent spin-wave theory treats magnons as open quantum modes. "Dynamics of Thermal Effects in the Spin-Wave Theory of Quantum Antiferromagnets" (Rivas et al., 2011) couples a Heisenberg antiferromagnet to a bosonic thermal bath, diagonalizes the LSWT Hamiltonian into $1/S$6 and $1/S$7 magnons, and derives a Lindblad-form master equation. The magnon decay rates are

$1/S$8

and the mode occupations relax exponentially toward thermal values. The staggered magnetization becomes explicitly time dependent,

$1/S$9

providing a genuinely dissipative, time-dependent spin-wave theory rather than an equilibrium finite-temperature expansion (Rivas et al., 2011).

In transport settings, the spin wave can be treated as a prescribed dynamical exchange texture coupled to itinerant electrons. "Magnon-driven chiral charge and spin pumping and electron-magnon scattering from time-dependent quantum transport combined with atomistic spin dynamics theory" (Suresh et al., 2019) uses a TDNEGF+ASD hybrid framework with a coherent traveling spin wave

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$0

coupled to electrons through

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$1

The key result is chiral pumping of charge and spin currents without dc bias and even without SOC, traced to retardation-induced nonadiabaticity rather than to the adiabatic spin-motive-force term alone (Suresh et al., 2019).

A much more macroscopic extension is "Spin waves in spin hydrodynamics" (Ambrus et al., 2022), which studies small perturbations of the spin polarization tensor in relativistic GLW hydrodynamics. Linearization around an unpolarized homogeneous background yields coupled equations for electric-like and magnetic-like spin variables, which reduce to a transverse wave equation

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$2

Only the transverse spin components propagate, and the spin-wave speed approaches one half of the speed of light in the ultra-relativistic limit. With dissipative corrections, the transverse modes acquire damping,

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$3

while the longitudinal components become diffusive (Ambrus et al., 2022).

5. Beyond linearity, finite size, and non-Hermitian dynamics

Several recent works show that time dependence alone does not rescue linear spin-wave theory when the relevant mode is intrinsically nonlinear or strongly interacting. "Rotor/spin-wave theory for quantum spin models with U(1) symmetry" (Roscilde et al., 2023) isolates the ${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$4 mode and shows that, once all nonlinearities are retained, it is exactly a ${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$5 quantum rotor rather than a harmonic boson. The Hamiltonian separates approximately as

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$6

with rotor sector

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$7

and finite-momentum spin-wave sector ${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$8. The resulting spectrum

${\cal U}(t)={\cal U}_0(t)\tilde{\cal U}(t)$9

reconstructs the Anderson tower of states, and the same separation provides a quantitatively accurate description of quench dynamics in regimes where ordinary linearization fails (Roscilde et al., 2023).

"Magnon spectrum of altermagnets beyond linear spin wave theory: Magnon-magnon interactions via time-dependent matrix product states vs. atomistic spin dynamics" (Garcia-Gaitan et al., 2024) pushes the issue from finite-size zero modes to genuine many-body continua. For a RuO$$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$0 effective spin model on a 4-leg cylinder, the TDMPS spectral function

$$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$1

shows that the upper magnon band is shifted and broadened relative to LSWT because it hybridizes with a three-magnon continuum through quartic HP terms. Classical ASD, although time dependent and nonlinear, does not reproduce the TDMPS spectrum, which the authors interpret as evidence that time dependence without full quantum many-body structure is insufficient in altermagnets and related antiferromagnetic systems (Garcia-Gaitan et al., 2024).

A different breakdown mechanism appears in "Breakdown of Linear Spin-Wave Theory in a Non-Hermitian Quantum Spin Chain" (Despres et al., 2023). There the non-Hermitian transverse-field Ising model is linearized around the $$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$2-polarized paramagnet, leading to a complex quadratic bosonic Hamiltonian and a complex Bogolyubov spectrum

$$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$3

For the excitation spectrum, this compares well with the exact Jordan–Wigner fermionic result in the large-field, large-dissipation regime. For quench dynamics, however, the normalized non-unitary evolution of bosonic correlators produces a finite-time divergence. The paper identifies the origin in the unbounded bosonic Hilbert space and the amplification of high-occupation sectors under non-Hermitian normalization, so the failure is much stronger than a gradual loss of quantitative accuracy (Despres et al., 2023).

6. Exact constraints, applications, and recurrent limitations

A decisive criterion for any time-dependent spin-wave theory is whether it respects the exact low-energy constraints of the underlying problem. In itinerant paramagnets, ordinary LSDA satisfies Larmor’s theorem, $$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$4, while "Spin waves with source-free time-dependent spin density functional theory" (Bologa et al., 2024) shows that the source-free xc magnetic-field construction violates this constraint and also gives the wrong long-wavelength behavior of ferromagnetic magnons. In the paramagnetic HEG, the source-free uniform mode becomes

$$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$5

and in the ferromagnetic HEG the long-wavelength mode becomes linear in $$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$6 rather than quadratic. The paper’s broader implication is explicit: enforcing $$\tilde{\cal H}(t)=\tilde{\cal H}_A(t)+\tilde{\cal H}_B(t),$$7 by projection is not sufficient for a reliable dynamical xc kernel (Bologa et al., 2024).

Across the broader field, the recurring approximations are equally clear. Quadratic localized-spin theories typically assume small-amplitude magnons, neglect nonlinear magnon-magnon scattering, and often omit damping or full demagnetizing fields. TDDFPT calculations usually work in ALSDA or ALDA and inherit the limitations of the underlying ground-state band structure, as illustrated by the persistent Ni problem. Transport-coupled theories may prescribe rather than self-consistently evolve the spin wave. Hydrodynamic theories are linearized around unpolarized homogeneous backgrounds. Tensor-network and exact methods remove some of these limitations but typically at restricted dimensionality or geometry (Borys et al., 2015, Cao et al., 2017, Suresh et al., 2019, Ambrus et al., 2022, Garcia-Gaitan et al., 2024).

The main applications already span several scales. In local-moment magnets, time-dependent spin-wave theory describes rotating magnets, field-driven reorganization, domain-wall scattering, magnon-induced wall motion, and periodic magnonic structures. In itinerant systems, it provides susceptibility-based access to magnons, Stoner continua, paramagnons, anisotropy gaps, SOC-induced nonreciprocity, and soft magnetic instabilities. In open or hybrid settings, it covers thermalization, damping, retardation-induced charge and spin pumping, and relativistic spin transport. A plausible synthesis is that the field has evolved from “magnons as harmonic bosons around a static order parameter” toward “magnons as collective poles, soft coordinates, or dynamical spectral features embedded in a time-dependent many-body background,” with the appropriate formulation determined by whether the dominant physics is geometric, susceptibility-based, dissipative, hydrodynamic, or genuinely beyond LSWT (Rückriegel et al., 2011, Gorni et al., 2022, Roscilde et al., 2023).