It is of considerable theoretical significance to study the effects of impurity on spin dynamics of quantum spin systems. In this paper, the dynamical properties of the one-dimensional quantum Ising model with symmetric and asymmetric link-impurity are investigated by the recursion method, respectively. The autocorrelation function C\left( t \right) = \overline \left\langle \sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right) \right\rangle and the associated spectral density \varPhi \left( \omega \right) = \displaystyle\int_ - \infty ^ + \infty \rm dt\rm e^\rm i\omega tC\left( t \right) are calculated. The Hamiltonian of the Ising model with link-impurity can be written as

\qquad\qquad\qquad\qquad\qquad H = - \displaystyle\frac12(J_j - 1\sigma _j - 1^x\sigma _j^x + J_j\sigma _j^x\sigma _j + 1^x) - \displaystyle\frac12J\sum\limits_i \ne j,j - 1^N \sigma _i^x\sigma _i + 1^x - \frac12B\sum\limits_i^N \sigma _i^z . where J is the nearest-neighbor exchange coupling of the main spin chain, B denotes the external transverse magnetic field, \sigma _i^\alpha \left( \alpha = x,y,z \right) are Pauli matrices at site i . The constant 1/2 is introduced for the convenience of theoretical deduction, and N is the number of spins. The so-called link-impurity J_j ( J_j - 1 ) is randomly introduced, which denotes the exchange coupling between the j th spin and the (j + 1)th spin (the (j – 1)th spin). The symmetric link-impurity and asymmetric link-impurity correspond to the case of J_j - 1 = J_j and J_j - 1 \ne J_j , respectively. The periodic boundary conditions are assumed in the theoretical calculation.

After introducing the link-impurity, the original competition between B and J in the pure Ising model is broken. The dynamic behavior of the system depends on synergistic effect of multiple factors, such as the mean spin coupling \bar J between J and the link-impurity, the asymmetry degree between J_j - 1 and J_j , and the strength of the external magnetic field. In calculation, the exchange couplings of the main spin chain are set to J \equiv 1 to fix the energy scale. We first consider the effects of symmetric link-impurity. The reference values can be set to J_j - 1 = J_j \lt J (e.g. 0.4, 0.6 or 0.8) or J_j - 1 = J_j \gt J (e.g. 1.2, 1.6, 2.0), which are called weak or strong impurity coupling. When the magnetic field B \geqslant J (e.g., B = 1 , 1.5 or 2.0), it is found that the dynamic behavior of the system exhibits a crossover from a collective-mode behavior to a central-peak behavior as the impurity strength J_j - 1 = J_j increases. Interestingly, for B \lt J (e.g. B = 0.4 or 0.7), there are two crossovers that are a collective-mode-like behavior to a double-peak behavior, then to a central-peak behavior as J_j - 1 = J_j increases.

For the case of asymmetric link-impurity, the impurity configuration is more complex. Using the cooperation between J_j - 1 and J_j , more freedoms of regulation can be provided and the dynamical properties are more abundant. For the case of B \leqslant J (e.g. B = 0.5 , 1.0), the system tends to exhibit a collective-mode behavior when the mean spin coupling \bar J is weak, and a central-peak behavior when \bar J are strong. However, when the asymmetry between J_j - 1 and J_j is obvious, the system tends to exhibit a double- or multi-peak behavior. For the case of B \gt J (e.g. B = 1.5 , 2.0), when \bar J is weak or the asymmetry between J_j - 1 and J_j is not obvious, the system tends to exhibit a collective-mode behavior. When \bar J is strong, it tends to show a central-peak behavior. However, when the asymmetry between J_j - 1 and J_j is evident, the bispectral feature (two spectral peaks appear at \omega _1 \ne 0 and \omega _2 \ne 0 ) dominates the dynamics. Under the regulating effect of link-impurities, the crossover between different dynamic behaviors can be easily realized, and it is easier to stimulate new dynamic modes, such as the double-peak behavior, the collective-mode-like behavior or bispectral feature one. The results in this work indicate that using link-impurity to manipulate the dynamics of quantum spin systems may be a new try.