Mobility edges of bosonic pairs in one-dimensional quasi-periodical lattices
Authors: Xu Zhi-Hao , Huangfu Hong-Li , Zhang Yun-Bo ,   Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
Published: December 2020

<sec>Mobility edge as one of the most important concepts in a disordered system in which there exists an energy dependent conductor-to-insulator transition has aroused great interest. Unlike an arbitrarily small disorder inducing the Anderson localization in one-dimensional random potential, the well-known Aubry-André model presents a metal-to-insulator transition without mobility edges. Some generalized Aubry-André models are proposed whose the mobility edges in compactly analytic forms are found. However, the existence of the many-body mobility edges in thermodynamic limit for an interacting disordered system is still an open question due to the dimension of the Hilbert space beyond the numerical capacity. In this paper, we demonstrate the existence of the mobility edges of bosonic pairs trapped in one dimensional quasi-periodical lattices subjected to strongly interactions. We believe that our theory will provide a new insight into the studying of the many-body mobility edges.</sec><sec>Two strongly interacting bosons are trapped in an incommensurate model, which is described as <inline-formula><tex-math id="M1">\begin{document}$\hat H = - J\sum\limits_j{} {\left( {\hat c_j^\dagger {{\hat c}_{j + 1}} + {\rm{h}}{\rm{.c}}{\rm{.}}} \right)} + 2\lambda \sum\limits_j{} {\dfrac{{\cos \left( {2{\text{π}}\alpha j} \right)}}{{1 - b\cos \left( {2{\text{π}}\alpha j} \right)}}} {\hat n_j} + \dfrac{U}{2}\sum\limits_j{} {{{\hat n}_j}\left( {{{\hat n}_j} - 1} \right)} ,$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M1.png"/></alternatives></inline-formula> where there exists no interaction, the system displays mobility edges at <inline-formula><tex-math id="M2">\begin{document}$b\varepsilon = 2(J - \lambda )$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M2.png"/></alternatives></inline-formula>, which separates the extended regime from the localized one and <i>b</i> = 0 is the standard Aubry-André model. By applying the perturbation method to the third order in a strong interaction case, we can induce an effective Hamiltonian for bosonic pairs. In the small <i>b</i> case, the bosonic pairs present the mobility edges in a simple closed expression form <inline-formula><tex-math id="M3">\begin{document}$b\left( {\dfrac{{{E^2}}}{U} - E - \dfrac{4}{E}} \right) = - 4\left(\dfrac{1}{E} + \lambda \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M3.png"/></alternatives></inline-formula>, which is the central result of the paper. In order to identify our results numerically, we define a normalized participation ratio (NPR) <inline-formula><tex-math id="M4">\begin{document}$\eta (E)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M4.png"/></alternatives></inline-formula> to discriminate between the extended properties of the many-body eigenvectors and the localized ones. In the thermodynamic limit, the NPR tends to 0 for a localized state, while it is finite for an extended state. The numerical calculations finely coincide with the analytic results for <i>b</i> = 0 and small <i>b</i> cases. Especially, for the <i>b</i> = 0 case, the mobility edges of the bosonic pairs are described as <inline-formula><tex-math id="M5">\begin{document}$\lambda = - 1/E$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M5.png"/></alternatives></inline-formula>. The extended regime and the one with the mobility edges will vanish with the interaction <i>U</i> increasing to infinity. We also study the scaling of the NPR with system size in both extended and localized regimes. For the extended state the NPR <inline-formula><tex-math id="M6">\begin{document}$\eta (E) \propto 1/L$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M6.png"/></alternatives></inline-formula> tends to a finite value with the increase of <i>L</i> and <inline-formula><tex-math id="M7">\begin{document}$L \to \infty $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M7.png"/></alternatives></inline-formula>, while for the localized case, <inline-formula><tex-math id="M8">\begin{document}$\eta (E) \propto {(1/L)^2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M8.png"/></alternatives></inline-formula> tends to zero when <inline-formula><tex-math id="M9">\begin{document}$L \to \infty $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M9.png"/></alternatives></inline-formula>. The <inline-formula><tex-math id="M10">\begin{document}$b \to 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M10.png"/></alternatives></inline-formula> limit is also considered. As the modulated potential approaches to a singularity when <inline-formula><tex-math id="M11">\begin{document}$b \to 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20182218_M11.png"/></alternatives></inline-formula>, the analytic expression does not fit very well. However, the numerical results indicate that the mobility edges of bosonic pairs still exist. We will try to consider the detection of the mobility edges of the bosonic pairs in the future.</sec>