Effects of link-impurity on spin dynamics of one-dimensional quantum Ising model
Authors: YUAN Xiaojuan ,   College of Physics and Electronic Engineering, Qilu Normal University, Jinan 250200, China
Published: December 2024

<sec>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 <inline-formula><tex-math id="M16">\begin{document}$C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $\end{document}</tex-math></inline-formula> and the associated spectral density <inline-formula><tex-math id="M17">\begin{document}$\varPhi \left( \omega \right) = \displaystyle\int_{ - \infty }^{ + \infty } {{\rm d}t{{\rm e}^{{\rm i}\omega t}}C\left( t \right)} $\end{document}</tex-math></inline-formula> are calculated. The Hamiltonian of the Ising model with link-impurity can be written as</sec> <sec><inline-formula><tex-math id="M18">\begin{document}$\qquad\qquad\qquad\qquad\qquad H = - \displaystyle\frac{1}{2}({J_{j - 1}}\sigma _{j - 1}^x\sigma _j^x + {J_j}\sigma _j^x\sigma _{j + 1}^x) - \displaystyle\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} . $\end{document}</tex-math></inline-formula>where <inline-formula><tex-math id="M19">\begin{document}$J$\end{document}</tex-math></inline-formula> is the nearest-neighbor exchange coupling of the main spin chain, <inline-formula><tex-math id="M20">\begin{document}$ B $\end{document}</tex-math></inline-formula> denotes the external transverse magnetic field, <inline-formula><tex-math id="M21">\begin{document}$\sigma _i^\alpha \left( {\alpha = x,y,z} \right)$\end{document}</tex-math></inline-formula> are Pauli matrices at site <inline-formula><tex-math id="M22">\begin{document}$ i $\end{document}</tex-math></inline-formula>. The constant 1/2 is introduced for the convenience of theoretical deduction, and <i>N</i> is the number of spins. The so-called link-impurity <inline-formula><tex-math id="M23">\begin{document}$ {J_j} $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M24">\begin{document}$ {J_{j - 1}} $\end{document}</tex-math></inline-formula>) is randomly introduced, which denotes the exchange coupling between the <i>j</i> th spin and the (<i>j</i> + 1)th spin (the (<i>j</i> – 1)th spin). The symmetric link-impurity and asymmetric link-impurity correspond to the case of <inline-formula><tex-math id="M25">\begin{document}$ {J_{j - 1}} = {J_j} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M26">\begin{document}$ {J_{j - 1}} \ne {J_j} $\end{document}</tex-math></inline-formula>, respectively. The periodic boundary conditions are assumed in the theoretical calculation.</sec><sec>After introducing the link-impurity, the original competition between <inline-formula><tex-math id="M27">\begin{document}$ B $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M28">\begin{document}$J$\end{document}</tex-math></inline-formula> in the pure Ising model is broken. The dynamic behavior of the system depends on synergistic effect of multiple factors, such as the mean spin coupling <inline-formula><tex-math id="M29">\begin{document}$ \bar J $\end{document}</tex-math></inline-formula> between <inline-formula><tex-math id="M30">\begin{document}$J$\end{document}</tex-math></inline-formula> and the link-impurity, the asymmetry degree between <inline-formula><tex-math id="M31">\begin{document}$ {J_{j - 1}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M32">\begin{document}$ {J_j} $\end{document}</tex-math></inline-formula>, and the strength of the external magnetic field. In calculation, the exchange couplings of the main spin chain are set to <inline-formula><tex-math id="M33">\begin{document}$J \equiv 1$\end{document}</tex-math></inline-formula> to fix the energy scale. We first consider the effects of symmetric link-impurity. The reference values can be set to <inline-formula><tex-math id="M34">\begin{document}$ {J_{j - 1}} = {J_j} \lt J $\end{document}</tex-math></inline-formula> (e.g. 0.4, 0.6 or 