Non-Adiabatic Longitudinal Fluctuations

• Non-adiabatic longitudinal vector fluctuations are dynamic deviations in the vector field’s longitudinal component, characterized by non-Lorentzian spectra and temperature-dependent relaxation.
• They result from spin anharmonicity and non-linear mode-mode coupling, leading to features such as quasielastic peaks, inelastic resonances, and logarithmic spectral anomalies.
• These fluctuations significantly impact itinerant antiferromagnets by modulating neutron scattering signatures and signaling precursors to magnetic phase transitions.

Non-adiabatic longitudinal vector fluctuations are departures from slow, equilibrium dynamics in the longitudinal component of a vector field or excitation, resulting in enhanced spectral features, nontrivial relaxation mechanisms, and significant dynamical coupling between longitudinal and transverse modes. These fluctuations are prevalent in diverse fields: itinerant magnetism, quantum transport, astrophysical plasmas, and early-universe cosmology. Their phenomenology manifests through non-Lorentzian spectra, anomalous damping rates, coupling-induced resonances, and amplified noise or instability in transport and collective observables.

1. Spectrum Structure in Itinerant Antiferromagnets

Longitudinal spin fluctuations (SF) in isotropic itinerant antiferromagnets possess a spectrum decomposed into quasielastic and inelastic components. The conventional Lorentzian form emerges only in the linear (Landau-damping) regime, characterized by a dynamical susceptibility of the form

$$\chi_l(k, \omega) = X_l(k)/[\omega_{sf}(k) - i\omega] \quad (2)$$

with spectral half-width $A(k) = \omega_{sf}(k) \sim |k|$ (Eq. (32)). However, accounting for non-linear mode-mode coupling via spin anharmonicity fundamentally alters the spectrum:

• Quasielastic component: Central peak or dip, shape strongly temperature and $\omega_{sf}$ dependent.
• Inelastic satellites: Resonances or antiresonances located near magnon frequencies $\omega = \pm \omega_m(k)$ (see Eq. (34)), with width scaling as $A(k)\sim k^2$ in the strong-coupling regime (Eq. (35)).
• Logarithmic anomalies: Inelastic satellites exhibit logarithmic singularities near magnon energies, $L(k,T)\sim \ln|\omega \pm \omega_m(k)|$ (Eq. (22)).

This non-adiabatic behavior reflects underlying dynamical interactions that cannot be captured by simple adiabatic or single-magnon models.

2. Spin Anharmonicity and Mode-Mode Coupling

Spin anharmonicity—non-linear three-mode interactions in the effective Ginzburg–Landau Hamiltonian (Eq. (3))—couples transverse (magnon) and longitudinal modes, with major consequences:

• Renormalized static susceptibilities: Longitudinal susceptibility $X_l'(k) = X_l(k) + Y (2m_t^2 + 3m_l^2)$ (Eq. (6)), introducing temperature dependence via fluctuating amplitudes.
• Enhanced relaxation channels: Bare Landau damping rate $\omega_{sf}(k)$ is supplemented by non-linear corrections $\Delta\omega_{sf}(k, T)$ (Eq. (7)), sourced by three-mode scattering.
• Dynamic mixing: Coupling yields new relaxation pathways for longitudinal excitations, resulting directly in the observed non-adiabatic spectrum.

This microscopic interaction mechanism is indispensable for the observed spectral features, such as central dips and satellite antiresonances.

3. Temperature Dependence and Relaxation Rates

Temperature controls both the amplitude of spin fluctuations and the efficiency of non-linear scattering:

• Low $T$ (weak coupling): Quasielastic spectrum dominates, narrow ($\sim|k|$ width), nearly Lorentzian.
• High $T$ (strong coupling): Central peak broadens ($\sim k^2$ width), or transitions to a central dip, flanked by intense inelastic features.
• Regime boundaries: Transition from peak to dip is governed by inequalities involving the coupling parameter $\check{S}_0(k,T)$ and $n(k) = \omega_{sf}(k)/\omega_m(k)$. For example, $3[1+\check{S}_0(k,T)] > n^2(k)$ yields a peak, while $(1+\check{S}_0(k,T))^3 < n^2(k)$ yields a dip (Eqs. (27–28)).

