Effects of trimodal random magnetic field on spin dynamics of quantum Ising chain

Abstract

<sec>It is of fundamental importance to know the dynamics of quantum spin systems immersed in external magnetic fields. In this work, the dynamical properties of one-dimensional quantum Ising model with trimodal random transverse and longitudinal magnetic fields are investigated by the recursion method. The spin correlation function <inline-formula><tex-math id="M2">\begin{document}$C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M2.png"/></alternatives></inline-formula> and the corresponding spectral density <inline-formula><tex-math id="M3">\begin{document}$\varPhi \left( \omega \right) = \displaystyle\int_{ - \infty }^{ + \infty } {{\rm{d}}t{{\rm{e}}^{{\rm{i}}\omega t}}C\left( t \right)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M3.png"/></alternatives></inline-formula> are calculated. The model Hamiltonian can be written as</sec><sec><inline-formula><tex-math id="M4">\begin{document}$ H = - \dfrac{1}{2}J\displaystyle\sum\limits_i^N {\sigma _i^x\sigma _{i + 1}^x} - \dfrac{1}{2}\displaystyle\sum\limits_i^N {{B_{iz}}\sigma _i^z} - \dfrac{1}{2}\sum\limits_i^N {{B_{ix}}\sigma _i^x} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M4.png"/></alternatives></inline-formula>,</sec><sec>where <inline-formula><tex-math id="M5">\begin{document}$\sigma _i^\alpha \left( {\alpha = x,y,z} \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M5.png"/></alternatives></inline-formula> are Pauli matrices at site <inline-formula><tex-math id="M6">\begin{document}$ i $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M6.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$J$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M7.png"/></alternatives></inline-formula>is the nearest-neighbor exchange coupling. <inline-formula><tex-math id="M8">\begin{document}$ {B_{iz}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M8.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ {B_{ix}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M9.png"/></alternatives></inline-formula> denote the transverse and longitudinal magnetic field, respectively. They satisfy the following trimodal distribution,</sec><sec><inline-formula><tex-math id="M10">\begin{document}$ \rho \left( {{B_{iz}}} \right) = p\delta ({B_{iz}} - {B_p}) + q\delta ({B_{iz}} - {B_q}) + r\delta ({B_{iz}}) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M10.png"/></alternatives></inline-formula>,</sec><sec><inline-formula><tex-math id="M11">\begin{document}$ \rho \left( {{B_{ix}}} \right) = p\delta ({B_{ix}} - {B_p}) + q\delta ({B_{ix}} - {B_q}) + r\delta ({B_{ix}}). $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M11.png"/></alternatives></inline-formula></sec><sec>The value intervals of the coefficients <inline-formula><tex-math id="M12">\begin{document}$p$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M12.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$q$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M13.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M14.png"/></alternatives></inline-formula> are all [0,1], and the coefficients satisfy the constraint condition <inline-formula><tex-math id="M15">\begin{document}$ p + q + r = 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M15.png"/></alternatives></inline-formula>.</sec><sec>For the case of trimodal random <inline-formula><tex-math id="M16">\begin{document}$ {B_{iz}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M16.png"/></alternatives></inline-formula> (consider <inline-formula><tex-math id="M17">\begin{document}$ {B_{ix}} \equiv 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M17.png"/></alternatives></inline-formula> for simplicity), the exchange couplings are assumed to be <inline-formula><tex-math id="M18">\begin{document}$J \equiv 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M18.png"/></alternatives></inline-formula> to fix the energy scale, and the reference values are set as follows: <inline-formula><tex-math id="M19">\begin{document}$ {B_p} = 0.5 < J $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M19.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ {B_q} = 1.5 > J $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M20.png"/></alternatives></inline-formula>. The coefficient <inline-formula><tex-math id="M21">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M21.png"/></alternatives></inline-formula> can be considered as the proportion of non-magnetic impurities. When <inline-formula><tex-math id="M22">\begin{document}$r = 0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M22.png"/></alternatives></inline-formula>, the trimodal distribution reduces into the bimodal distribution. The dynamics of the system exhibits a crossover from the central-peak behavior to the collective-mode behavior as <inline-formula><tex-math id="M23">\begin{document}$q$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M23.png"/></alternatives></inline-formula> increases, which is consistent with the value reported previously. As <inline-formula><tex-math id="M24">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M24.png"/></alternatives></inline-formula> increases, the crossover between different dynamical behaviors changes obviously (e.g. the crossover from central-peak to double-peak when <inline-formula><tex-math id="M25">\begin{document}$r = 0.2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M25.png"/></alternatives></inline-formula>), and the presence of