0.8) or <inline-formula><tex-math id="M35">\begin{document}$ {J_{j - 1}} = {J_j} \gt J $\end{document}</tex-math></inline-formula> (e.g. 1.2, 1.6, 2.0), which are called weak or strong impurity coupling. When the magnetic field <inline-formula><tex-math id="M36">\begin{document}$ B \geqslant J $\end{document}</tex-math></inline-formula> (e.g., <inline-formula><tex-math id="M37">\begin{document}$ B = 1 $\end{document}</tex-math></inline-formula>, 1.5 or 2.0), it is found that the dynamic behavior of the system exhibits a crossover from a collective-mode behavior to a central-peak behavior as the impurity strength <inline-formula><tex-math id="M38">\begin{document}$ {J_{j - 1}} = {J_j} $\end{document}</tex-math></inline-formula> increases. Interestingly, for <inline-formula><tex-math id="M39">\begin{document}$ B \lt J $\end{document}</tex-math></inline-formula> (e.g. <inline-formula><tex-math id="M40">\begin{document}$ B = 0.4 $\end{document}</tex-math></inline-formula> or 0.7), there are two crossovers that are a collective-mode-like behavior to a double-peak behavior, then to a central-peak behavior as <inline-formula><tex-math id="M41">\begin{document}$ {J_{j - 1}} = {J_j} $\end{document}</tex-math></inline-formula> increases.</sec><sec>For the case of asymmetric link-impurity, the impurity configuration is more complex. Using the cooperation between <inline-formula><tex-math id="M42">\begin{document}$ {J_{j - 1}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M43">\begin{document}$ {J_j} $\end{document}</tex-math></inline-formula>, more freedoms of regulation can be provided and the dynamical properties are more abundant. For the case of <inline-formula><tex-math id="M44">\begin{document}$ B \leqslant J $\end{document}</tex-math></inline-formula> (e.g. <inline-formula><tex-math id="M45">\begin{document}$ B = 0.5 $\end{document}</tex-math></inline-formula>, 1.0), the system tends to exhibit a collective-mode behavior when the mean spin coupling <inline-formula><tex-math id="M46">\begin{document}$ \bar J $\end{document}</tex-math></inline-formula> is weak, and a central-peak behavior when <inline-formula><tex-math id="M47">\begin{document}$ \bar J $\end{document}</tex-math></inline-formula> are strong. However, when the asymmetry between <inline-formula><tex-math id="M48">\begin{document}$ {J_{j - 1}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M49">\begin{document}$ {J_j} $\end{document}</tex-math></inline-formula> is obvious, the system tends to exhibit a double- or multi-peak behavior. For the case of <inline-formula><tex-math id="M50">\begin{document}$ B \gt J $\end{document}</tex-math></inline-formula> (e.g. <inline-formula><tex-math id="M51">\begin{document}$ B = 1.5 $\end{document}</tex-math></inline-formula>, 2.0), when <inline-formula><tex-math id="M52">\begin{document}$ \bar J $\end{document}</tex-math></inline-formula> is weak or the asymmetry between <inline-formula><tex-math id="M53">\begin{document}$ {J_{j - 1}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M54">\begin{document}$ {J_j} $\end{document}</tex-math></inline-formula> is not obvious, the system tends to exhibit a collective-mode behavior. When <inline-formula><tex-math id="M55">\begin{document}$ \bar J $\end{document}</tex-math></inline-formula> is strong, it tends to show a central-peak behavior. However, when the asymmetry between <inline-formula><tex-math id="M56">\begin{document}$ {J_{j - 1}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M57">\begin{document}$ {J_j} $\end{document}</tex-math></inline-formula> is evident, the bispectral feature (two spectral peaks appear at <inline-formula><tex-math id="M58">\begin{document}$ {\omega _1} \ne 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M59">\begin{document}$ {\omega _2} \ne 0 $\end{document}</tex-math></inline-formula>) 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.</sec>