In:
Acta Physica Sinica, Acta Physica Sinica, Chinese Physical Society and Institute of Physics, Chinese Academy of Sciences, Vol. 70, No. 19 ( 2021), p. 197501-
Abstract:
〈sec〉The dynamical properties of quantum spin systems are a hot topic of research in statistical and condensed matter physics. In this paper, the dynamics of one-dimensional quantum Ising model with both transverse and longitudinal magnetic field (LMF) is investigated by the recursion method. The time-dependent spin autocorrelation function 〈inline-formula〉〈tex-math id="M1"〉\begin{document}$C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M1.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M1.png"/〉〈/alternatives〉〈/inline-formula〉 and corresponding spectral density 〈inline-formula〉〈tex-math id="M2"〉\begin{document}$\varPhi \left( \omega \right)$\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M2.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M2.png"/〉〈/alternatives〉〈/inline-formula〉 are calculated. The Hamiltonian of the model system can be written as 〈/sec〉〈sec〉 〈inline-formula〉〈tex-math id="M3"〉\begin{document}$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_i^x\sigma _i^x} } - \dfrac{1}{2}\displaystyle\sum\limits_i^N {B_i^z\sigma _i^z}$\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M3.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M3.png"/〉〈/alternatives〉〈/inline-formula〉. 〈/sec〉〈sec〉This work focuses mainly on the effects of LMF (〈inline-formula〉〈tex-math id="M4"〉\begin{document}$ B_i^x $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M4.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M4.png"/〉〈/alternatives〉〈/inline-formula〉) on spin dynamics of the Ising system, and both uniform LMF and random LMF are considered respectively. Without loss of generality, the transverse magnetic field 〈inline-formula〉〈tex-math id="M5"〉\begin{document}$ B_i^z = 1 $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M5.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M5.png"/〉〈/alternatives〉〈/inline-formula〉 is set in the numerical calculation, which fixes the energy scale. 〈/sec〉〈sec〉The results show that the uniform LMF can induce crossovers between different dynamical behaviors (e.g. independent spins precessing, collective-mode behavior or central-peak behavior) and drive multiple vibrational modes (multiple-peaked behavior) when spin interaction (〈inline-formula〉〈tex-math id="M6"〉\begin{document}$ J $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M6.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M6.png"/〉〈/alternatives〉〈/inline-formula〉) is weak. However, the effect of uniform LMF is not obvious when spin interaction is strong. For the case of random LMF, the effects of bimodal-type and Gaussian-type random LMF are investigated, respectively. The dynamical results under the two types of random LMFs are quite different and highly dependent on many factors, such as the mean values (〈inline-formula〉〈tex-math id="M7"〉\begin{document}$ {B_1} $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M7.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M7.png"/〉〈/alternatives〉〈/inline-formula〉, 〈inline-formula〉〈tex-math id="M8"〉\begin{document}$ {B_2} $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M8.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M8.png"/〉〈/alternatives〉〈/inline-formula〉 and 〈inline-formula〉〈tex-math id="M9"〉\begin{document}$ {B_x} $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M9.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M9.png"/〉〈/alternatives〉〈/inline-formula〉) or the standard deviation (〈inline-formula〉〈tex-math id="M10"〉\begin{document}$ \sigma $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M10.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M10.png"/〉〈/alternatives〉〈/inline-formula〉) of random distributions. The nonsymmetric bimodal-type random LMF (〈inline-formula〉〈tex-math id="M11"〉\begin{document}$ {B_1} \ne {B_2} $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M11.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M11.png"/〉〈/alternatives〉〈/inline-formula〉) may induce new vibrational modes easily. The dynamical behaviors under the Gaussian-type random LMF are more abundant than under the bimodal-type random LMF. When 〈inline-formula〉〈tex-math id="M12"〉\begin{document}$ \sigma $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M12.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M12.png"/〉〈/alternatives〉〈/inline-formula〉 is small, the system undergoes two crossovers: from a collective-mode behavior to a double-peaked behavior, and then to a central-peak behavior as the mean value 〈inline-formula〉〈tex-math id="M13"〉\begin{document}$ {B_x} $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M13.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M13.png"/〉〈/alternatives〉〈/inline-formula〉 increases. However, when 〈inline-formula〉〈tex-math id="M14"〉\begin{document}$ \sigma $\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M14.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M14.png"/〉〈/alternatives〉〈/inline-formula〉 is large, the system presents only a central-peak behavior. 〈/sec〉〈sec〉For both cases of uniform LMF and random LMF, it is found that the central-peak behavior of the system is maintained when the proportion of LMF is large. This conclusion can be generalized that the emergence of noncommutative terms (noncommutative with the transverse-field term 〈inline-formula〉〈tex-math id="M15"〉\begin{document}$\displaystyle\sum\nolimits_i^N {B_i^z\sigma _i^z}$\end{document}〈/tex-math〉〈alternatives〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M15.jpg"/〉〈graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210631_M15.png"/〉〈/alternatives〉〈/inline-formula〉) in Hamiltonian will enhance the central peak behavior. Therefore, noncommutative terms, such as next-nearest-neighbor spin interactions, Dzyaloshinskii-Moryia interactions, impurities, four-spin interactions, etc., can be added to the system Hamiltonian to modulate the dynamical properties. This provides a new direction for the future study of spin dynamics.〈/sec〉
Type of Medium:
Online Resource
ISSN:
1000-3290
,
1000-3290
DOI:
10.7498/aps.70.20210631
Language:
Unknown
Publisher:
Acta Physica Sinica, Chinese Physical Society and Institute of Physics, Chinese Academy of Sciences
Publication Date:
2021
Bookmarklink