non-magnetic impurities favors low-frequency response. Owing to the competition between the non-magnetic impurities and transverse magnetic field, the system tends to exhibit multi-peak behavior in most cases, e.g. <inline-formula><tex-math id="M26">\begin{document}$r = 0.4$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M26.png"/></alternatives></inline-formula>, 0.6 or 0.8. However, the multi-peak behavior disappears when <inline-formula><tex-math id="M27">\begin{document}$r \to 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M27.png"/></alternatives></inline-formula>. That is because the system's response to the transverse field is limited when the proportion of non-magnetic impurities is large enough. Interestingly, when the parameters satisfy <inline-formula><tex-math id="M28">\begin{document}$ q{B_q} = p{B_p} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M28.png"/></alternatives></inline-formula>, the central-peak behavior can be maintained. What makes sense is that the conclusion is universal.</sec><sec>For the case of trimodal random <inline-formula><tex-math id="M29">\begin{document}$ {B_{ix}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M29.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M29.png"/></alternatives></inline-formula>, the coefficient <inline-formula><tex-math id="M30">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M30.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M30.png"/></alternatives></inline-formula> no longer represents the proportion of non-magnetic impurities when <inline-formula><tex-math id="M31">\begin{document}$ {B_{ix}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M31.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M31.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M32">\begin{document}$ {B_{iz}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M32.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M32.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M33">\begin{document}$ {B_{iz}} \equiv 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M33.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M33.png"/></alternatives></inline-formula>) coexist here. In the case of weak exchange coupling, the effect of longitudinal magnetic field on spin dynamics is obvious, so <inline-formula><tex-math id="M34">\begin{document}$J \equiv 0.5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M34.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M34.png"/></alternatives></inline-formula> is set here. The reference values are set below: <inline-formula><tex-math id="M35">\begin{document}$ {B_p} = 0.5 \lt {B_{iz}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M35.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M35.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M36">\begin{document}$ {B_q} = 1.5 \gt {B_{iz}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M36.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M36.png"/></alternatives></inline-formula>. When <inline-formula><tex-math id="M37">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M37.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M37.png"/></alternatives></inline-formula> is small (<inline-formula><tex-math id="M38">\begin{document}$r = 0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M38.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M38.png"/></alternatives></inline-formula>, 0.2 or 0.4), the system undergoes a crossover from the collective-mode behavior to the double-peak behavior as <inline-formula><tex-math id="M39">\begin{document}$q$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M39.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M39.png"/></alternatives></inline-formula> increases. However, the low-frequency responses gradually disappear, while the high-frequency responses are maintained as <inline-formula><tex-math id="M40">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M40.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M40.png"/></alternatives></inline-formula> increases. Take the case of <inline-formula><tex-math id="M41">\begin{document}$ r = 0.8 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M41.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M41.png"/></alternatives></inline-formula> for example, the system only presents a collective-mode behavior. The results indicate that increasing <inline-formula><tex-math id="M42">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M42.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M42.png"/></alternatives></inline-formula> is no longer conducive to the low-frequency response, which is contrary to the case of trimodal random <inline-formula><tex-math id="M43">\begin{document}$ {B_{iz}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M43.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M43.png"/></alternatives></inline-formula>. The <inline-formula><tex-math id="M44">\begin{document}$r$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M44.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M44.png"/></alternatives></inline-formula> branch only regulates the intensity of the trimodal random <inline-formula><tex-math id="M45">\begin{document}$ {B_{ix}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M45.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20230046_M45.png"/></alternatives></inline-formula>. Our results indicate that using trimodal random magnetic field to manipulate the spin dynamics of the Ising system may be a new try.</sec>