Spin friction of exchange-coupled local moments beyond the perturbative regime: nonlocal Gilbert damping at strong coupling, electronic criticality, and strong correlation, with retardation (inertia) and multi-orbital extensions

Local magnetic moments exchange-coupled to metallic conduction electrons experience dissipative 'spin friction' even without spin-orbit coupling. Lenzing, Krüger, and Potthoff [Phys. Rev. B 111, 014402 (2025)] derived the resulting Gilbert damping microscopically for classical moments in a single-band tight-binding host, showing it is nonlocal and time-dependent, exactly expressible through the nonlocal density of states for noninteracting electrons, and controllably enhanced in nonlocality by Hubbard correlations treated in the random-phase approximation (RPA). Their effective spin-only theory is, however, controlled only for weak exchange ($J\rho_{mm}\ll 1$), nearly adiabatic spin motion, and noninteracting or RPA-treated electrons. This problem asks for the microscopic — where possible nonperturbative — theory of spin friction and of the resulting relaxation dynamics in each regime where that perturbative description is uncontrolled: strong exchange coupling $J\rho_{mm}\gtrsim 1$ and the crossover to breakdown of the effective theory; at and above electronic ordering instabilities, where RPA predicts divergent, strongly nonlocal damping with a spatial pattern $\propto e^{i\boldsymbol{k}\cdot\boldsymbol{R}}$ that would render the naive dynamics nondissipative; strongly correlated conduction electrons with a controlled computation of the damping matrix $\alpha_{mm'}$ beyond RPA; realistic multi-orbital electronic structures with band degeneracies; and systematic higher-order retardation terms (magnetic inertia and nutation). The common objective is to establish the limits of validity of the spin-only effective description and the true dissipative spin dynamics beyond them.

Background

When a local magnetic moment — for example, that of a magnetic adatom on a metal surface — is exchange-coupled to the host's conduction electrons, moving the moment drags along a spin-polarized electron cloud that lags behind and exerts a retarding torque. This mechanism produces dissipative 'spin friction' even in the absence of spin-orbit coupling. Phenomenologically it is described by the Gilbert damping term in the Landau–Lifshitz–Gilbert (LLG) equation of magnetization dynamics, $\dot{\boldsymbol{S}}_m = \boldsymbol{B}_m \times \boldsymbol{S}_m + \boldsymbol{S}_m \times \sum_{m'} \alpha_{mm'}\,\dot{\boldsymbol{S}}_{m'}$, where $\boldsymbol{S}_m$ is a classical unit vector representing the moment at site $m$, $\boldsymbol{B}_m$ the effective field, and $\alpha_{mm'}$ the Gilbert damping; in standard phenomenology $\alpha$ is a single local scalar, but a microscopic treatment shows it is a matrix coupling different sites ($m'\neq m$, 'nonlocal damping') and depends on time since excitation.

The standard microscopic model is the $s$-$d$ exchange (Vonsovsky–Zener) or multi-impurity Kondo model with classical spins: a tight-binding band of quantum conduction electrons coupled locally, with exchange strength $J$, to a few classical spins. Lenzing, Krüger, and Potthoff derived the spin friction in this model both by linear response (Kubo) theory and by adiabatic response theory. For noninteracting electrons, the nonlocal Gilbert damping $\alpha_{mm'}$ is exactly expressible through the nonlocal tight-binding density of states near the Fermi energy, and the dimensionless product $J\rho_{mm}$ of exchange strength and local density of states $\rho_{mm}$ emerges as the control parameter for the reliability of the effective theory: the theory requires weak exchange $J\rho_{mm}\ll 1$ and breaks down when $\rho_{mm}$ diverges, e.g. at a van Hove singularity (a divergence of the density of states at a band saddle point). The derivation also truncates the memory (retardation) of the electron system at first order in the retardation time $\tau_{\rm ret}$, which is what yields a purely Gilbert-type damping.

