Effects of random longitudinal magnetic field on dynamics of one-dimensional quantum Ising model

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>