Fund Project:Project supported by the Natural Science Foundation of China (Grant No. 11747132), the Shandong Province Higher Educational Science and Technology Program, China (Grant No. J18KB104), and the Young Doctoral Support Program of Qilu Normal University, China (Grant Nos. 2017L0603, 2017L0604, QBJH19-0006)
Received Date:05 April 2021
Accepted Date:13 May 2021
Available Online:07 June 2021
Published Online:05 October 2021
Abstract: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 $C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $ and corresponding spectral density $\varPhi \left( \omega \right)$ are calculated. The Hamiltonian of the model system can be written as $H = - \dfrac{1}{2}J\displaystyle\sum\limits_i^N {\sigma _i^x\sigma _{i + 1}^x - \dfrac{1}{2}\displaystyle\sum\limits_i^N {B_i^x\sigma _i^x} } - \dfrac{1}{2}\displaystyle\sum\limits_i^N {B_i^z\sigma _i^z}$. This work focuses mainly on the effects of LMF ($ B_i^x $) 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 $ B_i^z = 1 $ is set in the numerical calculation, which fixes the energy scale. 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 ($ J $) is weak. However, the effect of uniform LMF is not obvious when spin interaction is strong. For the case of random LMF, the effects of bimodal-type and Gaussian-type random LMF are investigated, respectively. The dynamical results under the two types of random LMFs are quite different and highly dependent on many factors, such as the mean values ($ {B_1} $, $ {B_2} $ and $ {B_x} $) or the standard deviation ($ \sigma $) of random distributions. The nonsymmetric bimodal-type random LMF ($ {B_1} \ne {B_2} $) 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 $ \sigma $ 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 $ {B_x} $ increases. However, when $ \sigma $ is large, the system presents only a central-peak behavior. 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 $\displaystyle\sum\nolimits_i^N {B_i^z\sigma _i^z}$) 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. Keywords:Ising model/ longitudinal magnetic field/ spin autocorrelation function/ spectral density
3.常数纵场对动力学的影响首先讨论纵场取值为常数时系统的动力学性质. 为了不失一般性, 令横场$B_i^z = 1$. 纵场的取值可以小于1也可以大于1, 例如$ B_i^x = 0 $, 0.5, 1.0, 1.5和2.0. 当纵场取值为0时, 该模型蜕化为一维横向Ising模型[15,20]. 自旋耦合相互作用参数J取值为0.1, 0.5, 1.0和1.5, 既考虑了弱耦合($J < 1$)的情况又考虑了强耦合($J \geqslant 1$)的情况. 图1和图2分别给出了系统在不同参数下的自旋关联函数$C\left( t \right)$及相应的谱密度$\varPhi \left( \omega \right)$. 图 1 横场取值$B_i^z = 1$, 纵场取值$ B_i^x = 0 $, 0.5, 1.0, 1.5和2.0, (a)?(d)分别对应自旋耦合相互作用参数J = 0.1, 0.5, 1.0和1.5时的自旋关联函数 Figure1. Take the transverse magnetic field $B_i^z = 1$ and the longitudinal magnetic field $ B_i^x = 0 $, 0.5, 1.0, 1.5 and 2.0, respectively. Spin autocorrelation functions $C\left( t \right)$ for different values of spin interactions (e.g., J = 0.1, 0.5, 1.0 and 1.5) are given in (a)?(d), respectively.
图 2 横场取值$B_i^z = 1$, 纵场取值$ B_i^x = 0 $, 0.5, 1.0, 1.5和2.0, (a)?(d)分别对应自旋耦合相互作用参数J = 0.1, 0.5, 1.0和1.5时的谱密度. Figure2. Take the transverse magnetic field $B_i^z = 1$ and the longitudinal magnetic field $ B_i^x = 0 $, 0.5, 1.0, 1.5 and 2.0, respectively. The corresponding spectral density $\varPhi \left( \omega \right)$ for different values of spin interactions (e.g., J = 0.1, 0.5, 1.0 and 1.5) are given in (a)?(d), respectively.