Treating on-site Coulomb repulsion $U$ (the Hubbard model) at the random-phase approximation (RPA — summing the chain of electron-hole bubble and interaction ladder diagrams, a standard controlled scheme only at weak coupling) at half filling, the same authors found a strong enhancement of the nonlocality of spin friction with increasing $U$. Approaching the antiferromagnetic ordering transition at $U=U_c$, the RPA damping diverges and becomes strongly nonlocal, with a distance dependence $\alpha(\boldsymbol{R})\propto e^{i\boldsymbol{k}\cdot\boldsymbol{R}}$ at the ordering wave vector $\boldsymbol{k}=(\pi,\pi)$, which within the naive effective theory would imply spin dynamics without any dissipation — a self-consistent signal that the spin-only effective theory itself fails at the critical point.

Several adjacent lines exist but leave the core open. Time-dependent density-matrix renormalization group studies (tDMRG, a tensor-network method for the real-time dynamics of quantum many-body systems) of the quantum-spin Kondo impurity model show a transient, rapidly damped nutation (a small superimposed wobbling of the spin axis, analogous to that of a spinning top, arising when the spin carries effective inertia), indicating that higher-order retardation effects are real but are not yet incorporated systematically in an effective spin-only theory [Sayad, Rausch, Potthoff, EPL 116, 17001 (2016)]. Ab initio schemes for the real-space nonlocal Gilbert damping of bulk ferromagnets and their surfaces exist for noninteracting electrons [Nagyfalusi, Szunyogh, Palotás, Phys. Rev. B 109, 094417 (2024)], and Schwinger–Keldysh nonequilibrium field theory (a real-time, closed-time-path formulation of quantum field theory for systems driven out of equilibrium) has been used to derive intrinsically nonlocal, nonuniform damping tensors made anisotropic by spin-orbit coupling in metallic ferromagnets and insulator/metal heterostructures [Reyes-Osorio and Nikolić, Phys. Rev. B 109, 024413 (2024); Phys. Rev. B 110, 214432 (2024)], as well as a Keldysh-derived LLG equation with Gilbert damping for two adatoms on a Rashba spin-orbit-coupled two-dimensional electron gas (a two-dimensional carrier system in which structural inversion asymmetry produces linear-in-momentum spin-orbit coupling) [Huddie and Duine, arXiv:2601.19604]. All of these are perturbative in the exchange coupling and treat noninteracting electrons, so they do not touch the strongly coupled or strongly correlated regimes. A distinct 2026 discussion concerns the microscopic origin of magnetic inertia via magnon–phonon coupling or unquenched orbital angular momentum [Webb, Gan, Zhang, arXiv:2607.01398; Moussa et al., Phys. Rev. B 114, 024424 (2026)], mechanisms different from the retardation of exchange-induced spin friction. The unresolved target is the microscopic theory of spin friction — the damping matrix $\alpha_{mm'}$ and its higher-order retardation extensions — and the emergent relaxation dynamics, in the regimes where the weak-coupling, adiabatic, RPA-based effective spin-only theory of Lenzing et al. is uncontrolled.

