Delocalization-localization transitions in 1D non-Hermitian cross-stitch lattices

Abstract

<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>