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Physical review. B./Physical review. B

Magnon peak lineshape in the transverse dynamical structure factor of a ferromagnetically polarized easy-axis XXZ chain at low temperatures

P. N. Bibikov

Published Dec 19, 2023• 0 citations
Publisher

Abstract

The ferromagnetically polarized gapped XXZ spin chain is studied at low temperatures. Utilizing only the one- and two-magnon spectrums and focusing on the magnon-creation contribution to the transverse dynamical susceptibility, we represent the latter in the form of the Dyson equation. Then, following the well known correspondence between the imaginary part of magnetic susceptibility and dynamical structure factor, we get the low-temperature formula for the magnon-peak lineshape. The suggested approach is effective only if the processes related to magnon creations and to transitions from magnons to coupled magnon pairs are energetically separated. As it is shown in the paper, such separation is inherent in the easy-axis chains with rather strong anisotropy. We present several plots and discuss their lineshapes. As the supplemental result we obtain integral representations for the temperature-dependent magnon resonance shift and the parameter which is usually associated with the decay rate. The low-temperature behavior of the resonance shift is studied in details. All calculations are performed up to controllable error $o({e}^{\ensuremath{-}\ensuremath{\beta}{E}_{\mathrm{gap}}})$.

DOI
10.1103/physrevb.108.214421

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The ferromagnetically polarized gapped XXZ spin chain is studied at low temperatures. Utilizing only the one- and two-magnon spectrums and focusing on the magnon-creation contribution to the transverse dynamical susceptibility, we represent the latter in the form of the Dyson equation. Then, following the well known correspondence between the imaginary part of magnetic susceptibility and dynamical structure factor, we get the low-temperature formula for the magnon-peak lineshape. The suggested approach is effective only if the processes related to magnon creations and to transitions from magnons to coupled magnon pairs are energetically separated. As it is shown in the paper, such separation is inherent in the easy-axis chains with rather strong anisotropy. We present several plots and discuss their lineshapes. As the supplemental result we obtain integral representations for the temperature-dependent magnon resonance shift and the parameter which is usually associated with the decay rate. The low-temperature behavior of the resonance shift is studied in details. All calculations are performed up to controllable error o(e\ensuremath−\ensuremathβEgap)o({e}^{\ensuremath{-}\ensuremath{\beta}{E}_{\mathrm{gap}}})o(e\ensuremath−\ensuremathβEgap​).
Magnon peak lineshape in the transverse dynamical structure factor of a magnetically polarized easy-axis XXZXXZXXZ chain at low temperatures
The ferromagnetically polarized gapped XXZ spin chain is studied at low temperatures. Utilizing only the one- and two-magnon spectrums and focusing on the magnon-creation contribution to the transverse dynamical susceptibility, we represent the latter in the form of the Dyson equation. Then, following the well known correspondence between the imaginary part of magnetic susceptibility and dynamical structure factor, we get the low-temperature formula for the magnon-peak lineshape. The suggested approach is effective only if the processes related to magnon creations and to transitions from magnons to coupled magnon pairs are energetically separated. As it is shown in the paper, such separation is inherent in the easy-axis chains with rather strong anisotropy. We present several plots and discuss their lineshapes. As the supplemental result we obtain integral representations for the temperature-dependent magnon resonance shift and the parameter which is usually associated with the decay rate. The low-temperature behavior of the resonance shift is studied in details. All calculations are performed up to {\it controllable} error o(e−βEgap)o({\rm e}^{-βE_{gap}})o(e−βEgap​).
Abstract The Dyson equation for the real two-time commutator retarded one-magnon Green function of the ferromagnetically polarized XX chain is suggested following the Plakida–Tserkovnikov algorithm. Starting from this result a low-temperature integral representation for the corresponding magnon self-energy is obtained by the truncated form factor expansion, however, without any resummations. Within the suggested approach, the low-temperature asymptotics of the transverse dynamical structure factor may be readily studied. Some obtained line shapes are presented.
Suggesting two versions for the Plakida-Tserkovnikov algorithm breakdown in the low-temperature regime, we derive ordinary and nested Dyson equations for the transverse autocorrelator of the magnetically polarized Ising chain. Using them we get the corresponding low-temperature asymptotics for the autocorrelator. We show that the ordinary Dyson equation results in a correct o(e−βEgap)o({\rm e}^{-βE_{gap}})o(e−βEgap​) account of the magnon creation process, while the nested Dyson equation additionally gives the correct o(e−βEgap)o({\rm e}^{-βE_{gap}})o(e−βEgap​) contribution associated with transitions from magnons to bound two-magnon states. The obtained result may be useful for the extension of the suggested approach on the polarized XXZXXZXXZ-chain.
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Low-temperature asymptotics for the transverse dynamical structure factor for a magnetically polarized XXXXXX chain at small and negative frequencies
Using the truncated form factor expansion the low-temperature asymptotics for the transverse dynamical structure factor of the magnetically polarized XXXXXX chain is studied. Unlike the previous paper we do not use the representation of structure factor in terms of the corresponding magnetic susceptibility. This enables to obtain correct results at small and negative frequencies.
The electron spin resonance spectrum of the XXZ\mathit{XXZ}XXZ spin chain with finite length shows a double-peak structure at high temperatures around the electron paramagnetic resonance (EPR) frequency. This fact has been pointed out by direct numerical methods [S. El Shawish, O. C\'epas, and S. Miyashita, Phys. Rev. B 81, 224421 (2010); H. Ikeuchi, H. De Raedt, S. Bertaina, and S. Miyashita, Phys. Rev. B 92, 214431 (2015)]. The question of whether the double-peak structure survives in the thermodynamics is of particular interest. We study the size dependence of the line shape, including the even-odd effect. It is found that the peaks forming the double-peak structure are assigned to individual resonances, each of which is specified by the magnetizations of the resonating states $(M,M\ensuremath{-}1)$. To understand dependences, we decompose the spectrum into contributions from transitions specified by the magnetization, and we characterize the structure of the spectrum by individual contributions. We analyze the size dependence of each contribution individually by extending the moment method introduced by M. Brockman et al. to each component, and we find that the mean of each peak approaches the paramagnetic resonance point with 1/N1/N1/N (where NNN is the length of the chain), which indicates that the separation of the peaks of the double-peak structure also vanishes inversely with the system size. We also study the temperature dependence of the structure. At low temperatures, the spectrum has a single peak with a finite width at a position with a finite shift from the frequency of EPR, as pointed out by the analysis of field-theoretical works [M. Oshikawa and I. Affleck, Phys. Rev. Lett. 82, 5136 (1999)]. The study of the temperature dependence of the spectrum shows how the high-temperature spectrum changes to the low-temperature one with a drastic broadening of the spectrum.