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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 ) + 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 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 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.-
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) Schematic diagram of the one-dimensional non-Hermitian cross-stitch lattice; (b) CLS occupations of the $ U=1 $ class non-Hermitian cross-stitch lattice with $ A=1 $; (c) 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. Symmetric case of the $ U=1 $ class non-Hermitian cross-stitch lattice: (a) 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)|} $. Black solid lines represent the delocalization-localization transition. The spectrum $ \sigma_{f} $ is omitted, but its boundaries are indicated by black dashed lines. (c) Energy spectrum of $ \sigma_p $ with $ \lambda=5 $. (d) 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)分型维度$ D_2^{(l)} $随着能量实部$ {\rm{Re}}(E) $和无序强度$ \lambda $的变化; (b)能量虚部$ \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. Antisymmetric case of the $ U=1 $ class non-Hermitian cross-stitch lattice: (a) $D_2^{(l)} $ 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 $类非厄米十字晶格的局域化 (a)分型维度$ 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)} $的大小. (b)能量虚部$ \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. Antisymmetric case of $ U=2 $ non-Hermitian cross-stitch lattice: (a) $D_2^{(l)} $ of the spectrum as a function of $ \lambda $ and Re(E), 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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[1] Anderson P W 1958 Phys. Rev. 109 1492
Google Scholar
[2] Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clément D, Sanchez-Palencia L, Bouyer P, Aspect A 2008 Nature 453 891
Google Scholar
[3] Hu H, Strybulevych A, Page J H, Skipetrov S E, van Tiggelen B A 2008 Nat. Phys. 4 945
Google Scholar
[4] Pradhan P, Sridhar S 2000 Phys. Rev. Lett. 85 2360
Google Scholar
[5] Mott N 1987 J. Phys. C 20 3075
Google Scholar
[6] Wang Y, Zhang L, Sun W, Poon T F J, Liu X J 2022 Phys. Rev. B 106 L140203
Google Scholar
[7] Yamamoto K, Aharony A, Entin-Wohlman O, Hatano N 2017 Phys. Rev. B 96 155201
Google Scholar
[8] Aubry S, André G 1980 Ann. Isr. Phys. Soc. 3 133
[9] Longhi S 2019 Phys. Rev. Lett. 122 237601
Google Scholar
[10] Xu Z, Xia X, Chen S 2022 Sci. China: Phys. Mech. Astron. 65 227211
Google Scholar
[11] Biddle J, Das Sarma S 2010 Phys. Rev. Lett. 104 070601
Google Scholar
[12] Ganeshan S, Pixley J H, Sarma S D 2015 Phys. Rev. Lett. 114 146601
Google Scholar
[13] Leykam D, Flach S, Bahat-Treidel O, Desyatnikov A S 2013 Phys. Rev. B 88 224203
Google Scholar
[14] Zhang W, Addison Z, Trivedi N 2021 Phys. Rev. B 104 235202
Google Scholar
[15] Leykam D, Bodyfelt J D, Desyatnikov A S, Flach S 2017 Eur. Phy. J. B 90 1
Google Scholar
[16] Maimaiti W, Andreanov A 2021 Phys. Rev. B 104 035115
Google Scholar
[17] Bodyfelt D, Leykam D, Danieli C, Yu X, Flach S 2014 Phys. Rev. Lett. 113 236403
Google Scholar
[18] Danieli C, Bodyfelt J D, Flach S 2015 Phys. Rev. B 91 235134
Google Scholar
[19] Lee S, Andreanov A, Flach S 2023 Phys. Rev. B 107 014204
Google Scholar
[20] Lee S, Flach S, Andreanov A 2023 Chaos 33 073125
Google Scholar
[21] Ahmed A, Ramachandran A, Khaymovich I M, Sharma A 2022 Phys. Rev. B 106 205119
Google Scholar
[22] Liu C, Jiang H, Chen S 2019 Phys. Rev. B 99 125103
Google Scholar
[23] Liu C, Chen S 2019 Phys. Rev. B 100 144106
Google Scholar
[24] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[25] Zhao H, Miao P, Teimourpour M H, Malzard S, ElGanainy R, Schomerus H, Feng L 2018 Nat. Commun. 9 981
Google Scholar
[26] Jiang H, Lang L J, Yang C, Zhu S L, Chen S 2019 Phys. Rev. B 100 054301
Google Scholar
[27] Yao S, Song F, Wang Z 2018 Phys. Rev. Lett. 121 136802
Google Scholar
[28] Yao S, Wang Z 2018 Phys. Rev. Lett. 121 086803
Google Scholar
[29] Hatano N, Nelson D R 1996 Phys. Rev. Lett. 77 570
Google Scholar
[30] Hatano N, Nelson D R 1997 Phys. Rev. B 56 8651
Google Scholar
[31] Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384
Google Scholar
[32] Flach S, Leykam D, Bodyfelt J D, Matthies P, Desyatnikov A S 2014 Europhys. Lett. 105 30001
Google Scholar
[33] Miroshnichenko A E, Flach S, Kivshar Y S 2010 Rev. Mod. Phys. 82 2257
Google Scholar
[34] Evers F, Mirlin A D 2008 Rev. Mod. Phys. 80 1355
Google Scholar
[35] Macé N, Alet F, Laflorencie N 2019 Phys. Rev. Lett. 123 180601
Google Scholar
[36] Liu H, Lu Z, Xia X, Xu Z 2024 arXiv: 2311.03166 [cond-mat.dis-nn]
[37] Zeng Q B, Chen S, Lü R 2017 Phys. Rev. A 95 062118
Google Scholar
[38] Tang L Z, Zhang G Q, Zhang L F, Zhang D W 2021 Phys. Rev. A 103 033325
Google Scholar
[39] Kawabata K, Shiozaki K, Ueda M, Sato M 2019 Phys. Rev. X 9 041015
Google Scholar
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