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Abstract

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: $\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] $ with $A =\pm 1$ . When A= 1, the clean lattice supports two bands with dispersion relations $E_0=- t, $ $ E_1=4\cos (k - {\mathrm{i}}h) + t$ . The compact localized states (CLSs) within the flat band E ₀are localized in one unit cell, indicating that the system is characterized by the U= 1 class. Conversely, for A= –1, there are two flat bands in the system: $E_{\pm}=\pm\sqrt{t^2 + 4}$ . The CLSs within the flat bands are localized in two unit cells, indicating that the system is marked by the U= 2 class. After introducing quasi-periodic modulations $\varepsilon_n^{\beta}=\lambda_{\beta}\cos(2\pi\alpha n + \phi_{\beta})$ ( $\beta=\{a,b\}$ ), delocalization-localization transitions can be observed by numerically calculating the fractal dimension D ₂and imaginary part of the energy spectrum $\ln{|{\rm{Im}}(E)|}$ . 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 $U=1$ , the symmetric quasi-periodic modulation leads to two independent spectra $\sigma_f$ and $\sigma_p$ . The $\sigma_f$ retains its compact properties, while the $\sigma_p$ owns an extended-localized transition at $\lambda_{{\mathrm{c}}1}=4M$ with $M=\max\{{\mathrm{e}}^{h},\;{\mathrm{e}}^{ - h}\}$ . However, in the case of antisymmetric modulation, the system exhibits an exact mobility edge $\lambda_{{\mathrm{c}}2}=2\sqrt{2|E - t|M}$ . For the U= 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 $\lambda_{{\mathrm{c}}3}=4M$ . This work expands the understanding of localization properties in non-Hermitian flat-band systems and provides a new perspective on delocalization-localization transitions.

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Authors and contacts

Corresponding author:Xu Zhi-Hao,xuzhihao@sxu.edu.cn

Funds:Project supported by the National Natural Science Foundation of China (Grant No. 12375016), the Fundamental Research Program of Shanxi Province, China (Grant No. 20210302123442), the Beijing National Laboratory for Condensed Matter Physics (Grant No. 2023BNLCMPKF001), and the Fund for Shanxi “1331 Project” Key Subjects, China.

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Vol. 73, Issue 13 - 2024-07-05

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