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It is of considerable theoretical interest 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 $C(t)=\overline{\left\langle\sigma_j^x(t) \sigma_j^x(0)\right\rangle}$ and the associated spectral density $\Phi(\omega)=\int_{-\infty}^{+\infty} d t e^{i \omega t} C(t)$ are calculated. The Hamiltonian of the Ising model with link-impurity can be written as $H=-\frac{1}{2}\left(J_{j-1} \sigma_{j-1}^x \sigma_j^x+J_j \sigma_j^x \sigma_{j+1}^x\right)-\frac{1}{2} J \sum_{i \neq j, j-1}^N \sigma_i^x \sigma_{i+1}^x-\frac{1}{2} B \sum_i^N \sigma_i^z$. Where J is the nearest-neighbor exchange coupling of the main spin chain, B denotes the external transverse magnetic field, $\sigma_i^\alpha(\alpha=x, y, z)$ 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 Jj(Jj-1) is randomly introduced, which denotes the exchange coupling between the jth spin and the (j+1)th spin (the (j-1)th spin).The symmetric and asymmetric link-impurity correspond to the case of Jj-1=Jj and Jj-1≠Jj, 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 was broken. The dynamics 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 Jj-1 and Jj,and the strength of the external magnetic field. In calculation, the exchange couplings of the main spin chain are set $J \equiv 1$ to fix the energy scale. We first consider the effects of symmetric link-impurity, the reference values can be set Jj-1=Jj＜J(e.g., 0.4, 0.6 or 0.8) or Jj-1=Jj＞J(e.g., 1.2, 1.6, 2.0),which are called weak or strong impurity coupling. When the magnetic field B≥J(e.g.B=1, 1.5 or 2.0),it is found that the dynamics of the system exhibits a crossover from a collective-mode behavior to a central-peak behavior as the impurity strength Jj-1=Jj increase. Interestingly, for B＜J(e.g., B=4 or 0.7),there are two crossovers that is a collective-mode-like behavior to a double-peak behavior, then to a central-peak one as Jj-1=Jj increase.
For the case of asymmetric link-impurity, the impurity configuration is more complex. Using the cooperation between Jj-1 and Jj,more freedom of regulation can be provided and the dynamical properties are more abundant. For the case of B≤J(e.g., B=0.5, 1.0),the system tends to exhibit a collective-mode behavior when the mean spin coupling $\bar{J}$,are weak, and a central-peak behavior when $\bar{J}$ are strong. However, when the asymmetry between Jj-1 and Jj is obvious, the system tends to exhibit a double- or multi-peak behavior. For the case of B＞J(e.g., B=1.5, 2.0),when $\bar{J}$ are strong, it tends to exhibit a central-peak behavior. However, when the asymmetry between Jj-1 and Jj is evident, the bispectral feature (two spectral peak appear at $\omega_1 \neq 0$ and $\omega_2 \neq 0$) 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 indicate that using link-impurity to manipulate the dynamics of quantum spin systems may be a new try.-
Keywords:
- Ising model /
- Link-impurity /
- Spin correlation function /
- Spectral density
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