\begin{document}$C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $\end{document} and the corresponding spectral density \begin{document}$\varPhi \left( \omega \right) = \displaystyle\int_{ - \infty }^{ + \infty } {{\rm{d}}t{{\rm{e}}^{{\rm{i}}\omega t}}C\left( t \right)}$\end{document} are calculated. The model Hamiltonian can be written as\begin{document}$ H = - \dfrac{1}{2}J\displaystyle\sum\limits_i^N {\sigma _i^x\sigma _{i + 1}^x} - \dfrac{1}{2}\displaystyle\sum\limits_i^N {{B_{iz}}\sigma _i^z} - \dfrac{1}{2}\sum\limits_i^N {{B_{ix}}\sigma _i^x} $\end{document},where \begin{document}$\sigma _i^\alpha \left( {\alpha = x,y,z} \right)$\end{document} are Pauli matrices at site \begin{document}$ i $\end{document}, \begin{document}$J$\end{document}is the nearest-neighbor exchange coupling. \begin{document}$ {B_{iz}} $\end{document} and \begin{document}$ {B_{ix}} $\end{document} denote the transverse and longitudinal magnetic field, respectively. They satisfy the following trimodal distribution,\begin{document}$ \rho \left( {{B_{iz}}} \right) = p\delta ({B_{iz}} - {B_p}) + q\delta ({B_{iz}} - {B_q}) + r\delta ({B_{iz}}) $\end{document},\begin{document}$ \rho \left( {{B_{ix}}} \right) = p\delta ({B_{ix}} - {B_p}) + q\delta ({B_{ix}} - {B_q}) + r\delta ({B_{ix}}). $\end{document}The value intervals of the coefficients \begin{document}$p$\end{document}, \begin{document}$q$\end{document} and \begin{document}$r$\end{document} are all [0,1], and the coefficients satisfy the constraint condition \begin{document}$ p + q + r = 1 $\end{document}.For the case of trimodal random \begin{document}$ {B_{iz}} $\end{document} (consider \begin{document}$ {B_{ix}} \equiv 0 $\end{document} for simplicity), the exchange couplings are assumed to be \begin{document}$J \equiv 1$\end{document} to fix the energy scale, and the reference values are set as follows: \begin{document}$ {B_p} = 0.5 < J $\end{document} and \begin{document}$ {B_q} = 1.5 > J $\end{document}. The coefficient \begin{document}$r$\end{document} can be considered as the proportion of non-magnetic impurities. When \begin{document}$r = 0$\end{document}, the trimodal distribution reduces into the bimodal distribution. The dynamics of the system exhibits a crossover from the central-peak behavior to the collective-mode behavior as \begin{document}$q$\end{document} increases, which is consistent with the value reported previously. As \begin{document}$r$\end{document} increases, the crossover between different dynamical behaviors changes obviously (e.g. the crossover from central-peak to double-peak when \begin{document}$r = 0.2$\end{document}), and the presence of non-magnetic impurities favors low-frequency response. Owing to the competition between the non-magnetic impurities and transverse magnetic field, the system tends to exhibit multi-peak behavior in most cases, e.g. \begin{document}$r = 0.4$\end{document}, 0.6 or 0.8. However, the multi-peak behavior disappears when \begin{document}$r \to 1$\end{document}. That is because the system's response to the transverse field is limited when the proportion of non-magnetic impurities is large enough. Interestingly, when the parameters satisfy \begin{document}$ q{B_q} = p{B_p} $\end{document}, the central-peak behavior can be maintained. What makes sense is that the conclusion is universal.For the case of trimodal random \begin{document}$ {B_{ix}} $\end{document}, the coefficient \begin{document}$r$\end{document} no longer represents the proportion of non-magnetic impurities when \begin{document}$ {B_{ix}} $\end{document} and \begin{document}$ {B_{iz}} $\end{document} (\begin{document}$ {B_{iz}} \equiv 1 $\end{document}) coexist here. In the case of weak exchange coupling, the effect of longitudinal magnetic field on spin dynamics is obvious, so \begin{document}$J \equiv 0.5$\end{document} is set here. The reference values are set below: \begin{document}$ {B_p} = 0.5 \lt {B_{iz}} $\end{document} and \begin{document}$ {B_q} = 1.5 \gt {B_{iz}} $\end{document}. When \begin{document}$r$\end{document} is small (\begin{document}$r = 0$\end{document}, 0.2 or 0.4), the system undergoes a crossover from the collective-mode behavior to the double-peak behavior as \begin{document}$q$\end{document} increases. However, the low-frequency responses gradually disappear, while the high-frequency responses are maintained as \begin{document}$r$\end{document} increases. Take the case of \begin{document}$ r = 0.8 $\end{document} for example, the system only presents a collective-mode behavior. The results indicate that increasing \begin{document}$r$\end{document} is no longer conducive to the low-frequency response, which is contrary to the case of trimodal random \begin{document}$ {B_{iz}} $\end{document}. The \begin{document}$r$\end{document} branch only regulates the intensity of the trimodal random \begin{document}$ {B_{ix}} $\end{document}. Our results indicate that using trimodal random magnetic field to manipulate the spin dynamics of the Ising system may be a new try."> - 必威体育下载

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中国物理学会期刊
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Yuan Xiao-Juan
Acta Phys. Sin., 2023, 72(8): 087501.
doi: 10.7498/aps.72.20230046
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  • Received Date:10 January 2023
  • Accepted Date:11 February 2023
  • Available Online:17 February 2023
  • Published Online:20 April 2023

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