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AIP conference proceedings

Transverse susceptibilities of ferromagnetic spin system interacting with phonon reservoir in the spin-wave approximation

Mizuhiko Saeki

Published Jan 1, 2013• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.4826031

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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.
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.
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 spin-1/2 quantum Heisenberg ferromagnet with anisotropic spin exchange interactions in one dimension is investigated by means of the Green's function method and the spin-wave theory. In this paper we use the Holstein-Primakoff transformation to get the approximately Hamiltonian. We show that the magnetic field and anisotropy can have a rather drastic effect on the correlation functions and susceptibilities in the low-temperature region. Our results show that inverse correlation lengths kx and ky are independent of the temperature in the weak magnetic field. kx is less dependent on the anisotropy than ky. These results are consistent with the theoretical and experimental results.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
In the paper the Pair Approximation (PA) method for studies of the site-diluted spin-1/2 systems of arbitrary dimensionality with the long-range ferromagnetic interactions is adopted. The method allows to take into account arbitrary anisotropy of the interactions in the spin space, so it is not limited to purely Ising couplings. Within this approach, the Gibbs free energy is obtained, which allows to derive all the further interesting thermodynamic properties. In particular, we obtain an equation for the critical temperature of the second-order phase transitions for the model in question. In the study we focus our attention on the systems with ferromagnetic interactions decaying with the distance according to the power law J(r)∝r−nJ(r)\propto r^{-n}J(r)∝r−n. We discuss the dependence of the critical temperature on the concentration of magnetic component and the index nnn for selected one-, two- and three-dimensional lattices.We confirm the absence of the critical concentration for a diluted magnet with infinite interaction range. In the regime of the low concentrations of magnetic component, we find a non-linear increase of the critical temperature with the concentration in the form of Tc∝pn/dT_{c}\propto p^{n/d}Tc​∝pn/d, depending on the system dimensionality ddd and the index nnn, whereas n>dn > dn>d.
We study the transverse dynamical spin susceptibility of the two-dimensional U(1) spinon Fermi-surface spin liquid in a small applied Zeeman field. We show that both short-range interactions, present in a generic Fermi liquid, as well as gauge fluctuations, characteristic of the U(1) spin liquid, qualitatively change the result based on the frequently assumed noninteracting spinon approximation. The short-range part of the interaction lead to a collective ``spinon spin wave'' mode, which splits off from the two-spinon continuum at a small momentum and disperses downward. Gauge fluctuations renormalize the susceptibility, providing nonzero power-law weight in the region outside the spinon continuum and giving the spin wave a finite lifetime, which scales as momentum squared. We also study the effect of Dzyaloshinskii-Moriya anisotropy on the zero momentum susceptibility, which is measured in electron spin resonance (ESR), and obtain a resonance linewidth linear in temperature and varying as B2/3{B}^{2/3}B2/3 with magnetic field BBB at low temperatures. Our results form the basis for a theory of inelastic neutron scattering, ESR, and resonant inelastic x-ray scattering studies of this quantum spin-liquid state.