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SCIREA Journal of Physics

Resonance Absorption and Transverse Magnetization of an Anti-Ferromagnetic Spin System Interacting with a Phonon Reservoir in the Spin-Wave Region

Mizuhiko Saeki

Published Jan 13, 2026• 0 citations
Publisher

Abstract

A form of the transverse magnetic susceptibility is derived and the resonance absorption and transverse magnetization are studied for an anti-ferromagnetic spin system interacting with a phonon reservoir in the spin-wave region, employing the TCLE method of linear response in terms of the non-equilibrium thermo-field dynamics (NETFD), which is reformulated for the revised spin-phonon interaction taken to reflect the energy transfer between the spin system and phonon reservoir. Here, the TCLE method of linear response is a method in which the admittance of a physical system is directly derived from time-convolutionless equations with external driving terms. The approximate formulas of the resonance frequencies, peak-heights (heights of peak) and line half-widths in the resonance region of the power absorption and the amplitude of the expectation value of the transverse magnetization, which is referred as "the magnetization-amplitude", are derived for the anti-ferromagnetic system in a transversely rotating magnetic-field. For an anti-ferromagnetic system of one-dimensional infinite spins in the transversely rotating magnetic-field, the power absorption and magnetization-amplitude are investigated numerically in the region valid for the lowest spin-wave approximation. The approximate formulas of the resonance frequencies, peak-heights and line half-widths, are shown to coincide well with the results investigated calculating numerically the analytic results of the power absorption and magnetization-amplitude in the resonance region, and also are shown to satisfy "the narrowing condition" that as phonon reservoir is damped quickly, the peak-heights increase and the line half-widths decrease, and thus are verified numerically. In the resonance region of the power absorption and magnetization-amplitude, it is shown that as the temperature T becomes high, the resonance frequencies increase slightly, the peak-heights decrease and the line half-widths increase, and that as the wave number k becomes large, the resonance frequencies and peak-heights increase, and the line half-widths decrease, and also that as the spin-magnitude S becomes large, the resonance frequencies and peak-heights of the power absorption and magnetization-amplitude become large. The effects of the memory and initial correlation for the spin system and phonon reservoir, which are represented by the interference terms in the TCLE method and are referred as "the interference effects", are confirmed to increase the power absorption and magnetization-amplitude in the resonance region, and are shown to produce effects that cannot be disregarded for the high temperature, for the non-quickly damped reservoir or for the small wave number k.

