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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 } {dt{e^{i\omega t}}C\left( t \right)} $ are calculated. The Hamiltonian of the Ising model with link-impurity can be written as $ H = - \displaystyle\frac{1}{2}({J_{j - 1}}\sigma _{j - 1}^x\sigma _j^x + {J_j}\sigma _j^x\sigma _{j + 1}^x) - \frac{1}{2}J\sum\limits_{i \ne j,j - 1}^N {\sigma _i^x\sigma _{i + 1}^x} - \frac{1}{2}B\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 jth 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 $) dominates the dynamics. Under the regulating effect of link-impurities, the crossover between different dynamic behaviors can be easily realized, and it is easier to stimulate new dynamic modes, such as the double-peak behavior, the collective-mode-like behavior or bispectral feature one. The results in this work indicate that using link-impurity to manipulate the dynamics of quantum spin systems may be a new try. -
Keywords:
- Ising model /
- link-impurity /
- spin correlation function /
- spectral density
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图 1 对称型链接杂质在不同杂质耦合强度下的连分式系数$ {\varDelta _\nu } $, 其中横向磁场$ B = J \equiv 1 $, 杂质耦合强度取值为$ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0, 1.2, 1.4和1.6
Figure 1. Recurrants $ {\varDelta _\nu } $ for the symmetric type of link-impurity. The transverse magnetic field $ B = J \equiv 1 $, and the impurity coupling strength $ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0, 1.2, 1.4 and 1.6.
图 2 对称型链接杂质在不同杂质耦合强度下的自旋关联函数$C\left( t \right)$(a)及谱密度$\varPhi \left( \omega \right)$(b), 其中横向磁场$ B = J \equiv 1 $, 杂质耦合强度$ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0和1.2
Figure 2. Spin autocorrelation function $C\left( t \right)$ (a) and spectral density $\varPhi \left( \omega \right)$ (b) for the symmetric type of link-impurity under different impurity coupling strengths. The transverse magnetic field $ B = J \equiv 1 $, and the impurity coupling strength $ {J_{j - 1}} = {J_j} = 0.4 $, 0.6, 0.8, 1.0 and 1.2.
图 3 对称型链接杂质在不同杂质耦合强度下的谱密度$\varPhi \left( \omega \right)$, 图(a)—(d)中横向磁场的取值分别为$ B = 0.4 $, 0.7, 1.5和2.0, 主体格点自旋耦合$ J \equiv 1 $
Figure 3. Spectral densities $\varPhi \left( \omega \right)$ for symmetric type of link-impurity under different impurity coupling strength Without loss of generality, the parameter $ J \equiv 1 $, and the transverse magnetic field $ B = 0.4 $, 0.7, 1.5 and 2.0 in (a)–(d).
图 4 固定横场$ B = J \equiv 1 $, 非对称型链接杂质在不同杂质耦合强度下的谱密度$\varPhi \left( \omega \right)$, 其中固定$ {J_{j - 1}} = J' $, 图(a)—(d)中$ {J_{j - 1}} $分别取值为0.2, 0.5, 1.0和1.4; $ {J_j} = J'' $的取值从0.2变化到1.8
Figure 4. Spectral densities for non-symmetric type of link-impurity under different impurity coupling strength. The transverse magnetic field $ B = J \equiv 1 $, and the impurity coupling strength $ {J_{j - 1}} = 0.2 $, 0.5, 1.0 and 1.4 are set in panels (a)–(d), respectively. The other impurity coupling strength $ {J_j} $ changes from 0.2 to 1.8.
图 5 固定横场$ B = 0.5 = J/2 $, 非对称型链接杂质在不同杂质耦合强度下的谱密度, 其中固定$ {J_{j - 1}} = J' $, 图(a)—(f)中$ {J_{j - 1}} $分别取值为0.2, 0.5, 0.8, 1.2, 1.6, 和2.0, $ {J_j} = J'' $的取值从0.2变化到1.8
Figure 5. Spectral densities for non-symmetric type of link-impurity under different impurity coupling strength. The transverse magnetic field $ B = 0.5 = J/2 $, and the impurity coupling strength $ {J_{j - 1}} = 0.2 $, 0.5, 0.8, 1.2, 1.6 and 2.0 are set in panels (a)–(f), respectively. The other impurity coupling strength $ {J_j} $ changes from 0.2 to 1.8.
图 6 固定横场$ B = 1.5 = 1.5 J $, 给出非对称型链接杂质在不同杂质耦合强度下的谱密度, 固定$ {J_{j - 1}} = J' $, 图(a)—(d)中的$ {J_{j - 1}} $分别取值为0.4, 0.8, 1.2和1.6; $ {J_j} = J'' $的取值从0.2变化到1.8
Figure 6. Spectral densities for non-symmetric type of link-impurity under different impurity coupling strength. The transverse magnetic field $ B = 1.5 = 1.5 J $, and the impurity coupling strength $ {J_{j - 1}} = 0.4 $, 0.8, 1.2和1.6 are set in panels (a)–(d), respectively. The other impurity coupling strength $ {J_j} $ changes from 0.2 to 1.8.
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