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自旋系统的动力学性质是量子统计和凝聚态理论研究的热点. 本文利用递推方法, 通过计算自旋关联函数及谱密度, 研究了链接杂质对一维量子Ising模型动力学性质的调控效应. 研究表明, 链接杂质的出现打破了主体格点自旋耦合$ J $和外磁场$ B $之间原有的竞争关系, 系统的动力学最终取决于链接杂质和主体格点自旋耦合的平均效应$ \bar J $、链接杂质的不对称程度及外磁场$ B $的强度等多因素之间的协同作用. 对于对称型链接杂质($ {J_{j - 1}} = {J_j} $), 随着杂质耦合强度的增大, 在$ B \geqslant J $的情况下, 系统的动力学出现了由集体模行为到中心峰值行为的交跨; 在$ B \lt J $的情况下, 出现了由类集体模行为到双峰行为, 再到中心峰值行为的两次交跨. 对于非对称型链接杂质($ {J_{j - 1}} \ne {J_j} $), 其杂质位型较多, 可以提供更多的调控自由度, 尤其当其中某个杂质耦合强度如$ {J_{j - 1}} $(或$ {J_j} $)较小时, 通过调节另一个杂质耦合强度$ {J_j} $(或$ {J_{j - 1}} $)可以得到多种动力学行为之间的交跨; 在$ B \gt J $情况下, 非对称型链接杂质的调控机制更为复杂, 出现了与以往研究经验不符的交跨顺序, 且出现了双频谱这种新的动力学模式. 一般来讲, 当平均自旋耦合$ \bar J $较弱或非对称型链接杂质的不对称程度较低时, 系统倾向于呈现集体模行为; 当$ \bar J $较强时, 系统倾向于呈现中心峰值行为; 但当非对称型链接杂质的不对称程度较明显时, 谱密度倾向于呈双峰、多峰或双频谱特征. 研究表明, 链接杂质的调控结果更加丰富, 且具有独特的调控优势, 因此利用链接杂质来调控量子自旋系统的动力学不失为一种新的尝试.
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
Fig. 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
Fig. 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 $
Fig. 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
Fig. 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
Fig. 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
Fig. 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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