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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 ) + Je^{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) + Je^{ - 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$E_0=- t, E_1=4\cos (k - ih) + t$ . The compact localized states (CLSs) within the flat band E0 are 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 D2 and 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$U=1$ case, 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$\lambda_{c1}=4M$ with$M=\max\{e^{h},e^{ - h}\}$ . However, in the case of antisymmetric modulation, the system exhibits exact mobility edges$\lambda_{c2}=2\sqrt{2|E - t|M}$ . 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 as the modulation strength increases, with a critical point at$\lambda_{c3}=4M$ . This work should expand the understanding of localization properties in non-Hermitian flat-band systems and provides a new perspective on delocalization-localization transitions.-
Keywords:
- flat-band systems /
- non-Hermitian systems /
- disorder /
- localization
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图 1 干净晶格情况: (a) 一维非厄米十字晶格示意图. (b) $ U=1 $类非厄米十字晶格的CLS占据. (c) $ U=2 $类非厄米十字晶格的CLS占据. (d) 开边界条件下, $ U=1 $类十字晶格色散带中所有本征态的密度分布$ \rho_n^{(l)} $. 这里的参数选取$ h=0.6 $, $ t=2 $, $ L=100 $
Fig. 1. Crystalline Case:(a) A schematic diagram of the one-dimensional non-Hermitian cross-stitch lattice. (b) The CLS occupations of the $ U=1 $ class non-Hermitian cross-stitch lattice with $ A=1 $. (c) The CLS occupations of the $ U=2 $ class non-Hermitian cross-stitch lattice with $ A=-1 $. (d) Density distributions $ \rho_n^{(l)} $ for all the eigenstates in dispersive bands of the cross-stitch lattice under open bonudary conditions. Here, $ h=0.6 $, $ t=2 $, $ L=100 $
图 2 对称情况时, $ U=1 $ 类非厄米十字晶格的局域化性质: (a) $ \sigma_p $谱的分型维度$ D_2^{(l)} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化, 颜色表示分形维度$ D_2^{(l)} $的大小. (b) $ \sigma_p $谱的能量虚部$ \ln{|{\rm{Im}}(E)|} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化, 颜色表示$ \ln{|{\rm{Im}}(E)|} $的大小. 实线表示扩展-局域转变, $ \sigma_f $被省略, 边界用虚线表示. (c)$ \lambda=5 $时, $ \sigma_p $所对应的能谱. (d)$ \lambda=10 $时, $ \sigma_p $所对应的能谱. 这里$ L=1000 $
Fig. 2. The symmetric case of the $ U=1 $ class non-Hermitian cross-stitch lattice : (a) The real part of the spectrum $ \sigma_p $ as a function of $ \lambda $, where the color denotes the value of the fractal dimension $ D_2^{(l)} $. (b) $ \ln{|{\rm{Im}}(E)|} $ of $ \sigma_p $ as a function of $ \lambda $ and $ {\rm{Re}}(E) $, where the color denotes the value of $ \ln{|{\rm{Im}}(E)|} $. The black solid lines represent the delocalization-localization transition. The spectrum $ \sigma_{f} $ is omitted, but its boundaries are indicated by black dashed lines. (c) The energy spectrum of $ \sigma_p $ with $ \lambda=5 $. (d) The energy spectrum of $ \sigma_p $ with $ \lambda=10 $. Here, $ L=1000 $.
图 3 反对称情况时, $ U=1 $ 类非厄米十字型晶格的局域化: (a) 谱的分型维度$ D_2^{(l)} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化. (b) 谱的能量虚部$ \ln{|{\rm{Im}}(E)|} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化. 实线表示迁移率边. 这里$ L=1000 $
Fig. 3. The antisymmetric case of the $ U=1 $ class non-Hermitian cross-stitch lattice: (a) The real part of the spectrum as a function of $ \lambda $, where the color denotes the value of the fractal dimension $ D_2^{(l)} $. (b) $ \ln{|{\rm{Im}}(E)|} $ of the spectrum as a function of $ \lambda $ and $ {\rm{Re}}(E) $, where the color denotes the value of $ \ln{|{\rm{Im}}(E)|} $. The black solid lines represent the mobility edges. Here, $ L=1000 $.
图 4 反对称情况时, $ U=2 $类非厄米十字晶格的局域化: (a) 谱的分型维度$ D_2^{(l)} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化, 颜色表示分形维度$ D_2^{(l)} $的大小. (b) 谱的能量虚部$ \ln{|{\rm{Im}}(E)|} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化, 颜色表示$ \ln{|{\rm{Im}}(E)|} $的大小. 实线表示多重分形-局域的转变, 这里$ L=1000 $. (c) 多重分形区域的MIPR标度分析, 插图为局域区域的MIPR标度分析
Fig. 4. The antisymmetric case of $ U=2 $ non-Hermitian cross-stitch lattice: (a) The real part of the spectrum as a function of $ \lambda $, where the color denotes the value of the fractal dimension $ D_2^{(l)} $. (b) $ \ln{|{\rm{Im}}(E)|} $ of the spectrum as a function of $ \lambda $ and $ {\rm{Re}}(E) $, where the color denotes the value of $ \ln{|{\rm{Im}}(E)|} $. The black solid lines represent the multifractal-to-localized transition. Here, $ L=1000 $. (c) The MIPR scaling of multifractal regions for different $ \lambda $. The inset shows the MIPR scaling of localized regions for $ \lambda=10 $.
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