Problem Statement

Establish the microscopic theory of spin friction — the dissipative coupling of local classical magnetic moments to conduction electrons via the exchange interaction, quantified by the nonlocal Gilbert damping matrix $\alpha_{mm'}$ and, beyond first order in the retardation time, by the corresponding higher-order retardation coefficients — outside the regime in which the effective spin-only theory of Lenzing, Krüger, and Potthoff is controlled (weak exchange $J\rho_{mm}\ll 1$, nearly adiabatic spin motion, noninteracting or RPA-treated electrons). The common objective is to determine the spin friction and the emergent relaxation dynamics of the moments, with microscopic and where possible nonperturbative control, in each regime where that perturbative description is uncontrolled, and thereby the limits of validity of the spin-only effective theory itself. Concretely, the source's conclusions list several main lines of relevant future research; the directions pursued here are: (1) the behavior of spin friction and relaxation in the strong-exchange-coupling regime $J\rho_{mm}\gtrsim 1$ and the crossover to the regime where the effective theory fails, requiring nonperturbative exploration; (2) the relaxation dynamics at and above an electronic ordering instability (e.g. $U\to U_c$ and $U>U_c$ in the half-filled Hubbard model), where RPA predicts divergent, strongly nonlocal damping with spatial pattern $\propto e^{i\boldsymbol{k}\cdot\boldsymbol{R}}$ that would imply vanishing dissipation within the naive effective theory, including the role of fluctuations beyond RPA; (3) a controlled computation of $\alpha_{mm'}$ for strongly correlated conduction electrons beyond RPA or Hartree–Fock, and the impact of such correlations on long-time spin relaxation; (4) the extension to multi-orbital and more realistic material-specific electronic structures, possibly combined with ab initio band-structure calculations, and the quantitative consequences of band degeneracies for the nonlocal damping; and (5) the systematic inclusion of higher-order retardation effects beyond the first-order truncation that yields Gilbert damping — magnetic inertia and nutation — with an assessment of their quantitative importance on experimentally relevant time scales. These directions are independent rather than jointly required. Two further lines listed by the source are noted but not pursued here: geometrical spin torque in gapped and semi-metallic systems, and a theory of quantum (rather than classical) impurity spins. In each of these regimes, the task is to determine the effective dissipative spin dynamics from the underlying quantum-classical model — or, where no spin-only effective description exists (as expected at the ordering transition), to establish that with microscopic control — and thereby to delimit where the weak-coupling, adiabatic, RPA-based theory of the source applies.

References

1. Nicolas Lenzing, David Krüger, and Michael Potthoff, 'Microscopic theory of spin friction and dissipative spin dynamics', Physical Review B 111, 014402 (2025). DOI: 10.1103/PhysRevB.111.014402; arXiv:2410.16003. https://doi.org/10.1103/PhysRevB.111.014402
2. Nicolas Lenzing, David Krüger, and Michael Potthoff, 'Geometrical torque on magnetic moments coupled to a correlated antiferromagnet', Physical Review Research 5, L032012 (2023). DOI: 10.1103/PhysRevResearch.5.L032012; arXiv:2304.02071. https://doi.org/10.1103/PhysRevResearch.5.L032012
3. Mohammad Sayad, Roman Rausch, and Michael Potthoff, 'Inertia effects in the real-time dynamics of a quantum spin coupled to a Fermi sea', EPL 116, 17001 (2016). DOI: 10.1209/0295-5075/116/17001; arXiv:1609.05526. https://doi.org/10.1209/0295-5075/116/17001
4. Balázs Nagyfalusi, László Szunyogh, and Krisztián Palotás, 'Real-space nonlocal Gilbert damping from exchange torque correlation applied to bulk ferromagnets and their surfaces', Physical Review B 109, 094417 (2024). DOI: 10.1103/PhysRevB.109.094417; arXiv:2401.09938. https://doi.org/10.1103/PhysRevB.109.094417
5. Felipe Reyes-Osorio and Branislav K. Nikolić, 'Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory: Intrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling', Physical Review B 109, 024413 (2024). DOI: 10.1103/PhysRevB.109.024413; arXiv:2306.13013. https://doi.org/10.1103/PhysRevB.109.024413
6. Felipe Reyes-Osorio and Branislav K. Nikolić, 'Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or altermagnetic metallic overlayer: A Schwinger-Keldysh field theory approach', Physical Review B 110, 214432 (2024). DOI: 10.1103/PhysRevB.110.214432; arXiv:2312.09140. https://doi.org/10.1103/PhysRevB.110.214432
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10. Tarek Moussa, Darpa Narayan Basu, Ritwik Mondal, and Akashdeep Kamra, 'Unquenched orbital angular momentum as the origin of spin inertia', Physical Review B 114, 024424 (2026). DOI: 10.1103/9h9p-8qzd; arXiv:2603.29421. https://doi.org/10.1103/9h9p-8qzd
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