Thus, temperature drives a crossover from adiabatic to non-adiabatic longitudinal fluctuation regimes.

4. Inelastic Resonances and Antiresonances

In the presence of strong mode-mode coupling, the longitudinal spectrum develops notable inelastic features:

• Resonant enhancement: Satellite peaks at $\omega\approx \pm \omega_m(k)$, with their intensity and line shape tied to non-adiabatic coupling strength.
• Antiresonances: At higher coupling and temperature, satellites develop asymmetric line shapes and antiresonant profiles due to the logarithmic divergence in $L(k,T)$ (Eq. (22)).
• Non-propagating character: These inelastic longitudinal excitations do not exhibit magnon-like propagation, but are linked to dynamic scattering processes among the spin modes.

The full spectral response thus reflects non-adiabatic dynamics at energies near transverse magnon excitations.

5. Phenomenological Model and Theoretical Framework

The paper constructs an explicit phenomenological theory integrating linear response, Ginzburg–Landau dynamics, and non-linear mode coupling:

• Transverse and longitudinal susceptibilities: $\chi_t(k, \omega)$ and $\chi_l(k, \omega)$ (Eqs. (1–2)).
• Ginzburg–Landau Hamiltonian: Explicit three-mode (anharmonic) interactions (Eq. (3)), coupled time-dependent equations (Eqs. (4–5)).
• Spectral intensity formula: Full $I(k, T)$ incorporating both quasielastic (central) and inelastic (satellite) features (Eq. (17)).
• Mode-mixing and relaxation: Non-linear relaxation rate corrections, renormalized susceptibilities, and explicit dependence on temperature and wavevector.

This framework accurately captures both adiabatic and non-adiabatic longitudinal vector fluctuations and their dependence on physical parameters.

6. Connections to Experiment and Broader Implications

The non-adiabatic longitudinal vector fluctuations described in this model underlie:

• Neutron scattering spectral anomalies: Central peaks/dips and satellite features in the dynamic susceptibility observed as a function of wavevector and temperature.
• Magnetic instability in correlated electron systems: Enhanced longitudinal fluctuations serve as precursors to phase transitions, especially in systems nearing quantum criticality.
• Generalized mode-coupling in quantum solids: Similar processes manifest in unconventional superconductors, itinerant magnets, and low-dimensional quantum fluids.

The framework is broadly applicable for the interpretation of dynamical spin susceptibility measurements and for understanding nontrivial temperature-dependent magnetic behavior in itinerant electron systems.

Table: Evolution of Spectral Features with Increasing Non-Adiabaticity (as driven by temperature)

Regime	Central Feature	Inelastic Satellites
Weak coupling	Lorentzian peak (width ∼	k
Moderate coupling	Non-linear peak (width ∼ k²)	Emerging resonances at ω_m
Strong coupling	Central dip	Intense satellites/antiresonances

Increasing mode–mode coupling (by raising temperature or system anharmonicity) transforms the longitudinal SF spectrum from a simple Lorentzian peak into a complex non-adiabatic structure, dominated by dynamical interactions between spin components.

Summary

Non-adiabatic longitudinal vector fluctuations arise from non-linear interactions among spin components in isotropic itinerant antiferromagnets, producing spectral structures far richer than single-mode Lorentzian predictions. Spin anharmonicity leads to dynamic coupling between transverse and longitudinal modes, with explicit temperature dependence and non-linear corrections to relaxation rates. The resulting spectrum comprises central features (peaks or dips) and inelastic satellites, evolving from adiabatic to strongly non-adiabatic behavior as coupling increases. The phenomenological model given provides a quantitative foundation for understanding these effects in terms of time-dependent Ginzburg–Landau theory and dynamical susceptibilities, with direct experimental relevance for probing magnetic quantum matter.