It is of fundamental importance to know the dynamics of quantum spin systems immersed in external magnetic fields. In this work, the dynamical properties of one-dimensional quantum Ising model with trimodal random transverse and longitudinal magnetic fields are investigated by the recursion method. The spin correlation function C\left( t \right) = \overline \left\langle \sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right) \right\rangle and the corresponding spectral density \varPhi \left( \omega \right) = \displaystyle\int_ - \infty ^ + \infty \rmdt\rme^\rmi\omega tC\left( t \right) are calculated. The model Hamiltonian 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_iz\sigma _i^z - \dfrac12\sum\limits_i^N B_ix\sigma _i^x ,

where \sigma _i^\alpha \left( \alpha = x,y,z \right) are Pauli matrices at site i , Jis the nearest-neighbor exchange coupling. B_iz and B_ix denote the transverse and longitudinal magnetic field, respectively. They satisfy the following trimodal distribution,

\rho \left( B_iz \right) = p\delta (B_iz - B_p) + q\delta (B_iz - B_q) + r\delta (B_iz) ,

The value intervals of the coefficients p, q and r are all 0,1, and the coefficients satisfy the constraint condition p + q + r = 1 .

For the case of trimodal random B_iz (consider B_ix \equiv 0 for simplicity), the exchange couplings are assumed to be J \equiv 1 to fix the energy scale, and the reference values are set as follows: B_p = 0.5 < J and B_q = 1.5 > J . The coefficient r can be considered as the proportion of non-magnetic impurities. When r = 0, 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 q increases, which is consistent with the value reported previously. As r increases, the crossover between different dynamical behaviors changes obviously (e.g. the crossover from central-peak to double-peak when r = 0.2), 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. r = 0.4, 0.6 or 0.8. However, the multi-peak behavior disappears when r \to 1. 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 qB_q = pB_p , the central-peak behavior can be maintained. What makes sense is that the conclusion is universal.

For the case of trimodal random B_ix , the coefficient r no longer represents the proportion of non-magnetic impurities when B_ix and B_iz ( B_iz \equiv 1 ) coexist here. In the case of weak exchange coupling, the effect of longitudinal magnetic field on spin dynamics is obvious, so J \equiv 0.5 is set here. The reference values are set below: B_p = 0.5 \lt B_iz and B_q = 1.5 \gt B_iz . When r is small (r = 0, 0.2 or 0.4), the system undergoes a crossover from the collective-mode behavior to the double-peak behavior as q increases. However, the low-frequency responses gradually disappear, while the high-frequency responses are maintained as r increases. Take the case of r = 0.8 for example, the system only presents a collective-mode behavior. The results indicate that increasing r is no longer conducive to the low-frequency response, which is contrary to the case of trimodal random B_iz . The r branch only regulates the intensity of the trimodal random B_ix . Our results indicate that using trimodal random magnetic field to manipulate the spin dynamics of the Ising system may be a new try.