Delocalization-localization transitions in 1D non-Hermitian cross-stitch lattices
Authors: Liu Hui , Lu Zhan-Peng , Xu Zhi-Hao ,   State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China,   Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Published: May 2024

<sec>In this work, we investigate the influence of quasi-periodic modulation on the localization properties of one-dimensional non-Hermitian cross-stitch lattices with flat bands. The crystalline Hamiltonian for this non-Hermitian cross-stitch lattice is given by: </sec><sec><inline-formula><tex-math id="M232">\begin{document}$\hat{H}=\displaystyle\sum\limits_{n}\left[t(a_n^{\dagger} b_n + b_n^{\dagger}a_n ) + J{\mathrm{e}}^{h}\left(a_n^{\dagger}b_{n + 1} + a_n^{\dagger} a_{n + 1} + Ab_n^{\dagger}a_{n + 1} + Ab_n^{\dagger}b_{n + 1}\right) + J{\mathrm{e}}^{ - h} \left(Aa_{n + 1}^{\dagger}b_n + a_{n + 1}^{\dagger}a_n + b_{n + 1}^{\dagger}a_n + Ab_{n + 1}^{\dagger}b_n\right)\right] $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M232.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M232.png"/></alternatives></inline-formula>with <inline-formula><tex-math id="M216">\begin{document}$A =\pm 1$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M216.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M216.png"/></alternatives></inline-formula>. When <i>A</i> = 1, the clean lattice supports two bands with dispersion relations <inline-formula><tex-math id="M217">\begin{document}$E_0=- t, $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217.png"/></alternatives></inline-formula><inline-formula><tex-math id="M217-1">\begin{document}$ E_1=4\cos (k - {\mathrm{i}}h) + t$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217-1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217-1.png"/></alternatives></inline-formula>. The compact localized states (CLSs) within the flat band <i>E</i><sub>0</sub> are localized in one unit cell, indicating that the system is characterized by the <i>U</i> = 1 class. Conversely, for <i>A</i> = –1, there are two flat bands in the system: <inline-formula><tex-math id="M218">\begin{document}$E_{\pm}=\pm\sqrt{t^2 + 4}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M218.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M218.png"/></alternatives></inline-formula>. The CLSs within the flat bands are localized in two unit cells, indicating that the system is marked by the <i>U</i> = 2 class. After introducing quasi-periodic modulations <inline-formula><tex-math id="M219">\begin{document}$\varepsilon_n^{\beta}=\lambda_{\beta}\cos(2\pi\alpha n + \phi_{\beta})$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M219.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M219.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M220">\begin{document}$\beta=\{a,b\}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M220.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M220.png"/></alternatives></inline-formula>), delocalization-localization transitions can be observed by numerically calculating the fractal dimension <i>D</i><sub>2</sub> and imaginary part of the energy spectrum <inline-formula><tex-math id="M221">\begin{document}$\ln{|{\rm{Im}}(E)|}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M221.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M221.png"/></alternatives></inline-formula>. Our findings indicate that the symmetry of quasi-periodic modulations plays an important role in determining the localization properties of the system. For the case of <inline-formula><tex-math id="M222">\begin{document}$U=1$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M222.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M222.png"/></alternatives></inline-formula>, the symmetric quasi-periodic modulation leads to two independent spectra <inline-formula><tex-math id="M223">\begin{document}$\sigma_f$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M223.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M223.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M224">\begin{document}$\sigma_p$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M224.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M224.png"/></alternatives></inline-formula>. The <inline-formula><tex-math id="M229">\begin{document}$\sigma_f$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M229.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M229.png"/></alternatives></inline-formula> retains its compact properties, while the <inline-formula><tex-math id="M225">\begin{document}$\sigma_p$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M225.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M225.png"/></alternatives></inline-formula> owns an extended-localized transition at <inline-formula><tex-math id="M226">\begin{document}$\lambda_{{\mathrm{c}}1}=4M$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M226.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M226.png"/></alternatives></inline-formula> with <inline-formula><tex-math id="M230">\begin{document}$M=\max\{{\mathrm{e}}^{h},\;{\mathrm{e}}^{ - h}\}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M230.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M230.png"/></alternatives></inline-formula>. However, in the case of antisymmetric modulation, the system exhibits an exact mobility edge <inline-formula><tex-math id="M227">\begin{document}$\lambda_{{\mathrm{c}}2}=2\sqrt{2|E - t|M}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M227.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M227.png"/></alternatives></inline-formula>. For the <i>U</i> = 2 class, all the eigenstates remain localized under any symmetric quasi-periodic modulation. In the case of antisymmetric modulation, all states transition from multifractal to localized states as the modulation strength increases, with a critical point at <inline-formula><tex-math id="M228">\begin{document}$\lambda_{{\mathrm{c}}3}=4M$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M228.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M228.png"/></alternatives></inline-formula>. This work expands the understanding of localization properties in non-Hermitian flat-band systems and provides a new perspective on delocalization-localization transitions.</sec>