DOI
10.54647/physics140709

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A form of the transverse magnetic susceptibility is derived and the resonance absorption and transverse magnetization are studied for an anti-ferromagnetic spin system interacting with a phonon reservoir in the spin-wave region, employing the TCLE method of linear response in terms of the non-equilibrium thermo-field dynamics (NETFD), which is reformulated for the revised spin-phonon interaction taken to reflect the energy transfer between the spin system and phonon reservoir. Here, the TCLE method of linear response is a method in which the admittance of a physical system is directly derived from time-convolutionless equations with external driving terms. The approximate formulas of the resonance frequencies, peak-heights (heights of peak) and line half-widths in the resonance region of the power absorption and the amplitude of the expectation value of the transverse magnetization, which is referred as "the magnetization-amplitude", are derived for the anti-ferromagnetic system in a transversely rotating magnetic-field. For an anti-ferromagnetic system of one-dimensional infinite spins in the transversely rotating magnetic-field, the power absorption and magnetization-amplitude are investigated numerically in the region valid for the lowest spin-wave approximation. The approximate formulas of the resonance frequencies, peak-heights and line half-widths, are shown to coincide well with the results investigated calculating numerically the analytic results of the power absorption and magnetization-amplitude in the resonance region, and also are shown to satisfy "the narrowing condition" that as phonon reservoir is damped quickly, the peak-heights increase and the line half-widths decrease, and thus are verified numerically. In the resonance region of the power absorption and magnetization-amplitude, it is shown that as the temperature T becomes high, the resonance frequencies increase slightly, the peak-heights decrease and the line half-widths increase, and that as the wave number k becomes large, the resonance frequencies and peak-heights increase, and the line half-widths decrease, and also that as the spin-magnitude S becomes large, the resonance frequencies and peak-heights of the power absorption and magnetization-amplitude become large. The effects of the memory and initial correlation for the spin system and phonon reservoir, which are represented by the interference terms in the TCLE method and are referred as "the interference effects", are confirmed to increase the power absorption and magnetization-amplitude in the resonance region, and are shown to produce effects that cannot be disregarded for the high temperature, for the non-quickly damped reservoir or for the small wave number k.
A form of the transverse magnetic susceptibility of a ferromagnetic spin system with a uniaxial anisotropy energy and an anisotropic exchange interaction, interacting with a phonon reservoir, is derived in the spin-wave approximation in terms of the non-equilibrium thermo-field dynamics (NETFD) using the TCLE method. The transverse magnetic susceptibility for the ferromagnetic system of one-dimensional infinite spins is studied by assuming a damped oscillator model of the phonon reservoir. The higher-order effects in the spin-wave approximation are numerically investigated. It is shown that the higher-order parts cannot be neglected for the spin system of the small magnitude and for the high temperature system, but may be neglected for the spin system of the large magnitude.
For a quantal spin S (magnitude 1/2) interacting weakly with heat bath, the fluctuation-dissipation theorem, dispersion relation and sum rules are discussed by the method of TCL equations with external driving terms (the TCLE method). The transverse susceptibility of the quantal spin is numerically calculated using the TCLE method for the case of a boson-like bath including a quickly damped subsystem, comparing with the results obtained using the relaxation method under decoupled initial conditions of the spin and bath. It is shown that the interference terms in the TCLE method produce the effects which enhance the peaks of line shapes and abase the line shapes at the small frequency region. It is also shown that the peaks of line shapes are enhanced as the temperature becomes low, as shown for the case of a spin-like bath. For the case of a boson-like bath including a subsystem with finite damping constants, the transverse susceptibility is calculated using the TCLE method numerically.
The transverse susceptibility of a quantal spin (magnitude 1/2) interacting weakly with its surroundings (quantal heat bath) is discussed using the method of the TCL equation with external driving terms (the TCLE method) analytically and numerically. The susceptibility obtained by the TCLE method under decoupled initial conditions is shown to coincide with that obtained under thermal equilibrium initial conditions. The results obtained with the TCLE method are compared with those obtained with the relaxation method. It is shown that the susceptibility in the relaxation method includes the second term of the interference terms in the TCLE method, but not its first term. With numerical calculations for the case of a spin-like bath including a quickly damped subsystem, it is shown that the interference terms in the TCLE method give larger effects as the temperature becomes lower, that its first term enhances the peaks of line shapes slightly, and that the peaks of line shapes are enhanced as the temperature becomes low. A model which shows that the TCLE method is simpler than the relaxation method is investigated numerically using the TCLE method.
The transverse component of the dynamic form factor Sx(k,ω)\mathscr{S}^{x}(k, \omega )Sx(k,ω) of the Ising-like linear chain at various temperatures is studied by the perturbation theory about the transverse part of the Hamiltonian. The dynamic form factor Sx(k,ω)\mathscr{S}^{x}(k, \omega )Sx(k,ω) for both the ferromagnetic and antiferromagnetic couplings has the three-peak structure for k =0 at finite temperatures, and this structure persists even for any wave vector k and at any high temperature. Many interesting characters in Sx(k,ω)\mathscr{S}^{x}(k, \omega )Sx(k,ω) are obtained. The k -dependence of width of the absorption peak is remarkable in the antiferromagnetic case. The absorption peak has a finite width even at absolute zero except for k = π/2 in the antiferromagnetic case, while the shape of the absorption peak is reduced to a infinitely sharp one as the temperature is decreased to absolute zero in the ferromagnetic case.
This paper presents numerical results of the ESR absorption spectrum for S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY model in one dimension. We obtain detailed numerical results for a wide range of temperature and magnetic field for a large system of 400 sites, using the method developed by Derzhko et al. In the high-temperature regime, a similar behavior is observed for both the ferromagnet and the antiferromagnet; these line shapes have only one peak. On the other hand, in the low-temperature regime, line shapes show very different behaviors depending on the sign of the exchange coupling. While the line shape for the ferromagnet has only one peak as in the high-temperature regime, the line shape for antiferromagnet has two peaks and shows very strong magnetic-field dependence. We also discuss the physical meaning by comparing our numerical results with several theoretical results: the exact result in the high-temperature limit, Kubo-Tomita theory, field theory, mean-field theory, and elementary excitations.
The dynamics of an Ising system is studied assuming that a microwave radiation field is coupled to the magnetic system through the transverse spin components. The transverse dynamic susceptibility is calculated with the use of the Green-function formalism in an exact manner, without the use of any decoupling scheme. The method yields an exact eigenvalue equation for the resonance frequencies, valid for any dimension and lattice, and a physical meaning of these resonances can be given in terms of the low-lying excitations for an Ising system. Those excitations correspond to spin-flip transitions in elementary clusters which represent all the possible spin arrangements of nearest neighbors. The present method does not allow for a calculation of line intensities without the help of statistical mechanics to evaluate correlation functions.