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 = - \dfrac12J\displaystyle\sum\limits_i^N \sigma _i^x\sigma _i + 1^x - \dfrac12\displaystyle\sum\limits_i^N B_i^x\sigma _i^x - \dfrac12\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.